Author: 孤筝(lvbowen040427@163.com)

Chapter 1 Wave-Particle Duality and State Description

1.1 Formation and Applications of Quantum Mechanics

1.1.1 Old Quantum Theory

Photoelectric Effect and the Photon Hypothesis

• Photon energy: $E = h\nu$
• Threshold frequency: $\nu_0 = \dfrac{W_0}{h}$; no photoelectrons are emitted when $\nu < \nu_0$
• Photoelectric equation:
  $$ E_k^{\text{max}} = \frac{1}{2}\mu v^2_m = h\nu - W_0 $$
• The photoelectric effect demonstrates the particle nature of light.

Energy-Momentum Relation of Photons and Wave-Particle Unity

• Relativistic energy-momentum relation
  

  $$ E^2=(pc)^2+(m_0c^2)^2,\quad m_0=0\ \Rightarrow\ E=c\,\lVert\vec p\rVert $$

• Photon energy
  

  $$ E=h\nu=\frac{hc}{\lambda}=\hbar\omega $$

• Photon momentum (vector form)
  Let $\mathbf n$ be the unit vector in the propagation direction, then

  $$ \vec p=\frac{E}{c}\,\mathbf n=\frac{h}{\lambda}\,\mathbf n=\hbar\vec k,\quad \vec k=\frac{2\pi}{\lambda}\,\mathbf n $$

• Unified correspondence of wave and particle pictures
  

  $$ E\ \longleftrightarrow\ \hbar\omega,\qquad \vec p\ \longleftrightarrow\ \hbar\vec k $$

Bohr Structure of the Hydrogen Atom

• Quantization of orbital angular momentum: $$ L = n\hbar,\quad n=1,2,3,\dots $$
• Energy levels: $$ E_n = -\frac{13.6\ \text{eV}}{n^2} $$
• This successfully explains the line spectrum of hydrogen.

Bohr’s Postulates

• Electrons moving in stable orbits do not radiate energy.
• Electrons absorb or emit energy when they jump between energy levels: $$ \Delta E = h\nu $$

Compton Effect

• The wavelength of a high-energy photon increases after scattering from an electron: $$ \Delta\lambda = \lambda' - \lambda = \frac{h}{m_ec}(1-\cos\theta) $$
• The experiment confirms both the particle nature of light and conservation of momentum.

Blackbody Radiation

• Energy quantization assumption: the energy of the electromagnetic field takes discrete values $E=nh\nu$.
• Planck formula: $$ u(\nu,T)=\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/kT}-1} $$
• This successfully explains the blackbody spectrum and marks the birth of quantum theory.

1.1.2 Wave-Particle Duality of Microscopic Particles

de Broglie Hypothesis

• Microscopic particles have not only particle properties but also wave properties.
• Every particle with momentum $\vec p$ corresponds to a matter wave whose wavelength and frequency are related to its momentum and energy.

de Broglie Relations

• Wavelength: $$ \lambda = \frac{h}{p} $$
• Vector form: $$ \vec p = \hbar \vec k $$
• Frequency: $$ E = h\nu = \hbar\omega $$

1.2 States and Wave Functions

1.2.1 Uncertainty Principle

• The position and momentum of a microscopic particle cannot be measured simultaneously with arbitrary precision.
• Heisenberg uncertainty relation: $$ \Delta x \cdot \Delta p_x \geq \frac{\hbar}{2} $$
• Energy-time uncertainty relation: $$ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} $$
• Its origin lies in wave-particle duality and the non-commutativity of operators.

1.2.2 Wave Function

• To describe the state of a microscopic particle, we introduce the wave function $\psi(\vec r,t)$.
• Probabilistic interpretation: $|\psi(\vec r,t)|^2 dV$ gives the probability of finding the particle in the volume element $dV$.
• The wave function must satisfy the superposition principle and the Schrödinger equation.
• The total probability over all space is 1, so the probability distribution depends only on the relative magnitude of the wave function, not its absolute scale.
• Multiplying the wave function by a constant does not change the physical state it describes.
• Standard requirements for a wave function: single-valued, finite, and continuous.

1.2.3 Normalization of the Wave Function

• Normalization condition: $$ \int_{-\infty}^{\infty} \psi^* (\vec r,t) \psi (\vec r,t) dV = 1 $$
• How to normalize a wave function $$ \int_{-\infty}^{\infty} |\Psi(\vec r,t)|^2 dV = A^2 \int_{-\infty}^{\infty} |\psi(\vec r,t)|^2 dV = 1 $$ where $A$ is the normalization constant.

1.3 Schrödinger Equation

1.3.1 Wave Equation for a Free Particle

Concept
A free particle is a particle not subject to external forces. In quantum mechanics, its state is described by the wave function $\psi(\vec{r},t)$ and satisfies the Schrödinger equation.

Schrödinger equation for a free particle

where:

• $\hbar$: reduced Planck constant
• $m$: mass of the particle
• $\nabla^2$: Laplacian operator

Plane-wave solution

where:

• $\vec{k}$ is the wave vector, with $|\vec{k}| = k$
• $\omega$ is the angular frequency, satisfying $$ E = \hbar \omega = \frac{\hbar^2 k^2}{2m} $$

Momentum-wave vector relation

Plane-Wave Derivation of the Free-Particle Schrödinger Equation

1. Assume a plane-wave form

with $\vec{k}$ the wave vector, $\omega$ the angular frequency, and $A$ the amplitude constant.

2. Time derivative

Multiplying by $i\hbar$ gives

3. Spatial Laplacian

Thus,

4. Energy relation

5. Final equation

Remark

• This derivation uses only the plane-wave form and differentiation, without relying on operator definitions.
• It corresponds to the case $V=0$.

1.3.3 Stationary-State Schrödinger Equation and Stationary Wave Functions

Concept
A stationary-state wave function has separable time dependence:

Let $f(t)=e^{-i E t / \hbar}$.
Here $\phi(\vec{r})$ depends only on spatial coordinates, and $E$ is the total energy of the particle.

Derivation Starting from the time-dependent Schrödinger equation:

substitute $\psi(\vec{r},t) = \phi(\vec{r}) e^{-i E t / \hbar}$ to obtain

Time-independent form

Remark

• $\phi(\vec{r})$ is called a stationary-state wave function or eigenfunction.
• $E$ is the corresponding energy eigenvalue.

Derivation of the Schrödinger Equation from Operators

1. Start from classical energy

2. Introduce the de Broglie relations

and the plane-wave form

3. Operator representation

4. Kinetic-energy and Hamiltonian operators

5. Schrödinger equation

that is,

For a free particle ($V=0$), this reduces to

Principle of Superposition of States

Concept
If $\psi_1$ and $\psi_2$ are two possible states of the same system, then their linear combination

is also a possible state, where $c_1$ and $c_2$ are complex coefficients.

General expansion

with

• $c_n$ the expansion coefficients, or probability amplitudes;
• probabilities $|c_n|^2$ satisfying $$ \sum_n |c_n|^2 = 1 $$

Remark

• Superposition is one of the most fundamental principles of quantum mechanics.
• Different eigenstates may superpose, but a measurement yields only one eigenvalue.
• Interference in superposed states is one of the essential features that distinguishes quantum mechanics from classical mechanics.

Chapter 2 Simple Applications of the Schrödinger Equation

2.1 One-Dimensional Infinite Potential Well

2.1.1 Solving the Equation

1. Potential

2. Schrödinger equation in the well

which becomes

3. General solution

4. Boundary conditions

Hence $B=0$ and $kL = n\pi$ for $n=1,2,3,\dots$.

5. Eigenfunctions and eigenvalues

Remark

• The energy is quantized.
• The ground-state energy is nonzero, showing the zero-point energy.

2.2 Special Functions in Mathematical Physics

2.2.1 Orthogonality and Normalization

Orthogonality

Normalization

Orthonormality

2.2.2 Expansion in an Orthonormal Set

2.2.3 Fourier Series

where

2.2.4 Constructing Orthonormal Functions

The standard method is Gram-Schmidt orthogonalization:

2.2.5 Legendre Polynomials and Other Special Functions

Legendre polynomials

with orthogonality

Other common special functions

• Spherical harmonics $Y_l^m(\theta,\phi)$ appear in angular momentum problems.
• Bessel functions $J_n(x)$ appear in cylindrical symmetry problems.
• Hermite polynomials $H_n(x)$ appear in harmonic oscillator problems.

These special functions are solutions of the Schrödinger equation under different boundary conditions and symmetries.

2.3 Linear Harmonic Oscillator

2.4 Hydrogen Atom

2.4.1 Solution of the Equation (Separated into $r,\ \theta,\ \phi$ Parts)

1. Time-independent Schrödinger equation under a Coulomb potential

and

2. Separation of variables

This leads to three equations in $r$, $\theta$, and $\phi$ after separation.

3. Equation in $\phi$

4. Equation in $\theta$

with solutions proportional to the associated Legendre functions:

5. Angular part: spherical harmonics

which satisfy

6. Radial equation Let $u(r)=rR(r)$, then

7. Energy eigenvalues

with $l=0,1,\dots,n-1$.

8. Wave function

and

where $a_0=\dfrac{4\pi\varepsilon_0\hbar^2}{m e^2}$ is the Bohr radius.

2.4.2 Results and Discussion

1. Quantum numbers and their meanings

• $n$: principal quantum number
• $l$: orbital angular momentum quantum number
• $m$: magnetic quantum number

2. Degeneracy
For the Coulomb potential, the energy depends only on $n$. The degeneracy of the level with principal quantum number $n$ is $n^2$.

3. Spatial structure of the wave function

• The angular part is given by the spherical harmonics.
• The radial part $R_{nl}(r)$ has $n-l-1$ radial nodes.
• The ground state $(1,0,0)$ is spherically symmetric and has no radial node.

4. Summary
The hydrogen atom is solved by separating variables in spherical coordinates. The angular equations give spherical harmonics and angular quantum numbers, while the radial equation yields the discrete energy levels and radial eigenfunctions.

Chapter 3 Operator Representation of Dynamical Variables and Representation Theory

3.1 Relation Between Dynamical Variables and Operators

3.1.1 Mathematical Knowledge of Operators

1. Definition of an operator
   An operator is a rule acting on a function space or state space. In quantum mechanics, physical quantities are represented by operators, and the wave function is the object on which they act.

2. Linearity
   If

   $$ A(c_1\psi_1 + c_2\psi_2) = c_1 A\psi_1 + c_2 A\psi_2, $$

   then $A$ is a linear operator.

3. Commutation relations
   The commutator is defined by

   $$ [A,B] = AB - BA. $$

   If $[A,B]=0$, the two operators are said to commute.

4. Hermitian operators
   If

   $$ \langle \psi | A\varphi \rangle = \langle A\psi | \varphi \rangle, $$

   then $A$ is Hermitian. Hermitian operators have real eigenvalues and represent observables.

3.1.2 Dynamical Variables and Operators

1. Basic idea
   Every classical quantity $f(q,p)$ corresponds to a quantum operator $\hat f$.

2. Typical operator forms in the coordinate representation

   $$ \hat{x} = x, \qquad \hat{p} = -i\hbar \frac{\partial}{\partial x} $$

3. Fundamental commutation relation

   $$ [\hat{x}, \hat{p}] = i\hbar $$

4. Measurement and eigenvalue equations

   $$ \hat{A}\psi_a = a\psi_a $$

   Here $a$ is a possible measurement outcome, and $\psi_a$ is the corresponding eigenfunction.

3.2 Commutation Relations and the Uncertainty Principle

3.2.1 Commutation Relations

1. Definition

   $$ [A,B] = AB - BA $$

   If $[A,B]=0$, the two physical quantities can have simultaneous definite values.

2. Basic relation

   $$ [\hat{x}, \hat{p}_x] = i\hbar $$

3. Three-dimensional form

   $$ [\hat{x}_i, \hat{p}_j] = i\hbar \delta_{ij}, \quad [\hat{x}_i, \hat{x}_j]=0, \quad [\hat{p}_i, \hat{p}_j]=0 $$

4. Physical meaning
   Commutation relations determine whether two observables can be measured simultaneously with arbitrary precision.

3.2.2 Uncertainty Principle

1. Mathematical form

   $$ (\Delta A)^2 = \langle (A-\langle A \rangle)^2 \rangle,\qquad (\Delta B)^2 = \langle (B-\langle B \rangle)^2 \rangle $$

   which leads to

   $$ \Delta A \cdot \Delta B \geq \frac{1}{2}\left| \langle [A,B] \rangle \right| $$

2. Position-momentum uncertainty

   $$ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} $$

3. Energy-time uncertainty

   $$ \Delta E \cdot \Delta t \gtrsim \hbar $$

3.3 Representation Theory

3.3.1 Mathematical Basis

1. Concept of representation
   States and operators can be represented in different bases, such as the coordinate, momentum, and energy representations.

2. Expansion of a state

   $$ |\psi\rangle = \sum_n c_n |\phi_n\rangle,\qquad c_n = \langle \phi_n | \psi \rangle $$

3. Matrix elements

   $$ A_{mn} = \langle \phi_m | \hat{A} | \phi_n \rangle $$

4. Completeness and orthogonality

   $$ \sum_n |\phi_n\rangle \langle \phi_n| = I,\qquad \langle \phi_m | \phi_n \rangle = \delta_{mn} $$

3.3.2 Representations of States and Observables

1. Coordinate representation

   $$ \psi(x) = \langle x|\psi\rangle $$
   $$ \hat{x} \psi(x) = x \psi(x), \quad \hat{p}_x \psi(x) = -i\hbar \frac{\partial}{\partial x}\psi(x) $$

2. Momentum representation

   $$ \phi(p) = \langle p|\psi\rangle $$
   $$ \hat{p} \phi(p) = p \phi(p), \quad \hat{x} \phi(p) = i\hbar \frac{\partial}{\partial p}\phi(p) $$

3. Energy representation

   $$ |\psi\rangle = \sum_n c_n |E_n\rangle, \quad c_n = \langle E_n|\psi\rangle $$

4. Transformations between representations

   $$ \phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} dx $$
   $$ \psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) e^{ipx/\hbar} dp $$

3.4 Orbital Angular Momentum

3.4.1 Angular Momentum

1. Definition

   $$ \vec{L} = \vec{r} \times \vec{p},\qquad \hat{\vec{L}} = \hat{\vec{r}} \times \hat{\vec{p}} $$

2. Components

   $$ \hat{L}_x = y\hat{p}_z - z\hat{p}_y, \quad \hat{L}_y = z\hat{p}_x - x\hat{p}_z, \quad \hat{L}_z = x\hat{p}_y - y\hat{p}_x $$

3. Commutation relations

   $$ [\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z, \quad [\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x, \quad [\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y $$
   $$ \hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2 $$

3.4.2 Conservation of Angular Momentum

1. Condition for conservation

   $$ [\hat{H}, \hat{L}_i] = 0 \quad \Rightarrow \quad \hat{L}_i \ \text{is conserved} $$

2. Spherically symmetric potential

   $$ [\hat{H}, \hat{L}^2] = 0, \quad [\hat{H}, \hat{L}_z] = 0 $$

3.4.3 Calculation of Orbital Angular Momentum

1. Eigenvalue equations

   $$ \hat{L}^2 Y_{lm}(\theta,\varphi) = l(l+1)\hbar^2 Y_{lm}(\theta,\varphi) $$
   $$ \hat{L}_z Y_{lm}(\theta,\varphi) = m\hbar Y_{lm}(\theta,\varphi) $$

2. Eigenvalues

   $$ L = \sqrt{l(l+1)} \hbar,\qquad L_z = m\hbar $$

3. Physical meaning
   The quantum numbers $l$ and $m$ determine the magnitude of the orbital angular momentum and its $z$-component, respectively.

Chapter 4 Perturbation Theory and Its Applications

4.1 Time-Independent Perturbation Theory

4.1.1 Nondegenerate Perturbation Theory

1. Basic idea

   $$ \hat{H} = \hat{H}^{(0)} + \lambda \hat{H}' $$

2. Energy corrections

   $$ E_n^{(1)} = \langle \psi_n^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle $$
   $$ E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \psi_m^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}} $$

3. Wave-function correction

   $$ \psi_n^{(1)} = \sum_{m \neq n} \frac{\langle \psi_m^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} \psi_m^{(0)} $$

4.1.2 Degenerate Perturbation Theory

1. Origin of the problem
   If the zeroth-order energy corresponds to multiple orthogonal eigenstates, the state is degenerate, and the nondegenerate formulas fail.

2. Method

   $$ H'_{ij} = \langle \psi_i^{(0)} | \hat{H}' | \psi_j^{(0)} \rangle $$

   Diagonalize this matrix inside the degenerate subspace.

3. Result
   The first-order energy corrections are the eigenvalues of $H'_{ij}$, and the corrected states are the corresponding linear combinations.

4.2 Time-Dependent Perturbation Theory

1. Basic framework

   $$ \hat{H}(t) = \hat{H}^{(0)} + \hat{H}'(t) $$

2. State expansion

   $$ |\psi(t)\rangle = \sum_n c_n(t) e^{-iE_n^{(0)}t/\hbar} |\psi_n^{(0)}\rangle $$

3. Transition probability amplitude

   $$ c_f^{(1)}(t) = \frac{1}{i\hbar} \int_0^t \langle \psi_f^{(0)} | \hat{H}'(t') | \psi_i^{(0)} \rangle e^{i\omega_{fi} t'} dt' $$

   where $\omega_{fi} = (E_f^{(0)} - E_i^{(0)})/\hbar$.

4. Fermi’s golden rule

   $$ W_{i \to f} = \frac{2\pi}{\hbar} \, |\langle f | \hat{H}' | i \rangle|^2 \, \rho(E_f) $$

Summary

• Time-independent perturbation theory corrects energies and wave functions for static perturbations.
• Time-dependent perturbation theory describes transitions between energy levels, such as radiation absorption and emission.

Electron Spin

Experimental Discovery of Electron Spin

1. Stern-Gerlach experiment
   Passing a beam of silver atoms through a nonuniform magnetic field produces two trajectories, revealing an intrinsic angular momentum beyond orbital angular momentum.

2. Experimental conclusions

   • The spin quantum number is $s = 1/2$.
   • The two possible spin projections are $m_s = \pm 1/2$.
   • Spin contributes an additional magnetic moment: $$ \vec{\mu}_s = -g_s \frac{e}{2m_e} \vec{S}, \quad g_s \approx 2 $$

Theory of Electron Spin

1. Quantum description

   • Spin is intrinsic angular momentum and does not depend on spatial coordinates.
   • Its operators satisfy $$ [\hat{S}_i, \hat{S}_j] = i\hbar \epsilon_{ijk} \hat{S}_k $$

2. Physical meaning

   • Spin determines the magnetic behavior of electrons.
   • Quantized spin leads to Fermi-Dirac statistics and the Pauli exclusion principle.

Spin Angular Momentum

Spin Operators

1. Spin components

   $$ \hat{S}_x, \hat{S}_y, \hat{S}_z $$

   satisfying

   $$ [\hat{S}_x, \hat{S}_y] = i\hbar \hat{S}_z, \quad \text{cyclic symmetry} $$

2. Total spin operator

   $$ \hat{S}^2 = \hat{S}_x^2 + \hat{S}_y^2 + \hat{S}_z^2 $$

   with

   $$ \hat{S}^2 |\chi_s\rangle = s(s+1)\hbar^2 |\chi_s\rangle $$

Matrix Representation of Eigenfunctions

1. Spin-$1/2$ particles

   $$ |\uparrow\rangle = \begin{pmatrix}1\\0\end{pmatrix}, \quad |\downarrow\rangle = \begin{pmatrix}0\\1\end{pmatrix} $$

2. Pauli-matrix form of the spin operators

   $$ \hat{S}_x = \frac{\hbar}{2} \sigma_x, \quad \hat{S}_y = \frac{\hbar}{2} \sigma_y, \quad \hat{S}_z = \frac{\hbar}{2} \sigma_z $$

   where

   $$ \sigma_x = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \sigma_y = \begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\quad \sigma_z = \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix} $$

Theory of Angular Momentum Coupling

1. Spin-orbit coupling

   $$ \hat{H}_{\text{SO}} = \xi(r)\, \vec{L} \cdot \vec{S} $$

   which produces fine-structure splitting.

2. Total angular momentum

   $$ \vec{J} = \vec{L} + \vec{S}, \quad \hat{J}^2 = (\hat{L}+\hat{S})^2 $$

   with eigenstates $|j, m_j\rangle$ satisfying

   $$ \hat{J}^2 |j, m_j\rangle = j(j+1)\hbar^2 |j, m_j\rangle, \quad \hat{J}_z |j, m_j\rangle = m_j \hbar |j, m_j\rangle $$

3. Coupling result

   • $j = l \pm s$, $m_j = -j, -j+1, ..., j$.
   • Spin-orbit coupling is an important source of the fine structure of atomic spectra.

Principle of Indistinguishability

Systems of Identical Particles

Concepts and Principles

1. Definition of identical particles
   If two particles are completely identical in physical properties such as mass, charge, and spin, and cannot be distinguished by any experiment, they are called identical particles.

2. Principle of indistinguishability
   The physical laws are invariant under exchange of identical particles. Exchanging the positions and spins of any two identical particles leaves the Hamiltonian and observables unchanged.

Hamiltonian of a System of Identical Particles

1. Form

   $$ \hat{H} = \sum_{i=1}^N \hat{T}_i + \sum_{i

2. Symmetry

   $$ [\hat{H}, \hat{P}_{ij}] = 0 $$

   where $\hat{P}_{ij}$ is the exchange operator of particles $i$ and $j$.

Wave Functions of Identical-Particle Systems

1. Symmetry requirement

   $$ \hat{P}_{ij} \Psi(\dots, \vec{r}_i, \vec{r}_j, \dots) = \pm \Psi(\dots, \vec{r}_i, \vec{r}_j, \dots) $$
   • + for bosons, whose wave functions are symmetric
   • - for fermions, whose wave functions are antisymmetric

2. Construction of many-particle wave functions

   • Bosons: symmetrized sum
   • Fermions: antisymmetrized determinant (Slater determinant) $$ \Psi(\vec{r}_1, \dots, \vec{r}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \psi_1(\vec{r}_1) & \cdots & \psi_1(\vec{r}_N) \\ \vdots & \ddots & \vdots \\ \psi_N(\vec{r}_1) & \cdots & \psi_N(\vec{r}_N) \end{vmatrix} $$

Pauli Exclusion Principle

1. Content of the principle
   For identical fermions with half-integer spin, no two particles may occupy the same quantum state:

   $$ \Psi(\text{same quantum state}) = 0 $$

2. Physical meaning
   It explains the arrangement of electrons in atomic orbitals and underlies atomic structure, chemical properties, and Fermi-gas behavior.

3. Examples

   • In atoms, each orbital can hold at most two electrons with opposite spins.
   • In metals, electrons form a Fermi level that determines electrical and thermal properties.

Published on 2025-09-05 at 孤筝の温暖小家, last modified on 2025-09-05

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Author: 孤筝(lvbowen040427@163.com)

Fundamental Concepts of Digital Signal Processing

Signal Classification

1. Continuous-time signal: Analog signal, continuous in the time domain.
2. Discrete-time signal: Continuous in amplitude but discrete in time.
3. Amplitude-discrete signal: Discrete in amplitude but continuous in time.
4. Digital signal: Discrete in both amplitude and time.

Differences

The distinction between a discrete-time signal and a digital signal lies solely in the quantization error present in digital signals.

Implementation Methods of Digital Signal Processing

The primary target of digital signal processing is digital signals, achieved through numerical operations.

Software Implementation

Programs are written based on principles and algorithms and executed on general-purpose computers.

• Advantages: Flexibility
• Disadvantages: Slow computation speed, difficult to achieve real-time processing.
• Suitable for: Algorithm research and simulation.

Hardware Implementation

Hardware structures are designed according to specific requirements and algorithms, using basic components such as multipliers, adders, delay units, controllers, memory, and input/output interfaces.

• Advantages: Fast computation speed, capable of real-time processing.
• Disadvantages: Inflexibility

Hardware implementation refers to selecting an appropriate DSP chip, equipped with suitable software and language for the task, to achieve specific signal processing functions.

Dedicated Chips

Using Digital Signal Processing (DSP) chips is currently the fastest-growing and most widely applied method. DSP chips offer significant advantages over general-purpose microcontrollers:

• Internal multipliers and accumulators tailored for digital signal processing.
• Pipeline architecture, parallel structures, and multiple buses.
• Specialized instructions optimized for digital signal processing, enabling high-speed computation.

For ultra-high-speed real-time systems where DSP chips are insufficient, Field-Programmable Gate Arrays (FPGAs) or custom ASICs (Application-Specific Integrated Circuits) should be employed.

Characteristics of Digital Signal Processing

Compared to analog signal processing, digital signal processing offers:

1. Flexibility
2. High precision and stability
3. Ease of large-scale integration
4. Capability to perform functions unattainable by analog systems, such as storage, complex transformations, and operations.

Signal Dimensionality

A signal is typically a function of one or more independent variables.

• One-dimensional signal: Only one independent variable.
• Multidimensional signal: Two or more independent variables.

Discrete-Time Signals and Systems

Discrete-Time Signals

In practice, signals are generally analog. Uniform sampling (equally spaced sampling) converts them into discrete-time signals.

For an analog signal \( x_a(t) \), discrete time points \( t_n \).
With uniform sampling interval \( T \), \( t_n = nT \):

Here, \( x(n) \) is called a discrete-time signal, where \( n \) is an integer, forming a sequence:

Discrete-time signals are also referred to as sequences.

Representation Methods for Sequences

Set Notation

A set of numbers is denoted by \( \{\cdot\} \). A discrete-time signal can be represented as an ordered set of numbers.
The underlined element in the set indicates the sample value at \( n=0 \).

Formula Representation

Example:

Graphical Representation

The horizontal axis represents \( n \), and the vertical axis represents the value of \( x \), with dots atop vertical lines.

Common Standard Sequences

Unit Impulse Sequence \( \delta(n) \)

Also called the unit sample sequence, distinct from the unit impulse signal \( \delta(t) \).

Unit Step Sequence \( u(n) \)

Relationships:

Rectangular Sequence \( R_N(n) \)

\( N \) is the length of the rectangular sequence. It can be expressed using the unit step sequence:

Real Exponential Sequence

• Convergent sequence: \( |a| \lt 1 \)
• Divergent sequence: \( |a| \gt 1 \)

Sinusoidal Sequence

Here, \( \omega \) is the digital frequency (units: radians, \( rad \)), representing the rate of change (phase shift between adjacent samples).

Analog angular frequency \( \varOmega \)
If the sinusoidal sequence is derived from sampling an analog signal \( x_a(t) = \sin(\varOmega t) \):

The relationship between digital and analog frequencies is:

Given the sampling frequency \( F_s = \frac{1}{T} \):

Digital frequency is the normalized analog angular frequency relative to the sampling frequency.

Complex Exponential Sequence

Since \( n \) is an integer, both sinusoidal and complex exponential sequences are periodic with period \( 2\pi \).

Periodic Sequence

If for all \( n \), there exists a smallest positive integer \( N \) such that:

then the sequence \( x(n) \) is periodic with period \( N \).

Sequence Operations

Addition and Multiplication

Shifting, Flipping, and Scaling

Discrete-Time Systems

For a system with input \( x(n) \), output \( y(n) \), and operation \( T[\cdot] \):

Linear Systems

A system is linear if its input-output relationship satisfies the principle of superposition.

Additivity

Homogeneity (Scaling)

Time-Invariant Systems

A system is time-invariant if its operation \( T[\cdot] \) does not change over time, i.e., the system’s response is independent of when the input is applied.

Characteristics of Linear Time-Invariant (LTI) Systems

Total response = Zero-input response + Zero-state response

Unit Impulse Response

With zero initial state (no zero-input response):

For any input \( x(n) \):

Thus, the output is:

This is the convolution sum. For details, refer to Signals and Systems.

Causality of Systems

Definition: A system is causal if its output at time \( n \) depends only on the input at time \( n \) and prior inputs, not future inputs.

Necessary and Sufficient Condition:
The unit impulse response satisfies:

Stability of Systems

Definition: A system is stable if every bounded input produces a bounded output (BIBO stability).

Necessary and Sufficient Condition:
The unit impulse response is absolutely summable:

Linear Constant-Coefficient Difference Equations

Published on 2024-09-04 at 孤筝の温暖小家, last modified on 2024-09-04

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

Fundamentals of Hydrostatics

Gas Properties

Compressibility

Bulk Modulus

Definition: The increase in pressure required to produce a unit relative change in volume.

For a given mass of gas, volume is inversely proportional to density, i.e.,

Substituting back yields

Bulk modulus of water at room temperature: $2.1 \times 10^9N/m^2$

Under normal conditions, water can be considered an incompressible fluid.

Viscosity

Newton’s Law of Viscosity

The frictional resistance generated by fluid motion is proportional to the contact area

$\tau$: Frictional stress, the frictional resistance per unit area

$\vec{n}$: Normal direction of the contact surface

$\mu$: Proportionality constant, known as the coefficient of viscosity of the fluid, with units of $N \cdot s/m^2$

$\frac{du}{d \vec{n}}$: Velocity gradient

The coefficient of viscosity varies for different fluid media and changes with temperature, being largely independent of pressure.

The coefficient of viscosity for gases increases with rising temperature.

Sutherland’s Formula

One of the approximate formulas describing the relationship between air viscosity coefficient and temperature is Sutherland’s Formula.

$\mu_0$: Air viscosity coefficient at temperature $288.15K$

$C$: Constant, with a value of $110.4K$

Kinematic Viscosity Coefficient

$\nu$: Kinematic viscosity coefficient, unit is $m^2/s$

$\mu$: Viscosity coefficient

$\rho$: Density

Thermal Conductivity

Definition: When there is a temperature gradient in a certain direction within a gas, heat will transfer from the higher-temperature region to the lower-temperature region. This property is referred to as the thermal conductivity of the gas.

The amount of heat transferred per unit time is proportional to the heat transfer area and the temperature gradient along the direction of heat flow, expressed as:

$q$: Heat per unit time through a unit area, unit: $kJ/(m^2 \cdot s)$

$\frac{\partial T}{\partial \vec{n}}$: Temperature gradient, unit: $K/m$

$\lambda$: Thermal conductivity coefficient, unit: $kJ/(m \cdot K \cdot s)$

The negative sign indicates that the direction of heat transfer is always opposite to the direction of the temperature gradient.

Fluid Classification

Continuum Hypothesis

Ideal Fluid

不考虑粘性，在这种模型中，流体微团不承受粘性力作用。常用于气体。

忽略粘性的气体称为理想气体。

Pressure Isotropy

The pressure at a point in an ideal fluid is independent of the orientation of the pressure surface; it is merely a continuous function of spatial coordinates.

Incompressible Fluid

不考虑气体压缩性或弹性，可认为体积弹性模数无穷大，或流体密度为常数。常用于液体。

求解不可压流体的流动规律，只需要服从力学定律，不需要考虑热力学关系。

对流速较低的气体，也可按不可压流体处理流动问题。

Adiabatic Fluid

A model that does not consider the heat transfer properties of a fluid, treating the thermal conductivity coefficient of the fluid as zero. Low-speed flowing air typically has very small thermal conductivity values and can be regarded as adiabatic.

A gas model that disregards the heat conduction effects between gas microelements is referred to as an adiabatic gas.

Perfect Gas

For any state, there exists a certain functional relationship between the pressure, density, and temperature of a gas

Equation of State for a Perfect Gas

$\overline{R}$: Universal gas constant, $8315m^2/(s^2 \cdot K)$

$m$: Relative molecular mass of a specific gas

When $R=\frac{\overline{R}}{m}$,

$R$ is the gas constant, approximately $287.035m^2/(s^2 \cdot K)$ for air

Forces on a Fluid Particle

Stress

Shear Stress (Friction)

Volume Force

• Gravity
• Electromagnetic Force
• Centrifugal Force

Static Equilibrium Equation

In a ==static fluid==, take a point $P$ with pressure $p$.

Construct a Cartesian coordinate system where the pressure at any point in the fluid is

Build a rectangular parallelepiped centered at $P$ with edges parallel to the coordinate axes and lengths $dx, dy, dz$.

Observing the $x$-axis direction, the forces on the two faces are respectively

The body force on the fluid element in the $x$-axis direction is

Here, $f_x$ is the component of the body force per unit mass in the $x$-axis direction.

Since the fluid is static, the fluid element is in equilibrium.

The force balance equation in the $x$-axis direction is

Simplifying gives

$\because$ The total differential of $p$ is

$\therefore$

Define the ==body force potential function==

Its total differential is

Where $\frac{\partial \varOmega}{\partial x}=f_x$, $\frac{\partial \varOmega}{\partial y}=f_y$, $\frac{\partial \varOmega}{\partial z}=-f_z$.

From the above relations, we obtain

Integrating both sides over $x, y, z$ gives

When the pressure $p_a$ at a known point A, the difference in body force potential $\varOmega_a-\varOmega$ between two points, and the density $\rho$ of the static fluid (uniform everywhere) are known, the pressure at any point can be determined from its body force potential $\varOmega$:

Corollary: An isobaric surface in the fluid must also be an equipotential surface of the body force.

Atmosphere

Atmospheric Layers

Lower Atmosphere

• Altitude: Sea level – 85 km
• Characteristics: Uniform composition, nitrogen accounts for 78.1% of the total volume, oxygen accounts for 21% of the total volume

Troposphere

• Altitude
  • Equator: 16~18 km
  • Mid-latitudes: 10~12 km
  • Polar regions: 7~10 km
• Mass: Accounts for 75% of the total atmospheric mass
• Characteristics: Features vertical air currents, storms, and thunderstorms. Temperature decreases rapidly with increasing altitude.

Convective Boundary Layer

Transition layer, with a thickness ranging from several hundred meters to one or two kilometers.

Stratosphere

• Altitude: Troposphere ~32 km
• Mass: Accounts for about one-quarter of the atmospheric mass
• Characteristics: No weather phenomena, horizontal air flow, constant temperature (average about 216.65 K)

Mesosphere

• Altitude: 32~85 km
• Mass: 1/3000
• Temperature: First rises then falls, can drop below 106 K at 85 km.

Upper Atmosphere

• Altitude: Above 85 km
• Characteristics: Non-uniform composition, directly absorbs solar radiation

Thermosphere

• Altitude: 85~500 km
• Temperature: Increases with altitude, reaching up to 1370 K during the day at 500 km.
• Characteristics: Directly exposed to solar shortwave radiation

Outer Atmosphere

• Altitude: 500+ km, where the atmosphere gradually merges with interstellar space
• Mass: $1/10^{11}$
• Characteristics: The atmosphere is too thin to be defined by temperature. Air molecules can escape into space.

Upper Atmosphere and Ionosphere

• The upper atmosphere is dissociated into electrons and ions by solar shortwave radiation, forming the ionosphere.
• Above 100 km altitude, the air becomes a good conductor.
• Above 150 km, the air is too thin to transmit sound.

D Layer

• 高度：60~80 km

E Layer

• 高度：100~120 km

$F_1$ Layer

• Height: 180~220 km

$F_2$ Layer

• Altitude: 300~350 km

Flow Field

Flow field: The space filled with moving fluid

Flow parameters: Physical quantities used to characterize fluid motion, such as velocity, density, pressure, etc.

Fluid mechanics methods: Lagrangian method, Eulerian method

Lagrange Method

Focuses on particles (motion)

• Studies the variation of motion parameters and trajectories of individual particles in the flow field over time.
• Synthesizes the changes in motion parameters of all fluid particles to derive the overall flow field motion characteristics.

Eulerian Method

Focuses on spatial points (fixed)

• Studies how the motion parameters of fluid particles change over time as they pass through fixed spatial points.
• By combining the variation of motion parameters at all spatial points in the flow field, the overall flow field motion can be understood.

In the Eulerian method, the motion parameters of the flow field are generally functions of spatial coordinates and time.

Taking velocity as an example:

The four variables are independent.

In a general three-dimensional space, a Cartesian coordinate system is established, and scalar parameters are decomposed into the $x$, $y$, and $z$ directions for separate analysis.

Differentiating to obtain the acceleration components:

$\because$ $\frac{dx}{dt} = v_x$, $\frac{dy}{dt} = v_y$, $\frac{dz}{dt} = v_z$

$\therefore$

From this, it can be seen that acceleration is a function of time and position:

Local Acceleration

The first term on the right side of the equation represents the rate of change of velocity of a fluid particle at a fixed spatial point over time, known as local acceleration. (Relationship between velocity and time)

Local acceleration is caused by the temporal variability of velocity in the flow field.

Convective Acceleration

The latter three terms reflect the rate of change in velocity as a fluid particle moves along the velocity vector direction from one point in space to an adjacent point at the same instant, known as convective acceleration. (Relationship between velocity and displacement)

Convective acceleration is caused by the non-uniformity of the flow field.

Unsteady Flow Field

In the flow field, the physical quantities at least at one spatial point vary with time.

Steady Flow Field

In a steady flow field, the physical quantities at any spatial point do not change with time.

Pathline

The collection of all spatial points traversed by a marked moving fluid particle in a flow field over a period of time is called the pathline of that fluid particle.

Streamlines

In a flow field, curves that are tangent to the velocity vector at every spatial point are called streamlines.

Streamlines are curves formed by different fluid particles at the same instant, indicating the direction of velocity for those particles at that moment.

Characteristics:

• In a steady flow field, the streamlines do not change over time.

• Unsteady flow fields have streamlines that vary with time.

• In a steady flow field, the streamline passing through a given spatial point coincides with the trajectories of all fluid particles passing through that point.

• Generally, streamlines do not intersect (at the same instant and spatial point, there cannot be two velocity directions).

• At points where the velocity is zero, streamlines can intersect. Such points are typically called stagnation points.

• At points where the velocity is infinite, streamlines can intersect. Such points are typically called singularities.

• Streamlines can be tangent to each other, and beyond the point of tangency, the two lines coincide.

• Every point in the flow field has a streamline passing through it. The collection of all streamlines is called the streamline pattern or simply the flow pattern.

Streamline Differential Equation

Let the velocity at a point $M(x,y,z)$ on a streamline be $\vec{v}$, and the infinitesimal segment length of the streamline at point $M$ be $ds$, decomposed into $v_x,v_y,v_z$ and $dx,dy,dz$ in the Cartesian coordinate system.

At any point on the streamline, the direction of velocity is the same as the tangent direction of the streamline, so

Here, $\vec{i}$ is the unit normal vector in the $x$-axis direction, and similarly for the $y$ and $z$ axes.

The above equation is the differential equation of the streamline.

When the velocity distribution is known, the shape of the streamline passing through any point in the flow field can be determined.

Stream Tube

In a flow field, a closed curve C that is not a streamline is considered. Drawing streamlines through every point on C, the collection of these streamlines forms a tubular surface known as a stream tube.

Analysis of Fluid Particle Motion

Motion Forms

Rigid Body Motion

• Translational Motion
• Rotational Motion About an Axis

Fluid Motion

• Translational motion
• Rotational motion about an axis
• Deformation motion
  • Linear deformation
  • Shear deformation

Two-Dimensional Analysis

![Pasted image 20240902212258.png][1]

Consider an arbitrary rectangular fluid element ABCD in the flow field, with side lengths of $\delta_x,\delta_y$ respectively, both being infinitesimal quantities.

Let $v_x,v_y$ be the component velocities of the fluid element at point A, and assume the component velocities are continuous functions of spatial coordinates. The velocities at points B and D can be expressed using Taylor series expansions around point A.

$\because$ The side lengths of the fluid element are sufficiently small

$\therefore$ Higher-order infinitesimals can be neglected

When the fluid element moves, in addition to the overall motion, point B also moves relative to point A.

The relative velocity in the $x$-direction is $v_{Bx}-v_x=\frac{\partial v_x}{\partial x}\delta_x$, and similarly for the $y$-direction $\frac{\partial v_x}{\partial x}\delta_x$.

The relative velocity of D with respect to A is $v_{Dx}-v_x=\frac{\partial v_x}{\partial y}\delta_y,v_{Dy}-v_y=\frac{\partial v_y}{\partial y}\delta_y$.

Linear Deformation Motion

![Pasted image 20240902212840.png][2]

The relative velocities $\frac{\partial v_x}{\partial x}\delta_x$ and $\frac{\partial v_y}{\partial y}\delta_y$ represent the linear deformation rates of rectangle ABCD’s edges. During time interval $dt$:

The relative rate of area change is:

After neglecting higher-order terms:

Extending to three-dimensional space similarly yields:

Angular Deformation Motion

![Pasted image 20240902214219.png][3]

The relative velocities $\frac{\partial v_y}{\partial x}\delta x,\frac{\partial v_x}{\partial y}\delta y$ represent the rotation of edges AB and AD around point A.

Defining counterclockwise rotation as positive,

The angular velocity of edge AB is

Similarly, the angular velocity of edge AD is

Microelement Rotation Angular Velocity about the $z$-axis

Definition: The average value of the angular velocities of two mutually perpendicular lines in the projection of a microelement on the $xOy$ plane rotating about the $z$-axis. (Half the sum of angular velocities)

Angular Deformation Rate

Definition: Half of the change in angle per unit time between two mutually perpendicular lines in the projection of a fluid element on the $xOy$ plane. (Half of the angular velocity difference)

When extended to three-dimensional space, the angular velocities and angular deformation rates of the fluid element along the three axes can be similarly derived.

Omitted.

Divergence

Definition: The sum of the directional derivatives of each velocity component in its respective direction is called the three-degree of the velocity vector.

Physical meaning: It quantifies the relative volume change rate of a fluid element during motion.

==Assumption==: The density of the fluid remains unchanged (the fluid motion is considered incompressible).

The volume flow rate emitted from a point is defined as

This equals the net volume outflow per unit time from a unit volume control volume at a certain point in space, which is also equal to the relative volume change rate of the fluid element during motion.

Curl

Definition: Twice the angular velocity of rotation.

Velocity Potential

In fluid mechanics, fluid motion can be classified based on whether fluid particles have rotational motion:

• Rotational flow
• Irrotational flow

When the flow is considered irrotational, $\omega=0$,

The above system of equations is the necessary and sufficient condition for $v_xdx+v_ydy+v_zdz$ to constitute the total differential of some function $\phi(x,y,z)$. That is,

$\phi$ is called the velocity potential or velocity potential function.

When using cylindrical coordinates,

Scalars

Pressure

Density

Temperature

Viscosity Coefficient

Vector

Flow Velocity

Shear Stress

Ideal Gas Equation of State

$R=8.314J \cdot mol^{-1} \cdot K^{-1}$ is the universal molar gas constant.

$R'$ is the ==specific gas constant==.

For ideal air $R'=287J/(kg \cdot K)$

Aerodynamics and Moments

Aerodynamic Force $R$: Resultant

The force exerted by air on an object

• Pressure $p$: Pressure
• Shear stress $\tau$: Shear stress

The resultant of pressure and shear stress is the force exerted by air on the object, known as the aerodynamic force.

Wind-axis system

• Lift $L$: Lift, the vertical component
• Drag $D$: Drag, the horizontal component

Freestream

Freestream refers to the undisturbed incoming flow in front of the aircraft, i.e., the natural flow of air without interference from the aircraft or other objects.

The directions of lift and drag are determined by the freestream direction.

Angle of Attack (AOA)

Angle of attack (abbreviated as AOA, commonly represented by the Greek letter α) is a term in aerodynamics. It refers to the angle between the chord line of an airfoil and the direction of the free stream (or relative wind flow). For an aircraft, the angle of attack is defined as the angle between the aircraft’s longitudinal axis and the direction of the relative wind. A positive angle of attack occurs when the airfoil is tilted upward, while a negative angle of attack occurs when it is tilted downward.

Body Axis System

• Normal force $N$: Perpendicular to the airfoil direction
• Axial force $A$: Parallel to the airfoil direction

力矩 $M$：Moment

The moment that makes the aircraft pitch up is positive, and the moment that makes the aircraft pitch down is negative.

Dynamic Pressure $q$

The dynamic pressure generated by free stream $V_{\infty},\rho_{\infty}$

Unit is $Pa$, same as pressure

Characteristic Geometric Dimension $S$

For three-dimensional objects, it represents the area; for two-dimensional objects, it represents the perimeter.

Dimensionless Parameters

Three-dimensional objects commonly use uppercase $C$, while two-dimensional objects commonly use lowercase $c$.

Lift Coefficient

Drag Coefficient

Normal Force Coefficient

Axial Force Coefficient

Aerodynamic Coefficients

Moment Coefficient

Pressure Coefficient

$p$：某点静压 (Static pressure at a point)

$p_{\infty}$：自由来流静压 (Freestream static pressure)

Friction Coefficient

$\tau$: Shear stress at a point, which is the derivative of shear stress with respect to area. Dimensions are the same as pressure.

Two Centers

Center of Pressure (COP)

Center of Pressure (COP): The point of intersection between the line of action of the resultant force of fluid pressure on a plane or curved surface within the fluid and that plane or surface. The aerodynamic force $R$ produces zero moment $\vec{0}$ about this point.

Aerodynamic Center (AC)

Aerodynamic Center (English: aerodynamic center, abbreviated as AC) in aerodynamics refers to a fixed point on an airfoil where the pitching moment does not vary with changes in angle of attack, i.e.,

The Difference Between Aerodynamic Center and Center of Pressure

The center of pressure is a special point where the force system is synthesized, resulting in zero net moment at that point, with the center of pressure located behind the aerodynamic center. On the other hand, the aerodynamic center is the point where the net moment remains unchanged.

The position of the center of pressure changes with variations in the angle of attack. As the angle of attack increases, the lift increases, causing the center of pressure to move forward. This simultaneously reduces the distance between the center of pressure and the aerodynamic center. The increased lift multiplied by the shortened moment arm precisely equals the unchanged moment, which is exactly what the definition of the aerodynamic center requires.

Published on 2024-09-02 at 孤筝の温暖小家, last modified on 2024-09-02

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

Physical Quantities

Internal Energy U

Q is the heat absorbed by the system, W is the work done by the system

Entropy S

Entropy is a measure of the total amount of energy that cannot perform work in terms of dynamics. That is, when the total entropy increases, the ability to perform work decreases, and the measure of entropy is an indicator of energy degradation.
In a ==reversible process==

Q: In a reversible process, the system absorbs heat at a constant temperature.

焓 H

Free Energy F

Internal energy of the system - energy that cannot do work = energy that can do work (free energy)

Gibbs Function (Free Enthalpy) G

Chemical Potential $\mu$

Where:

• n: Amount of substance ($mol$)
• $G_m$: Molar Gibbs function

Equation

State Function Total Differential (Reversible Process)

Total Differential of Internal Energy (Fundamental Equation of Thermodynamics)

i.e., $U=U(S,V)$

$\because$ The second-order partial derivatives of U are equal
$\therefore$ We obtain the first Maxwell relation:

Enthalpy Total Differential

Differential of Free Energy

Gibbs Function Total Differential

Total Differential of Entropy

Maxwell Relations

The Maxwell Relations are important equations in thermodynamics that describe the relationships between state variables.
Here are the four Maxwell Relations:

1. First Maxwell Relation:

   $$ \left ( \frac{\partial T}{\partial V} \right)_S = - \left ( \frac{\partial P}{\partial S} \right)_V $$

2. Second Maxwell Relation:

   $$ \left ( \frac{\partial T}{\partial P} \right)_S = \left ( \frac{\partial V}{\partial S} \right)_P $$

3. Third Maxwell Relation:

   $$ \left ( \frac{\partial S}{\partial V} \right)_T = \left ( \frac{\partial P}{\partial T} \right)_V $$

4. Fourth Maxwell Relation:

   $$ \left ( \frac{\partial S}{\partial P} \right)_T = -\left ( \frac{\partial V}{\partial T} \right)_P $$

Corollary:

For an ideal gas $pV=nRT$, substituting yields:

Equation of State

Definition: The functional relationship equation between temperature and state parameters.

Ideal Gas Equation of State

• n: Amount of substance
• R: Gas constant
• N: Total number of particles in the system
• k: Boltzmann constant

Van der Waals Equation

• a、b are constants

Common Coefficients

Coefficient of Volume Expansion $\alpha$

Pressure Coefficient $\beta$

Isothermal Compressibility $\kappa_T$

Heat Capacity C

Constant Volume Heat Capacity $C_V$

Heat Capacity at Constant Pressure $C_p$

Multipartite Heat Capacity $C_n$

Adiabatic Equation for Ideal Gas

$\gamma$：[[#Adiabatic Index]] C: Constant

==The Second Law of Thermodynamics==

1. Two adiabatic lines cannot intersect.
2. T: External temperature
3. The external temperature equals the system temperature (T) if and only if the process is reversible.

Efficiency of Reversible Heat Engine $\eta$

[[#Carnot’s Theorem]]

• $T_2$: Temperature of the high-temperature heat reservoir
• $T_1$: Temperature of the low-temperature heat reservoir
• $0 \lt \eta \lt 1$

Energy Equation in Throttling Process

For the throttling process, the following energy balance equation can be written:

where, $H_1$ and $H_2$ are the enthalpies before and after throttling, respectively. Proof: The system is adiabatic, $\Delta U=W$ $U_2-U_1=p_1V_1-p_2V_2$ $U_2+p_2V_2=U_1+p_1V_1$ $H_2=H_1$

Joule-Thomson Coefficient

[[#Throttling Process]]

Fundamental Equations of Thermodynamics for Open Systems ($+\mu dn$)

Total Differential of Internal Energy

Enthalpy Total Differential

Differential of Free Energy

Gibbs Function Total Differential

Grand Potential

The Clausius-Clapeyron Equation

Where:

• $L=T(S_m^\beta-S_m^\alpha)$ is the latent heat of phase transition.
• $S_m$ is the molar entropy
• $s$ is the specific entropy (entropy per unit mass)
• $v$ is the specific volume (volume per unit mass)

Ehrenfest Equations

In [[#second-order phase transitions]], the Clapeyron equation becomes an indeterminate form of $\frac{0}{0}$. By applying L’Hôpital’s rule and taking partial derivatives of the numerator and denominator with respect to $T$, we obtain

Alternatively, taking partial derivatives with respect to $𝑝$ yields

These two equations are known as the Ehrenfest equations.

First-Order Phase Transition Equation

Formula

For first-order phase transitions, at the transition temperature $T_c$, the Gibbs free energy $G$ itself is continuous, but its first derivatives with respect to temperature and pressure exhibit discontinuities:

• $\left ( \frac{\partial G}{\partial T} \right)_P = -S$
• $\left ( \frac{\partial G}{\partial P} \right)_T = V$

At the transition point $T_c$, these first derivatives undergo jumps:

Here, $S$ represents entropy, $V$ represents volume, and the subscripts 1 and 2 denote the two phases before and after the phase transition.

Proof

The characteristic of a first-order phase transition lies in the discontinuity of certain first-order derivatives of the thermodynamic potential function at the transition point. Taking entropy as an example:

1. Continuity of Gibbs Free Energy: At the transition temperature $T_c$, the Gibbs free energy $G$ is continuous: $$ G_1 (T_c, P) = G_2 (T_c, P) $$
2. Discontinuity of Entropy: Taking the partial derivative of $G$ with respect to temperature yields the expression for entropy: $$ S = -\left ( \frac{\partial G}{\partial T} \right)_P $$ At $T_c$, the jump in entropy is: $$ \Delta S = S_2 - S_1 = -\left ( \frac{\partial G_2}{\partial T} \right)_P + \left ( \frac{\partial G_1}{\partial T} \right)_P \neq 0 $$

This demonstrates the discontinuity of entropy at the phase transition point.

Second-Order Phase Transition Equations

Formula

For second-order phase transitions, at the critical temperature $T_c$, the Gibbs free energy $G$ and its first derivatives (such as entropy and volume) are continuous, but its second derivatives (such as heat capacity, compressibility, and thermal expansion coefficient) are discontinuous:

• $\left ( \frac{\partial^2 G}{\partial T^2} \right)_P = \frac{\partial S}{\partial T} = \frac{C_P}{T}$
• $\left ( \frac{\partial^2 G}{\partial P^2} \right)_T = \frac{\partial V}{\partial P} = -\kappa_T V$

At the phase transition point $T_c$, these second derivatives exhibit jumps:

Here, $C_P$ is the constant-pressure heat capacity, and $\kappa_T$ is the isothermal compressibility.

Proof

The characteristics of second-order phase transitions lie in the discontinuity of certain second derivatives of the thermodynamic potential function at the phase transition point. Taking heat capacity as an example:

1. Continuity of Entropy: At the phase transition temperature $T_c$, the entropy $S$ is continuous: $$ S_1 (T_c, P) = S_2 (T_c, P) $$
2. Discontinuity of Heat Capacity: Taking the partial derivative of entropy with respect to temperature yields the expression for heat capacity: $$ C_P = T \left ( \frac{\partial S}{\partial T} \right)_P $$ At $T_c$, the jump in heat capacity is: $$ \Delta C_P = C_{P 2} - C_{P 1} = T \left ( \left ( \frac{\partial S_2}{\partial T} \right)_P - \left ( \frac{\partial S_1}{\partial T} \right)_P \right) \neq 0 $$

This indicates that heat capacity exhibits discontinuity at the phase transition point.

Differential of the Gibbs Function for Multicomponent Systems

Gibbs Phase Rule

Where:

• $f$: Degrees of freedom in a multicomponent multiphase system, i.e., the number of independently variable intensive properties
• $k$: Number of components
• $\varphi$: Number of phases

de Broglie Relations

[[#de Broglie Wave]]

Planck Constant

Reduced Planck Constant (Dirac Constant):

Uncertainty Relation

Or

==Generally==, take

Three-Dimensional Free Particle Quantum State Count

==Temporarily disregarding quantum spin==

When particles move within a macroscopically sized container, their momentum and energy values are quasi-continuous.

Cartesian Momentum Representation

Find the number of quantum states for free particles within a volume $V=L^3$, in the momentum range from $p_x$ to $p_x+dp_x$, $p_y$ to $p_y+dp_y$, and $p_z$ to $p_z+dp_z$.
Solution 1:
From the relationship between momentum and quantum numbers for [[#Three-Dimensional Free Particles]], the possible number of $p_x$ states is:

This means $dn_x$ is characterized by $dp_x$.
The same applies to the other two directions.

Solution 2:
From the definition of [[#Phase Cell]], the phase cell size for a three-dimensional free particle is $\Delta q_1\cdots \Delta q_r\Delta p_1\cdots \Delta p_r\approx h^3$. In this problem, the $\mu$-space volume is $d\Omega=Vdp_xdp_ydp_z$.
The number of phase cells that can fit into this volume is the number of quantum states for free particles.

Momentum Representation in Spherical Coordinates

Within volume $V$, for momentum magnitude ranging from $p$ to $p+dp$, direction from $\theta$ to $d\theta$, and $\varphi$ to $\varphi+d\varphi$, the number of possible states for a free particle is

There are three quantum numbers $p,\theta,\varphi$ in total, corresponding to three degrees of freedom.

Energy Representation

Integrating over $\theta,\varphi$ ($0 \lt \theta \lt \pi,0 \lt \varphi \lt 2\pi$) gives:

Here the momentum can be in any direction, with 1 degree of freedom. $\because$ $\varepsilon=\frac{p^2}{2m}$ $\therefore$ Within volume $V$, the number of possible quantum states for free particles in the energy range from $\varepsilon$ to $\varepsilon+d\varepsilon$ is

where $D(\varepsilon)$ represents the number of possible states per unit energy interval, known as the ==density of states==.

Considering Quantum Spin

[[#Spin Angular Momentum]] Number of quantum states = Number of momentum quantum states obtained above * Possible spin states of the particle eg. If the particle’s spin quantum number is $\frac{1}{2}$, $S_z=2s+1=1$, the above result should be multiplied by 2.

Definitions and Theorems

Basic Definitions

Equilibrium State

The definition of equilibrium state focuses on two points:

1. There is no macroscopic exchange of energy or matter between the system and its surroundings (distinguished from steady state). Microscopic exchanges are allowed.
2. The macroscopic properties of all parts of the system remain unchanged over a long period of time. Equilibrium state is thermodynamic equilibrium, with fluctuations present.

Relaxation time: The time required for a system to return to equilibrium from a non-equilibrium state.

Methods for Describing Equilibrium Properties: Macroscopic Description

• State parameters: Macroscopic variables that determine equilibrium properties
• State functions: Macroscopic variables determined by state parameters

Extensive and Intensive Quantities

In thermodynamics, the macroscopic variables of a homogeneous system can be divided into two categories: extensive and intensive quantities.

广延量（Extensive Variables）

广延量是指依赖于系统规模或大小的热力学变量。这些变量随着系统的大小成比例地增加或减少。

• 质量（Mass）：整个系统的质量。
• 体积（Volume, V）：系统占据的空间体积。
• 能量（Energy, E）：包括内部能量、动能和势能。
• 熵（Entropy, S）：表示系统微观状态的不确定性和混乱度。
• 粒子数（Number of particles, N）：系统中粒子的总数。
• 热量（Heat, Q）：系统所包含或传递的热量。
• 电荷（Charge, Q）：系统中的电荷总量。

强度量（Intensive Variables）

强度量是指与系统的大小无关的热力学变量，这些变量在系统内的任一点都保持一致，不随系统规模而变化。

• 温度（Temperature, T）：表示系统的热状态。
• 压强（Pressure, P）：系统单位面积上所受的力。
• 化学势（Chemical potential, μ）：系统中每增加一个粒子所需要的自由能。
• 浓度（Concentration）：单位体积内粒子的数量。
• 电场（Electric field, E）：单位电荷所受的力。
• 磁场（Magnetic field, H）：单位磁偶极子所受的力矩。

The Relationship Between Extensive and Intensive Quantities

Extensive and intensive quantities are closely related. Typically, extensive quantities can be calculated through the product or integral of intensive quantities. For example:

• Internal energy (U): Can be determined by the system’s temperature, volume, and particle count.
• Volume: Can be calculated by integrating the density of each unit volume within the system.
• Heat and work: Are the products of intensive quantities (such as temperature and pressure) and extensive quantities (such as entropy and volume change).

Temperature

Law of Thermal Equilibrium: If object A is in thermal equilibrium with objects B and C respectively, then B and C must also be in thermal equilibrium when they are in thermal contact.

• Definition of temperature: Objects in mutual thermal equilibrium must possess a physical quantity intrinsic to the object itself, which is defined as temperature.
• Temperature scale: A numerical representation method for temperature.
• Three essential elements: Thermometric property of the thermometric substance, fixed points, and scale division.

Work in Thermodynamic Processes

Quasi-static process: $W=\int_{V_1}^{V_2}pdV$, where $V_1$ is the initial volume of the system and $V_2$ is the final volume.

Reversible process: A quasi-static process without dissipation

In a reversible process, the work done by the surroundings on the system can be expressed using the system’s state parameters.

State space: A space constructed with independent state parameters as coordinate axes.

In state space, a point represents an equilibrium state of the system, and a curve represents a quasi-static process.

Process equation: The functional relationship between independent state parameters during a quasi-static process.

Expressions for work (considering only reversible processes):

• Fluid volume change process: −𝑝d𝑉
• Surface film area change process: 𝜎d𝐴
• Elastic wire length change: 𝐹d𝐿
• Polarization work (ignoring dielectric volume change):
• Magnetization work (ignoring magnetic medium volume change):

Work in special non-static processes:

• Isochoric (constant volume) $𝑊=0$
• Isobaric (constant pressure) $𝑊=−𝑝 \Delta 𝑉$

Carnot’s Theorem

Among all heat engines operating between two fixed temperatures, reversible heat engines have the highest efficiency. Corollary: If two reversible heat engines operate at the same temperatures, their efficiencies are equal. [[#Reversible Heat Engine Efficiency $ eta$]]

Principle of Entropy Increase

Under adiabatic conditions, processes that decrease entropy are impossible (entropy never decreases).

Proof: From the second law of thermodynamics:

For an adiabatic system, $dQ$=0, thus $dS\ge 0$.

Corollary: In an isolated (adiabatic) system, irreversible processes always proceed in the direction of increasing entropy, i.e., $dS \gt 0$.

Throttling Process

In thermodynamics, the throttling process (or Joule-Thomson process) is a typical adiabatic irreversible process characterized by a significant pressure drop as the fluid passes through a throttling device (such as an orifice, valve, or porous plug), while the enthalpy remains constant. Below is a detailed analysis of this process:

Characteristics of Throttling Process

1. Adiabatic Nature: The throttling process is typically considered adiabatic because there is insufficient time for heat exchange during rapid flow.
2. Isenthalpic Property: The most important characteristic is that the throttling process is isenthalpic, meaning the enthalpy ($H$) remains unchanged before and after throttling. This property plays a key role in the throttling process. [[#Energy Equation in Throttling Process]]
3. Irreversibility: The throttling process is an irreversible process, and the entropy of the system usually increases.
4. Pressure Drop: During the throttling process, the pressure of the fluid drops significantly.

Relationship Between Enthalpy, Temperature, and Pressure

For an ideal gas, enthalpy is solely a function of temperature, so the temperature remains constant during a throttling process (a special case of the Joule-Thomson effect). However, for real gases, the situation is more complex—the temperature may increase or decrease depending on the gas’s ==Joule-Thomson coefficient== ([[#Joule-Thomson Coefficient]]), defined as:

• If $\mu_{JT} \gt 0$, the gas temperature decreases during throttling.
• If $\mu_{JT} \lt 0$, the gas temperature increases during throttling.

Applications of Throttling Process

The throttling process is crucial in many industrial applications, such as:

• Refrigeration Systems: Utilizing throttling valves (e.g., expansion valves) to reduce the temperature and pressure of refrigerants, thereby achieving cooling effects.
• Gas Separation: Cooling certain gases to their liquefaction temperature through throttling to separate different gas components.

Characteristic Functions

Characteristic functions (also known as thermodynamic potential functions or thermodynamic potentials) are used to describe the state of a thermodynamic system. Various thermodynamic properties and relationships can be derived from these functions.
There are four main characteristic functions: internal energy, Helmholtz free energy, enthalpy, and Gibbs free energy.

• Internal energy $U(S,V)$: For adiabatic isochoric systems, U remains constant
• Enthalpy $H(S,p)$: For adiabatic isobaric systems, H remains constant
• Free energy $F(T,V)$: For isothermal isochoric systems, F remains constant
• Gibbs function $G(T,p)$: For isothermal isobaric systems, G remains constant

Radiation

Equilibrium Radiation

Equilibrium Radiation refers to the electromagnetic radiation characteristics of a system in a state of thermodynamic equilibrium. In this state, the processes of radiation and absorption within the system reach a dynamic balance, and the spectrum and intensity of electromagnetic radiation are solely determined by the system’s temperature.

Characteristics of Equilibrium Radiation

1. Thermal Equilibrium:
   • The system is in a state of thermal equilibrium, meaning the temperature is uniform across all parts, with no net heat flow.
2. Blackbody Radiation:
   • Under thermal equilibrium, the radiation exhibits blackbody characteristics. Blackbody radiation is an idealized form of radiation that perfectly absorbs and re-emits all frequencies, with its spectrum depending solely on temperature.

Radiative Flux Density

辐射通量密度（Radiative Flux Density），也称为辐射出射度或辐射强度，是指单位时间内通过单位面积的辐射能量。它是描述辐射场的重要物理量，在热力学、气象学、天文学等领域有广泛应用。

Definition

The radiant flux density E is expressed as:

where:

• E is the radiant flux density, measured in watts per square meter ($W/m^2$).
• $d\Phi$ is the radiant energy flux passing through area $dA$, measured in watts ($W$).
• $dA$ is the area, measured in square meters ($m^2$).

==Or==

where:

• $c$: speed of light
• $u$: radiant energy density, $u=\frac{U}{V}$, the equilibrium radiant energy per unit volume

Applications of the Stefan-Boltzmann Law

For an ideal black body, its radiant flux density is proportional to the fourth power of its temperature, as described by the Stefan-Boltzmann law:

where:

• $E$ is the radiant flux density of the black body.
• $\sigma$ is the Stefan-Boltzmann constant, with a value of $5.67 \times 10^{-8} \, W \, m^{-2} \, K^{-4}$.
• $T$ is the absolute temperature of the black body, in Kelvin ($K$).

Directionality of Radiant Flux Density

辐射通量密度是一个矢量量，考虑其方向性时称为辐射强度 $I$，表示为单位立体角上的辐射通量密度：

其中：

• $I$ 是辐射强度，单位是瓦特每平方米每立体角（$W/m^2/sr$）。
• $d\Omega$ 是立体角，单位是球面度（sr）。
• $\theta$ 是辐射方向与法线之间的夹角。

Blackbody Radiation (Cavity Radiation)

黑体辐射（Blackbody Radiation）是指一个理想化的物体（称为黑体）在热平衡状态下发出的电磁辐射。黑体是一个理想化的概念，它具有完全吸收和完全辐射所有频率电磁波的能力。黑体辐射的特性只取决于黑体的温度，而与其材料或表面性质无关。

Characteristics of Blackbody Radiation

1. Perfect Absorption:
   • A blackbody can completely absorb all incident electromagnetic radiation, regardless of wavelength or direction. Thus, a blackbody exhibits no reflection or transmission at any wavelength.
2. Perfect Emission:
   • A blackbody emits electromagnetic radiation at any temperature, radiating across all wavelengths. This radiation is solely determined by the blackbody’s temperature.

Practical Applications of Blackbody Radiation

1. Cosmic Background Radiation:
   • The cosmic microwave background radiation closely approximates blackbody radiation at a temperature of 2.725 K, serving as crucial evidence for the Big Bang theory.
2. Stellar Spectra:
   • The radiation from stars approximates blackbody radiation, allowing their surface temperatures to be deduced by analyzing their spectra.
3. Infrared Thermometry:
   • Based on the principle of blackbody radiation, the surface temperature of an object can be inferred by measuring the intensity of the infrared radiation it emits.
4. Thermal Imaging:
   • Thermal imaging devices utilize the principle of blackbody radiation, detecting infrared radiation emitted from an object’s surface to create temperature-based images.

Criteria for Thermodynamic Equilibrium

Isolated System

For an isolated (adiabatic) system, S increases or remains constant, reaching $S_\max$ at equilibrium. If a small disturbance occurs, $\Delta S \lt 0$, expand it to a second-order Taylor series $\Delta S=\delta S+\frac{1}{2}\delta^2 S$ At equilibrium, the conditions $\delta S=0,\delta^2 S \lt 0$ must be satisfied.

Isothermal-Isochoric System

等温等容系统 F decreases or remains constant, reaching $F_\min$ at equilibrium. If a small disturbance occurs, $\Delta F \gt 0$, which can be expanded as a second-order Taylor series $\Delta F=\delta F+\frac{1}{2}\delta^2 F$. At equilibrium, the conditions $\delta F=0,\delta^2 F \gt 0$ are satisfied.

Isothermal-Isobaric System

In an isothermal-isobaric system, G decreases or remains constant, reaching $G_\min$ at equilibrium. If a slight perturbation occurs, $\Delta G \gt 0$, which can be expanded as a second-order Taylor series: $\Delta G=\delta G+\frac{1}{2}\delta^2 G$. At equilibrium, the conditions $\delta G=0$ and $\delta^2 G>0$ must be satisfied.

Equilibrium Conditions for Two-Phase Systems in a Unit

Classification of Phase Transitions

[[#Ehrenfest Equations]] Ehrenfest (Paul Ehrenfest) classified phase transitions based on the continuity of thermodynamic potential functions and their derivatives at the phase transition point. Ehrenfest categorized phase transitions into first-order and second-order phase transitions.

First-Order Phase Transition

A first-order phase transition refers to a phase transition where the first derivatives of the thermodynamic potential (such as Gibbs free energy), like entropy and volume, are discontinuous at the transition point.

Typical first-order phase transitions include melting, vaporization, and sublimation. [[#First-Order Phase Transition Equations]]

Second-order Phase Transition

A second-order phase transition refers to a phase transition process where the first derivative of the thermodynamic potential function is continuous at the transition point, but its second derivative exhibits discontinuity.

Typical second-order phase transitions include the transition of superconductors and liquid crystal phase transitions.
[[#Second-order Phase Transition Equations]]

Multicomponent System Phase Equilibrium Conditions

Equilibrium Conditions for Isothermal-Isobaric Systems

The same applies to other systems

Gibbs Paradox

Description: The entropy increases abruptly from $2nR\ln 2$ to 0 when transitioning from two arbitrarily similar gases to the same gas.
Reason: Indistinguishability of identical particles.

The Third Law of Thermodynamics

Nernst’s Theorem

The change in entropy of a condensed system during an isothermal process approaches zero as the thermodynamic temperature tends to zero.

Principle of the Unattainability of Absolute Zero

It is impossible to cool an object to thermodynamic zero through a finite number of steps.

Corollaries

1. $\lim_{T \to 0}C_n=0$ This corollary is proven by Einstein’s quantum statistics in the [[#solid]] section.
2. $\lim_{T \to 0}\alpha=0,\lim_{T \to 0}\beta=0$
3. At absolute zero, the entropy value can be 0

Classical Description of Particle Motion States

$\mu$ Space

Describes the mechanical motion state of particles using position and momentum coordinates. The $2r$ variables $q_1,q_2,...,q_r;p_1,p_2,...,p_r$ form a $2r$-dimensional space with Cartesian coordinates, known as $\mu$ space.

Degrees of Freedom of a Particle: The minimum number of coordinates required to determine the spatial position or configuration position of a particle.

Phase Cell

The volume of a particle’s motion state in $\mu$ space

For a particle with $r$ degrees of freedom, the size of the phase cell is $\Delta q_1\cdots \Delta q_r\Delta p_1\cdots \Delta p_r$ When it satisfies the [[#Uncertainty Relation]] $\Delta q_i\Delta p_i\approx h$, the size of the phase cell $\approx h^r$

Free Particle

A free particle is a particle that moves freely without the influence of any force. In the absence of an external field, molecules of an ideal gas or free electrons in a metal can be approximately regarded as free particles.

Momentum of a free particle: $p_x=m\dot{x}$, the momentum of the particle in the x-axis direction. Here, m is the mass of the particle, and $\dot{x}$ is the velocity of the particle in the x-axis direction (the first derivative of position with respect to time). Energy of a free particle:

Linear Harmonic Oscillator

Classical mechanics tells us that a particle with mass $m$ under the action of an elastic force $F = -Ax$ will perform simple harmonic motion along the $x$-axis near the origin, known as a linear harmonic oscillator.

• Angular frequency of oscillation: $$\omega=\sqrt{\frac{A}{m}}$$
• $A$: Elastic coefficient

Energy of a Linear Harmonic Oscillator with One Degree of Freedom

The sum of kinetic and potential energy

Written in the standard form of an ellipse equation

Rotor (Degree of Freedom: 2)

Consider the motion of a particle \( P \) with mass \( m \) attached to the origin \( O \) by a light rod of fixed length.

• In Cartesian coordinates, the position of the particle is determined by coordinates \( x, y, z \).

Particle energy (kinetic energy):

• Using spherical coordinates \( r, \theta, \varphi \) to describe the particle’s position: \( x = r\sin \theta \cos \varphi, y = r\sin \theta \sin \varphi, z = r\cos \theta \)

Particle energy:

• Introducing conjugate momenta \( p_\theta = mr^2\dot{\theta}, p_\varphi = mr^2\sin^2\theta \dot{\varphi} \), and the moment of inertia \( I \) of the particle relative to \( O \).
• \( 0 \lt \theta \lt \pi, 0 \lt \varphi \lt 2\pi \), two degrees of freedom.

Diatomic Molecules

The two-body problem can be reduced to a single-body problem. In statistical physics, the rotation of a diatomic molecule about its center of mass is treated as a rotor.

According to classical mechanics, in the absence of external forces, the total angular momentum of the rotor $\vec L=\vec r\times \vec p$ is a conserved quantity, meaning neither its magnitude nor direction changes over time. Since $\vec r$ is perpendicular to $\vec L$, the motion of the particle lies in a plane perpendicular to $\vec L$. If we choose the $z$-axis to be parallel to $\vec L$, the particle’s motion must lie within the $xy$-plane. This is equivalent to fixing $\theta=\frac{\pi}{2}, p_\theta=0$.

The particle’s energy in this case is:

where: $L^2=\vec{L}\cdot \vec{L}$
Here, the particle has 1 degree of freedom, namely $\varphi$.

Quantum Description of Particle Motion States

de Broglie Wave

Microscopic particles (photons, electrons, protons, neutrons, and even atoms, molecules, etc.) universally exhibit wave-particle duality. A free particle with energy $\varepsilon$ and momentum $\vec{p}$ is associated with a plane wave of circular frequency $\omega$ and wave vector $\vec{k}$, known as the de Broglie wave.

[[#de Broglie Relations]]

Quantum State

In quantum mechanics, the motion state of microscopic particles is called the quantum state. The quantum state is characterized by a set of quantum numbers, and the number of quantum numbers is equal to the degrees of freedom of the particle.

==Linear Harmonic Oscillator Energy==

[[#Energy of a Linear Harmonic Oscillator with 1 Degree of Freedom]] In ==quantum physics==, the possible energy values of a linear harmonic oscillator with angular frequency $\omega$ are:

Here, n is the quantum number representing the oscillator’s motion state and energy, with only one such number—meaning the linear harmonic oscillator has 1 degree of freedom. Clearly, the values of $\varepsilon_n$ are discrete.

Energy Levels

Discrete energies are called energy levels.

In $\mu$ space, surfaces with equal energy are called ==equipotential surfaces==. Since these surfaces are discontinuous, each equipotential surface is referred to as an energy level. [[#$ mu$ space]]

Rotor Energy

[[#Diatomic Molecule]] In quantum physics, the value of $L^2$ can only take discrete values:

For a given angular momentum $L$, its projection $L_z$ along its eigen-direction (taken as the $z$-axis) can only take discrete values:

Clearly, there are $2l+1$ possible values for $m_l$.

A rotor’s motion state with 2 degrees of freedom is characterized by two quantum numbers $l,m_l$. In classical theory, the spatial orientation of the motion plane is arbitrary, whereas in quantum theory, $m_l$ can only take the above discrete values, known as spatial quantization.

Thus, the rotor’s energy is also quantized:

Degeneracy $\omega_l$

$\because$ It is observed that: the energy depends only on the quantum number $l$, and the quantum state depends on the two quantum numbers $m_l, l$. $\therefore$ There are $2l+1$ quantum states for the energy level $\varepsilon_l$.

Generally speaking, if a given energy level has more than one quantum state, it is called degenerate. If an energy level has only one quantum state, it is called non-degenerate. If an energy level has k quantum states, its degeneracy is k.

Between two adjacent [[#Energy Levels]], there are several [[#Phase Cells]], and the number of phase cells is the degeneracy of the energy level.

Spin Angular Momentum

Certain fundamental particles possess intrinsic angular momentum, known as spin angular momentum $S$.

$s$: ==Spin quantum number==. Can be integer or half-integer, an inherent property of particles that can be summed. [[#Bosons and Fermions]] The state of spin angular momentum is determined by its magnitude (spin quantum number $s$) and its projection along its eigenaxis. Using $z$ to denote the eigenaxis, the possible values of $S_z$ are

totaling $2s+1$ values.

Energy of Free Particles

Boundary Conditions

To determine the possible states of motion of a particle, it is necessary to know the boundary conditions of the de Broglie wave at the container walls. Typically, standing wave conditions or periodic boundary conditions are employed.

For a one-dimensional free particle in a container of length $L$, the periodic boundary condition requires that the possible states of motion must satisfy that an integer multiple of the de Broglie wavelength $\lambda$ equals the container length $L$. That is,

• $n_x$: The projection of the quantum number $n$ in the $x$-direction. For a one-dimensional free particle, the quantum number is simply $n_x$.

One-Dimensional Free Particle

• Wave vector $k_x$: $$ k_x=\frac{2\pi}{L}n_x,n_x=0,\pm 1,\pm 2,\cdots $$
• Momentum $p_x$: $$ p_x=\frac{2\pi\hbar}{L}n_x,n_x=0,\pm 1,\pm 2,\cdots $$
• Energy $\varepsilon_{n_x}$ $$ \varepsilon_{n_x}=\frac{p_x^2}{2m}=\frac{2\pi^2\hbar^2}{m}\cdot\frac{n_x^2}{L^2},n_x=0,\pm 1,\pm 2,\cdots $$

Three-Dimensional Free Particle

• Momentum: $$ \begin{cases} p_x=\frac{2\pi\hbar}{L}n_x,n_x=0,\pm 1,\pm 2,\cdots\\ p_y=\frac{2\pi\hbar}{L}n_y,n_y=0,\pm 1,\pm 2,\cdots\\ p_z=\frac{2\pi\hbar}{L}n_z,n_z=0,\pm 1,\pm 2,\cdots \end{cases} $$
• Energy: $$ \varepsilon=\frac{1}{2m}(p_x^2+p_y^2+p_z^2)=\frac{2\pi^2\hbar^2}{m}\frac{n_x^2+n_y^2+n_z^2}{L^2} $$
• Quantum numbers: $n_x,n_y,n_z$ (three in total, corresponding to three degrees of freedom)
• Energy levels: Determined by $n_x^2+n_y^2+n_z^2$

Description of the Microscopic Motion States of a System

Microstates

The microscopic state of a system is its mechanical motion state.
It manifests as different occupation patterns where particles occupy different energy levels and phase cells.

System

==Limited to systems composed of identical and nearly independent particles==

• System composed of identical particles: A system consisting of the same type of particles with identical intrinsic properties (same mass, charge, spin, etc.). Examples include a free electron gas composed of free electrons, helium gas composed of $^4He$ atoms, etc. ^f1b50b
• System composed of nearly independent particles: The interaction between particles is very weak, and the average energy of interaction is much smaller than the average energy of a single particle, so the interaction between particles can be ignored, and the total energy of the system can be expressed as the sum of the energies of individual particles. Examples include systems composed of ideal gases. ^24ebad $$ E=\sum^{N}_{i=1}\varepsilon_i $$ Where:
• $\varepsilon_i$: Energy of the i-th particle
• $N$: Total number of particles

Classical Mechanics Description

In classical physics, identical particles [[#^f1b50b]] are distinguishable. Determining the microscopic state of motion of a system requires determining the mechanical state of motion (individual quantum state) of each particle.

Number of variables: $2Nr$
$r$: degrees of freedom of a single particle

Bosons and Fermions

In nature, microscopic particles can be divided into two categories, known as bosons and fermions.

[[#Spin Angular Momentum]]

Fermion

Spin quantum number is a half-integer

Examples: electron, proton, neutron, $\mu$ meson (all with $\frac{1}{2}$ spin); $^2H$ atom, $^3H$ nucleus (with $\frac{3}{2}$ spin), etc.

Boson

Spin quantum number is an integer

Examples: photon (1), $\pi$ meson (0), $^1H$ atom (1), $^4He$ atom (4), etc.

Pauli Exclusion Principle

In a system containing multiple identical and nearly independent fermions, an individual quantum state can accommodate at most one fermion.

Distribution

A set composed of the number of particles at each energy level, this set is called a distribution.

$a_l$: The number of particles at energy level $\varepsilon_l$ Satisfies:

A distribution encompasses a large number of possible microstates, and a microstate corresponds to a distribution.

Three Systems

Boltzmann System

A system composed of identical [[#^f1b50b]]nearly independent particles[[#^24ebad]] that are ==distinguishable==, with no restriction on the number of particles occupying a single quantum state, is called a Boltzmann system.

Bose System

A system composed of bosons, not constrained by the Pauli exclusion principle. In a Bose system consisting of multiple identical and nearly independent bosons, the number of bosons occupying the same individual quantum state is unrestricted.

Fermi System

A system composed of fermions, obeying the Pauli exclusion principle.

Three Types of Statistics

Boltzmann Statistics (Classical Statistics)

Distinguishable identical particles, with no restriction on the number of particles per phase cell.

Bose Statistics

Indistinguishable identical particles, with no restriction on the number of particles per phase cell.

Fermi Statistics

Indistinguishable identical particles, with a maximum of one particle per phase cell.

Principle of Equal Probability

For an isolated system in equilibrium, all possible microstates of the system are equally probable.

Most Probable Distribution

According to the principle of equal a priori probabilities, for an isolated system in equilibrium, the distribution with the greatest number of microscopic states has the highest probability of occurrence, known as the most probable distribution.

Boltzmann Distribution

The most probable distribution of particles in a Boltzmann system is called the Maxwell-Boltzmann distribution, or simply the Boltzmann distribution.

• Approximate equation: $\ln m!=m (\ln m-1),m\gg 1$
• Denote $\Omega_{m.B.}$ as $\Omega$
  Under Boltzmann statistics, this distribution contains the most microstates, with the maximum number of microstates $\Omega$.
  $$ a_l=\omega_le^{-\alpha-\beta\varepsilon_l}=e^{-\alpha-\beta\varepsilon_l}\frac{\Delta\omega_l}{h_0^r} $$
  The latter is the ==classical expression of the Boltzmann distribution==.
  $$ N=\sum_la_l=\sum_l\omega_le^{-\alpha-\beta\varepsilon_l}=\sum_se^{-\alpha-\beta\varepsilon_s} $$
  $$ E=\sum_l\varepsilon_la_l=\sum_l\omega_l\varepsilon_le^{-\alpha-\beta\varepsilon_l}=\sum_s\varepsilon_se^{-\alpha-\beta\varepsilon_s} $$
  $$ f_s=\frac{a_l}{\omega_l}=e^{-\alpha-\beta\varepsilon_l} $$
• $a_l$: The most probable number of particles (average number of particles) at energy level $\varepsilon_l$, which should satisfy the condition $a_l\gg 1$.
• $\alpha$: Lagrange multiplier.
• $\beta$: Lagrange multiplier, generally determined by experimental conditions.
• $\omega_l$: The number of quantum states at energy level $\varepsilon_l$, i.e., the degeneracy. Should satisfy the condition $\omega_l\gg 1$.
• $s$: A quantum state at energy level $\varepsilon_l$.
• $\sum_l$: Sum over quantum number $l$.
• $\sum_s$: Sum over all quantum states.
• $f_s$: The average number of particles in quantum state $s$.

Bose Distribution

The most probable distribution of particles in a Bose system is called the Bose-Einstein distribution, abbreviated as the Bose distribution.

Fermi Distribution

The most probable distribution of a Fermi system.

Relationship Among the Three Distributions

If $\alpha$ satisfies the condition ([[#Classical Limit Condition]], non-degeneracy condition)

then the $\pm$ in the Bose and Fermi distributions can be neglected, and both distributions transition to the Boltzmann distribution. In this case, for all $l$, we have

Localization

• In nature, some systems can be considered as composed of localized particles. For example, atoms or ions in a crystal oscillate slightly around their equilibrium positions.
• Although these particles are indistinguishable by their quantum nature, they can be distinguished based on their positions, so ==localized particles== can be treated as distinguishable particles.
• Systems composed of localized particles are called localized systems. Examples include paramagnetic solids and nuclear spin systems.

Localized systems and Bose (Fermi) systems that satisfy classical limit conditions both obey the Boltzmann distribution. However, their microscopic state numbers are different.

Statistical Expressions of Thermodynamic Quantities

Internal Energy

Particle Partition Function

Abbreviated as partition function

Applicable scope: Nearly independent particle system, particles are distinguishable, and the number of particles each phase cell (quantum state) can accommodate is not limited.

By eliminating $\alpha$ from the above three equations, we obtain the ==statistical expression of internal energy==:

Total Differential of Internal Energy

• $\sum_la_ld\varepsilon_l$: The change in internal energy caused by the alteration of energy levels due to changes in external parameters while the particle distribution remains unchanged, which represents the work done by the surroundings on the system during the process.
• $\sum_l\varepsilon_lda_l$: The change in internal energy caused by the redistribution of particles while the energy levels remain unchanged, which represents the heat absorbed by the system from the surroundings during the process.
• In an infinitesimal quasi-static process, the heat absorbed by the system from the surroundings equals the increase in internal energy due to the redistribution of particles among energy levels. Clearly, heat has no microscopic counterpart and is a unique macroscopic quantity in thermal phenomena. $dQ$ is not a total differential but merely an infinitesimal quantity.

Entropy

$\beta$ is a function of temperature, let

• $k$: Boltzmann constant, $k=1.381\times10^{-23}J\cdot K^{-1}$
• $N_A$: Avogadro’s number, $N_A=6.022\times10^{23}mol^{-1}$
• $R$: Molar gas constant, $R=8.314J\cdot K^{-1}\cdot mol^{-1}$

Integrating gives:

The integration constant is chosen to be 0.

From the Boltzmann distribution

we obtain

Simplifying the entropy gives

Boltzmann Relation

The more microstates a macrostate corresponds to, the greater its disorder and the larger its entropy.

• For distinguishable particle systems (localized systems), $\Omega=\Omega_{M.B.}$
• For Bose (Fermi) systems satisfying the classical limit condition, $\Omega=\frac{\Omega_{M.B.}}{N!}$

Free Energy

Equation of State for Ideal Gas

Classical Limit Conditions

[[#Three Distribution Relations]]

1. The smaller $N/V$, the more rarefied the gas;
2. The higher the temperature $T$;
3. The larger the molecular mass $m$ The easier it is to satisfy the classical limit conditions. Another expression: $$ n\lambda^3=e^{-\alpha}\ll 1 $$

Maxwell’s Velocity Distribution Law

Under general conditions, gases satisfy the classical limit condition.

Based on the Boltzmann distribution, the translational motion of the center of mass of gas molecules is studied, and the velocity distribution law of gas molecules is derived.

Maxwell Velocity Distribution Law

$f(v_x,x_y,x_z)$ should satisfy

$n=\frac{N}{V}$: number of molecules per unit volume

Maxwell’s Speed Distribution Law

Using the spherical polar coordinates volume element $v^2\sin \theta dv d \theta d \varphi$ to replace the Cartesian volume element $dv_xdv_ydv_z$, integrating over $\theta,\varphi$ gives the number of molecules per unit volume with speeds in the range $dv$ as:

This equation is called the molecular speed distribution of gases.

Most Probable Speed $v_p$

The speed distribution function has a maximum value. The speed at which the speed distribution function reaches its maximum is called the most probable speed. If the speed is divided into equal intervals, the interval containing $v_p$ has the highest number of molecules.

Average speed $\overline{v}$: the average value of speed $v$

Root-mean-square speed $v_s$: the square root of the average value of $v^2$

$M$: molar mass

Wall Collision Number (Effusion)

The number of molecules colliding with a unit area of the container wall per unit time is called the wall collision number. The number of particles escaping per unit time through a unit area hole in the container wall is called effusion. Clearly: Wall collision number = Effusion

$dA$ is an infinitesimal area element on the container wall, with its normal direction along the $x$-axis. Let $d\varGamma dA dt$ represent the number of molecules that collide with area $dA$ within time $dt$, with velocities in the range $dv_xdv_ydv_z$.

Integrating over velocities,

we obtain

Equipartition Theorem (Classical Statistics)

For a classical system in equilibrium at temperature $T$, the average value of each independent quadratic term in the particle energy equals $\frac{1}{2}kT$.

Monatomic Ideal Gas

The above equation has three squared terms. Applying the equipartition theorem gives:

Thus, the internal energy of a monatomic ideal gas is:

Using the thermodynamic relation $C_p-C_V=Nk$, the isobaric heat capacity is obtained as:

Adiabatic Index

[[#Ideal Gas Adiabatic Equation]]

Diatomic Molecular Gas

The energy of a diatomic molecule has five quadratic terms (momentum in three directions of the molecule, the motion between the two atoms can be reduced to a rotor, two degrees of freedom, i.e., two quadratic terms).

Solids

Atoms in a solid can undergo small vibrations around their equilibrium positions. Assuming each atom’s vibration is an independent simple harmonic motion, the atomic model of a solid simplifies to a three-dimensional linear oscillator.

The energy of an atom in one degree of freedom is:

with two quadratic terms.
The average energy per atom is:

The internal energy of the solid is:

The heat capacity at constant volume is:

In experiments, $C_p$ is usually measured. Using the thermodynamic relation:

$C_V$ can be derived.
At room temperature and high temperatures, $C_V$ matches predictions, but at low temperatures, the heat capacity drops rapidly with decreasing temperature—a phenomenon unexplained by classical theory.
[[#Principle of Unattainability of Absolute Zero]]

Metal

There are free electrons in metals. If the equipartition theorem is applied to the electrons, the heat capacity of the free electrons would be of the same order of magnitude as the heat capacity of the ionic vibrations. Experimental results show that above $3K$, the heat capacity of free electrons is negligible compared to the heat capacity of ionic vibrations.

Quantum Statistics of Localized/Nearly Independent Particle Systems Satisfying Classical Limit Conditions

Diatomic Ideal Gas

An ideal gas is a ==non-localized system==. Since it satisfies the [[#Classical Limit Condition]], it can be discussed using the Boltzmann distribution.

Under certain approximations, the motion of diatomic molecules can be divided into translation $t$, vibration $v$, and rotation $r$.

Translational Energy

Consistent with the results obtained from the equipartition theorem of classical statistics.

Vibrational Energy and Characteristic Temperature

$U^v$:

• The first term is temperature-independent and represents the ==zero-point energy== of $N$ oscillators.
• The second term is the ==thermal excitation energy== of $N$ oscillators at temperature $T$.

Introducing the ==vibrational characteristic temperature== $\theta_v$, which satisfies:

Substituting into the above system of equations yields:

$\because$ The ==characteristic temperature of diatomic molecules== is on the order of $10^3K$, $\therefore$ In the room temperature range, $T\ll \theta_v$. Within this range, the contribution of vibrational degrees of freedom to heat capacity is nearly zero.
At room temperature, the energy level spacing $\hbar \omega$ of diatomic molecules is much larger than $kT$. Due to the discrete energy levels, oscillators must acquire energy $\hbar \omega$ to transition to an excited state.

• In the case of $T \ll \theta_v$, the probability of an oscillator acquiring thermal energy $\hbar \omega$ and transitioning to an excited state is extremely low. Thus, on average, almost all oscillators remain frozen in the ground state.
  When the gas temperature rises, they hardly absorb any energy (unable to transition to higher energy levels). This explains why ==vibrational degrees of freedom do not participate in energy equipartition at room temperature==.
• In the case of $T \gg \theta_v$, $C_V^v \to Nk$ (can be proven using L’Hôpital’s rule).

Rotational Energy

Rotational energy levels

Degeneracy

Partition function

Characteristic rotational temperature

==At room temperature== $\frac{\theta_r}{T}\ll 1$, integration yields

From this we obtain

Consistent with classical statistics.

Heteronuclear Diatomic Molecules

Monatomic Ideal Gas

Entropy

==Monatomic Ideal Gas==

Using the approximation $\ln N!=N(\ln N-1)$, it simplifies to

The entropy in the above equation is an extensive quantity and represents absolute entropy.

Vapor Pressure Equation

Sackur-Tetrode equation

Chemical Potential

$\because$ For an ideal gas, $\frac{N}{V}(\frac{h^2}{2\pi mkT})^3/2\ll 1$
$\therefore$ ==The chemical potential of an ideal gas is negative==.

Solids

The thermal motion of atoms in a solid can be regarded as the vibration of $3N$ oscillators. Einstein assumed that these $3N$ oscillators all have the same frequency. Let $\omega$ denote the angular frequency of the oscillators, the energy levels of the oscillators are

The partition function

Internal energy

Heat capacity

Introducing the ==Einstein characteristic temperature== $\theta_E=\frac{\hbar \omega}{k}$, the heat capacity can be rewritten as

• When $T \gg \theta_E$, the approximation $e^{\theta_E/T}-1\approx \theta_E/T$ can be made, yielding $$ C_V=3Nk $$ which is consistent with the result from the equipartition theorem. At this point, the energy level spacing is much smaller than $kT$, and the quantization effect can be ignored.
• When $T \ll \theta_E$, the approximation $e^{\frac{\theta_E}{T}}-1 \approx e^{\frac{\theta_E}{T}}$ holds, $$ C_V=3Nk(\frac{\theta_E}{T})^2e^{-\frac{\theta_E}{T}} $$ As $T \to 0$, $C_V \to 0$. This proves the third law of thermodynamics[[#推论]].

Bose/Fermi Systems

[[#Bose Systems]], [[#Fermi Systems]].

Grand Partition Function for Bosonic Systems

Average Particle Number

Internal Energy

Grand Partition Function for Fermi Systems

Average Particle Number

Internal Energy

Generalized Force

Equation of State

Entropy

Grand Potential

[[#Grand Potential]]

Lagrange Multiplier

title: Bose-Einstein Condensation description: keywords: tags:

System Properties

Consider a system composed of $N$ identical, nearly independent bosons, with temperature $T$ and volume $V$. For clarity, assume the particles have zero spin.

According to the Bose distribution, the number of particles in each energy level is

$\because$ The number of particles in an energy level cannot be negative, $\therefore$ Let $\varepsilon_0$ denote the lowest energy level, $\varepsilon_0 \gt \mu$. If the lowest energy level is 0, then $\mu \lt 0$.

Chemical Potential

The chemical potential of a Bose system is less than 0. The chemical potential increases as the temperature decreases. When the temperature drops to a certain critical temperature $T_C$, $\mu \to 0$. $T_C$ is determined by the following equation:

Condensation Phenomenon

When $T \le T_C$, the ==particle number density== accumulated at the lowest energy level $\varepsilon=0$ is given by

At absolute zero, particles will occupy the lowest energy states as much as possible. For bosons, the number of particles that can occupy a single quantum state is unrestricted. Therefore, at absolute zero, all bosonic particles will be in the lowest energy level $\varepsilon=0$.

The above equation shows that:

When $T \lt T_C$, a macroscopic number of particles condense into the lowest energy level. This phenomenon is called ==Bose-Einstein condensation==.

At $T=T_C$, $C_V$ reaches its maximum value of $1.925Nk$.
When the temperature is sufficiently high,

Critical Condition for Bose-Einstein Condensation in an Ideal Bose Gas

$\lambda$: thermal de Broglie wavelength of atoms

Photon Gas (Bose Gas)

From the particle perspective, the radiation field inside an empty cavity can be regarded as a photon gas. The radiation field inside the empty cavity can be decomposed into the superposition of infinitely many monochromatic plane waves.

Based on the [[#de Broglie relations]], and the photon $\omega=ck$, we obtain the ==energy-momentum relation for photons==

• Photons are bosons with a spin quantum number of 1, and they obey Bose-Einstein statistics when in equilibrium.
• Since the cavity walls continuously emit and absorb photons, the number of photons in the photon gas is not conserved. When deriving the Bose distribution, only the condition that $E$ is constant exists, and not the condition that $N$ is constant. Therefore, only one Lagrange multiplier $\beta$ should be introduced, with $\alpha=0$.

Photon Gas Statistical Distribution

Thermodynamic Potential

From

we obtain the thermodynamic potential of the photon gas as $\mu=0$.

Number of Quantum States

The spin quantum number of a photon is 1, and its spin projection along the momentum direction can take two possible values, $\pm \hbar$, corresponding to left and right circular polarization. Considering that the photon spin has 2 projections, as shown in the [[#Energy Representation]] section of [[#Number of Quantum States for a Three-Dimensional Free Particle]],

Within a cavity of volume $V$, in the momentum range from $p$ to $p+dp$,

the number of quantum states for photons is

Within a cavity of volume $V$, in the angular frequency range from $\omega$ to $\omega+d \omega$,

the number of quantum states for photons is

Energy Distribution Function

In a cavity of volume $V$, within the frequency range from $\omega$ to $\omega+d \omega$,

The average number of photons

The energy of the radiation field

Rayleigh-Jeans Law

In the low-frequency range where $\frac{\hbar \omega}{kT}\ll 1$, $e^{\frac{\hbar \omega}{kT}}\approx 1+\frac{\hbar \omega}{kT}$, the above equation can be approximated as

This equation is known as the ==Rayleigh-Jeans Law==, which fits well at low frequencies but leads to the ultraviolet catastrophe at high frequencies.

Wien’s Law

In the high-frequency range where $\frac{\hbar \omega}{kT}\gg 1$, $e^{\frac{\hbar \omega}{kT}}-1\approx e^{\frac{\hbar \omega}{kT}}$, the above equation can be approximated as

This formula matches Wien’s Law, providing good agreement at high frequencies but poor agreement at low frequencies.

Inside a cavity of volume $V$

The internal energy of equilibrium radiation is

Grand Partition Function

Photon Gas Internal Energy

Consistent with [[#Internal Energy Distribution Function]]

Pressure

Comparing the above two equations gives

Entropy

When $T \to 0$, $S \to 0$, consistent with [[#Third Law of Thermodynamics]]

Radiation Flux Density

The formula for [[#Radiation Flux Density]] derived from [[#Radiation]] is

Based on the concept of [[#Wall Collision Rate (Effusion)]], the radiation flux density of a photon gas is obtained as

Free Electron Gas in Metals (Fermi Gas)

The free electrons in metals form a highly degenerate Fermi gas.

Average Electron Number

According to the Fermi distribution, at temperature $T$, the average number of electrons occupying a quantum state with energy $\varepsilon$ is:

The electron spin has 2 possible projections along its momentum direction. Based on the [[#Energy Representation]] in [[#Three-Dimensional Free Particle Quantum State Number]],

Within volume $V$, the number of electron quantum states in the energy range from $\varepsilon$ to $\varepsilon+d \varepsilon$ is:

Thus, the average number of electrons in this range is:

Total Number of Electrons

Integrating over $d \varepsilon$ gives

Given the number of electrons $N$, temperature $T$, and volume $V$, the chemical potential $\mu$ can be determined from the above equation.

System Properties at $T=0$

Average Electron Number per Quantum State

Fermi Level

The chemical potential at zero temperature is also known as the Fermi level, $\mu(0)$, which is the chemical potential when $T=0$.

From

we obtain

Fermi Momentum

Let

we obtain

This momentum is called ==Fermi momentum==.

Fermi Velocity

This velocity is called the ==Fermi velocity==.

Fermi Temperature

This temperature is called the ==Fermi temperature==.

Energy of Electron Gas

Electron Average Energy

Electron Gas Pressure

Comparison of Bose and Fermi Gases at $T=0$

In stark contrast to an ideal Bose gas, where all particles occupy the zero-energy, zero-momentum ground state with zero pressure at absolute zero temperature, a Fermi gas possesses extremely high average energy and momentum, generating substantial pressure at absolute zero.

Published on 2024-06-06 at 孤筝の温暖小家, last modified on 2024-06-06

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