Quantum physics final exam review

📘 The Knowledge System of Quantum Physics

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🌅 Chapter 1: The Birth of Quantum Concepts

The Predicament of Classical Physics

• Blackbody radiation problem: classical theory predicts the ultraviolet catastrophe
  • Wien’s displacement law: $\lambda_{\text{max}} T = b$
  • Stefan-Boltzmann law: $I = \sigma T^4$
• Photoelectric effect problem: cannot be explained by classical wave theory

The Introduction of Quantum Concepts

• Planck’s quantization: $E = nh\nu$, energy is quantized
• Einstein’s light quantum: $E_k = h\nu - \phi$, the particle nature of light
• de Broglie matter waves: $\lambda = \frac{h}{p}$, the wave nature of particles

Early Quantum Models

• Bohr model of the atom: $E_n = -\frac{m e^4}{2 \hbar^2 n^2} = -\frac{13.6\,\text{eV}}{n^2}$
• Compton scattering: $\lambda' - \lambda = \frac{h}{mc}(1 - \cos \theta) = 2 \lambda \sin^2 \frac{\theta}{2}$

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🌊 Chapter 2: Wave Functions and the Schrödinger Equation

Introducing the Wave Function

• Wave-particle duality: particles have both wave-like and particle-like properties
• Definition of the wave function: $\psi(x,t) = A e^{i(kx - \omega t)}$ expressed with the position vector r: $$\psi(\vec{r}, t) = \Psi_0 e^{-\frac{i}{\hbar}(Et-\vec{p} \cdot \vec{r})}$$
• Statistical interpretation (normalization): $|\psi(x,t)|^2$ is the probability density of the particle at x

Properties of the Wave Function

• Normalization condition: $\int |\psi(\vec{r}, t)|^2 \, dV = 1$
• Standard conditions: single-valued, finite, continuous, square-integrable
• Boundary conditions: $\psi$ and $\frac{d\psi}{dx}$ are continuous
• Uniqueness of the wave function: multiplying by a constant still describes the same physical state of the particle

The Schrödinger Equation

• Time-dependent equation: $i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$
• Stationary-state equation: $\hat{H} \psi = E \psi$
• Hamiltonian operator: $\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\vec{r})$
• • Probability current density: $\vec{j} = \frac{\hbar}{2mi}(\psi^* \nabla \psi - \psi \nabla \psi^*)$
• Probability conservation equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot \vec{j} = 0$, $\rho = |\psi|^2$
• Role of the Schrödinger equation:
  • Describes the time evolution of a quantum system
  • Solving the equation yields the wave function (write down the stationary-state equation -> solve the differential equation -> determine the parameters from the properties)
  • The square of the wave function gives the probability distribution of the particle’s position

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🔧 Chapter 3: Operator Theory and Measurement

Introducing Operators

• Definition of operators: physical quantity → Hermitian operator (note: all operators in quantum mechanics are linear)
• Common operators:
  • Position: $\hat{x} = x$
  • Momentum: $\hat{p} = -i\hbar \frac{d}{dx}$
  • Energy (Hamiltonian): $\hat{H} = \frac{\hat{p}^2}{2m} + V(x)$
• Hermitian operators:
  • Defining equation: $\hat{A}^\dagger = \hat{A}$
  • Or expressed as: $∫ φ^*(x) Âψ(x) dx = ∫ [Âφ(x)]^* ψ(x) dx$

Eigenvalue Problems

• Eigenvalue equation: $\hat{A} \psi_n = a_n \psi_n$ (analogous to eigenvectors and eigenvalues in linear algebra)
• Hermitian properties:
  • Eigenvalues are real
  • Eigenfunctions are orthogonal: $\langle\psi_m|\psi_n\rangle = \delta_{mn}$
  • $\int \psi_m^*(x) \psi_n(x) \, dx = \delta_{mn} = 0$ (if m ≠ n)
  • Eigenfunctions are complete: any wave function can be expanded in its eigenfunctions

State Expansion and Measurement

• State expansion: $|\psi\rangle = \sum c_n |\phi_n\rangle$
• Expansion coefficients: $c_n = \langle \phi_n | \psi \rangle$
• Integral form:
  $\psi(x) = \int c_n \phi_n(x) \, dx$, $c_n = \int \phi_n^*(x) \psi(x) \, dx$, cn is the projection coefficient of the wave function onto the eigenstate
• Measurement postulate:
  • The measurement result can only be an eigenvalue $a_n$
  • Measurement probability: $P(a_n) = |c_n|^2$
  • Measurement average: $\langle A \rangle = \sum_n a_n |c_n|^2 = \int \psi^* \hat{A} \psi \, dx$

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🎯 Chapter 4: Uncertainty and Commutation Relations

The Uncertainty Principle

• Heisenberg relation: $\Delta x \cdot \Delta p \ge \frac{\hbar}{2}$
• Energy-time relation: $\Delta E \cdot \Delta t \ge \frac{\hbar}{2}$

Commutation Relations

• Definition of the commutator: $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$
• Fundamental commutation relation: $[\hat{x}, \hat{p}] = i\hbar$
• Angular momentum commutation relations:
  • $[\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k$ (cyclic)
  • $$[\hat{L}^2, L_i] = 0$$
• Angular momentum eigenvalue equations:
  • $$\hat{L}^2 |l, m\rangle = \hbar^2 l(l+1) |l, m\rangle$$
  • $$\hat{L}_z |l, m\rangle = \hbar m |l, m\rangle$$
• Commutation and measurement: commuting operators can be measured precisely at the same time

Complete Set Theory

• Definition of a complete set: the minimal set of commuting operators needed to completely determine a state
• Completeness condition: $\sum_n |\phi_n\rangle\langle\phi_n| = I$

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📦 Chapter 5: Typical Quantum Systems

Potential Well Problems

• Infinite potential well:

  • Wave function: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin \left(\frac{n\pi x}{L} \right)$ orthogonal
  • Energy levels: $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$
  • E1 is the zero-point energy, i.e., the particle can never be completely at rest

• Tunneling effect: $T \sim e^{-2\kappa a}$

The Harmonic Oscillator

• Energy level formula: $E_n = \hbar \omega(n + \frac{1}{2})$
• Zero-point energy: the ground-state energy is non-zero

The Hydrogen Atom

• Quantum numbers: $n, l, m$
• Explicit differential form of the eigenvalue equation: $-\frac{\hbar^2}{2mr^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\Phi + \frac{\hat{L}^2}{2mr^2}\Phi - \frac{e^2}{4\pi\varepsilon_0 r}\Phi = E\Phi$
• Energy levels: $E_n = -\frac{13.6\,\text{eV}}{n^2}$

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🌀 Chapter 6: Spin and Identical Particles

Electron Spin

• Spin operator: $\hat{S}_z = \pm \frac{\hbar}{2}$
• Pauli matrices: describe spin-1/2 systems
  

Statistics of Identical Particles

• Fermions: half-integer spin, antisymmetric wave function, obey the Pauli exclusion principle
• Bosons: integer spin, symmetric wave function, do not obey the Pauli exclusion principle

Two-State Systems

• Ammonia molecule model: $|+\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |2\rangle)$

In the state space spanned by , , write the Hamiltonian operator as a matrix:

$$\hat{H} = \begin{bmatrix} E_0 & -A \\ -A & E_0 \end{bmatrix}$$

Let ; solving this eigenvalue problem gives two eigenstates:

$$\begin{aligned} |\psi_+\rangle &= \frac{1}{\sqrt{2}} (|1\rangle + |2\rangle) \quad \text{(symmetric state)} \\ |\psi_-\rangle &= \frac{1}{\sqrt{2}} (|1\rangle - |2\rangle) \quad \text{(antisymmetric state)} \end{aligned}$$

The corresponding energy eigenvalues are:

$$E_+ = E_0 - A, \quad E_- = E_0 + A$$

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🎪 Summary of Core Ideas

The Basic Postulates of Quantum Mechanics

1. State postulate: the wave function completely describes the quantum state
2. Operator postulate: physical quantities correspond to Hermitian operators
3. Measurement postulate: measurement outcomes are eigenvalues, with probability $|c_n|^2$
4. Evolution postulate: states evolve according to the Schrödinger equation
5. Superposition postulate: any state can be written as a linear superposition of eigenstates

Characteristics of the Quantum World

• Wave-particle duality: both a wave and a particle
• Quantization: physical quantities take discrete values
• Probabilistic nature: measurement outcomes are probabilistic
• Uncertainty: not all physical quantities can be known precisely at the same time
• Superposition: a system can be in a superposition of multiple states
• Entanglement: quantum correlations exist between particles

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Translated from the Chinese original.