Quantum Mechanics Formulas

Cheatsheet Content

### Infinite Square Well #### Standard Well (0 to L) - **Energy Levels:** $$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} = \frac{n^2h^2}{8mL^2} \quad (n = 1, 2, 3, \ldots)$$ *Use for box from x = 0 to x = L* - **Wavefunctions:** $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) \quad (0 ### Superposition & Time Evolution #### Time Evolution - **General Eigenstate:** $$\psi_n(x,t) = \psi_n(x)e^{-iE_nt/\hbar}$$ - **Superposition:** $$\Psi(x,t) = \sum_n c_n\psi_n(x)e^{-iE_nt/\hbar}$$ *Phase factor: $e^{-iE_nt/\hbar} = e^{-i\omega_n t}$ where $\omega_n = E_n/\hbar$* #### Orthogonality Condition - **For two states to be orthogonal:** $$\langle\psi(t)|\psi(0)\rangle = 0$$ - **Equal Superposition (1/√2 each):** $$\Psi(t) = \frac{1}{\sqrt{2}}[\psi_1 e^{-iE_1t/\hbar} + \psi_2 e^{-iE_2t/\hbar}]$$ *Inner product:* $$\langle\Psi(t)|\Psi(0)\rangle = \cos\left(\frac{(E_2-E_1)t}{\hbar}\right)$$ *Orthogonal when:* $$\cos\left(\frac{(E_2-E_1)t}{\hbar}\right) = 0$$ *First orthogonality time:* $$\boxed{t = \frac{\pi\hbar}{2(E_2-E_1)}}$$ #### Period of Oscillation - **For superposition of two states:** - **Period:** $$T = \frac{2\pi\hbar}{E_2 - E_1}$$ - **Frequency:** $$f = \frac{E_2 - E_1}{2\pi\hbar}$$ - **Angular frequency:** $$\omega = \frac{E_2 - E_1}{\hbar}$$ #### Phase Evolution - **Phase of wavefunction:** $$\phi(t) = -\frac{E}{\hbar}t$$ - **Phase change after time t:** $$\Delta\phi = -\frac{E}{\hbar}t$$ - **For 2π phase change:** $$t = \frac{2\pi\hbar}{E}$$ #### Special Time Values - **For n-th state in ISW (0 to L):** - **Energy:** $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$ - **Period of 2π phase:** $$T_n = \frac{2\pi\hbar}{E_n} = \frac{4mL^2}{n^2\pi\hbar}$$ ### Expectation Values & Operators #### General Expectation Value - **In Position Space:** $$\langle \hat{A} \rangle = \int_{-\infty}^{\infty} \psi^*(x)\hat{A}\psi(x)dx$$ - **Using Eigenstate Expansion:** $$\langle \hat{A} \rangle = \sum_n |c_n|^2 a_n$$ *where $\hat{A}|n\rangle = a_n|n\rangle$* #### Position Operator - **Expectation Value:** $$\langle x \rangle = \int_{-\infty}^{\infty} x|\psi(x)|^2 dx$$ - **In Superposition:** $$\langle x \rangle = \sum_{n,m} c_n^*c_m\langle n|x|m\rangle$$ - **Parity Selection Rule (Symmetric Potential):** - $\langle \text{even}|x|\text{even}\rangle = 0$ - $\langle \text{odd}|x|\text{odd}\rangle = 0$ - $\langle \text{even}|x|\text{odd}\rangle \neq 0$ *For superposition of ground + first excited:* $$\langle x \rangle = \alpha_0\alpha_1[\langle 0|x|1\rangle + \langle 1|x|0\rangle]$$ #### Momentum Operator - **Definition:** $$\hat{p} = -i\hbar\frac{d}{dx}$$ - **Expectation Value:** $$\langle p \rangle = -i\hbar\int_{-\infty}^{\infty} \psi^*\frac{d\psi}{dx}dx$$ - **Real Wavefunction Rule:** $$\boxed{\text{If } \psi(x) = \psi^*(x) \text{ (real)} \Rightarrow \langle p \rangle = 0}$$ *Proof: Hermiticity of p̂* #### Energy (Hamiltonian) - **Expectation Value:** $$\langle H \rangle = \sum_n |c_n|^2 E_n$$ *For equal coefficients: $|c_n| = 1/\sqrt{N}$* $$\langle H \rangle = \frac{1}{N}\sum_n E_n$$ #### Variance - **General Formula:** $$\sigma_A^2 = \langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2$$ - **Standard Deviation:** $$\sigma_A = \sqrt{\text{Variance}} = \Delta A$$ - **For Eigenstate:** $$\sigma_A^2 = 0 \quad \text{(definite value)}$$ - **For Superposition:** $$\sigma_A^2 > 0 \quad \text{(quantum uncertainty)}$$ #### Matrix Representation - **For N-dimensional Hilbert Space:** - **Expectation value:** $$\langle H \rangle = \sum_n |c_n|^2 E_n$$ - **Variance:** $$\sigma^2 = \sum_n |c_n|^2 E_n^2 - \left(\sum_n |c_n|^2 E_n\right)^2$$ - **Block Diagonal Matrix Example:** $$H = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$ *Upper 2×2 block eigenvalues:* $$\lambda = \frac{\text{Tr}}{2} \pm \sqrt{\left(\frac{\text{Tr}}{2}\right)^2 - \det}$$ *For this example: $\lambda = 2 \pm 1$ → Eigenvalues: 1, 3, 2* #### Measurement & Collapse - **Before measurement:** $|\psi\rangle = \sum_n c_n|n\rangle$ - **Probability of measuring $a_k$:** $P_k = |c_k|^2$ - **After measuring $a_k$:** $$|\psi\rangle \rightarrow |k\rangle \quad \text{(collapse)}$$ *For sequential measurements: Must use collapsed state for next measurement!* ### Delta Function Potential #### Attractive Delta: V(x) = -αδ(x), α > 0 - **Bound State Energy:** $$\boxed{E = -\frac{m\alpha^2}{2\hbar^2}}$$ *Exactly ONE bound state* - **Bound State Wavefunction:** $$\psi(x) = \sqrt{\kappa}e^{-\kappa|x|}$$ *where* $$\kappa = \frac{m\alpha}{\hbar^2}$$ - **Decay Length:** $$\lambda = \frac{1}{\kappa} = \frac{\hbar^2}{m\alpha}$$ #### Boundary Conditions - **General Delta at x = x₀: V(x) = Aδ(x - x₀)** - **Condition 1 (Continuity):** $$\psi(x_0^+) = \psi(x_0^-)$$ - **Condition 2 (Derivative Jump):** $$\boxed{\psi'(x_0^+) - \psi'(x_0^-) = \frac{2mA}{\hbar^2}\psi(x_0)}$$ - **Sign Convention:** - A > 0: Repulsive (positive jump) - A ### 3D Box & Degeneracy #### Cubic Box (Lₓ = Lᵧ = L_z = L) - **Energy Eigenvalues:** $$\boxed{E_{n_x,n_y,n_z} = \frac{\pi^2\hbar^2}{2mL^2}(n_x^2 + n_y^2 + n_z^2)}$$ *where $n_x, n_y, n_z = 1, 2, 3, \ldots$* - **Ground State:** $$(n_x, n_y, n_z) = (1,1,1)$$ $$E_{111} = \frac{3\pi^2\hbar^2}{2mL^2}$$ *Non-degenerate (only one state)* - **First Excited State:** $$(2,1,1), (1,2,1), (1,1,2)$$ $$E = \frac{6\pi^2\hbar^2}{2mL^2}$$ *3-fold degenerate* - **Energy Ratio:** $$\frac{E_{\text{1st excited}}}{E_{\text{ground}}} = \frac{6}{3} = 2$$ #### Wavefunctions $$\psi_{n_x,n_y,n_z}(x,y,z) = \left(\frac{2}{L}\right)^{3/2}\sin\frac{n_x\pi x}{L}\sin\frac{n_y\pi y}{L}\sin\frac{n_z\pi z}{L}$$ *Normalization: $\left(\frac{2}{L}\right)^{3/2}$* #### Degeneracy Counting - **Given sum N = n²ₓ + n²ᵧ + n²_z:** - **All different (a, b, c):** $$\text{Degeneracy} = 3! = 6$$ - **Two equal (a, a, b):** $$\text{Degeneracy} = \frac{3!}{2!} = 3$$ - **All equal (a, a, a):** $$\text{Degeneracy} = \frac{3!}{3!} = 1$$ #### Specific Energy Levels | Sum | States | Degeneracy | |-----|--------|------------| | 3 | (1,1,1) | 1 | | 6 | (2,1,1) perms | 3 | | 9 | (2,2,1) perms | 3 | | 11 | (3,1,1) perms | 3 | | 12 | (2,2,2) | 1 | | 14 | (3,2,1) perms | 6 | *Not all sums are achievable! (Number theory constraint)* #### Rectangular Box (Lₓ ≠ Lᵧ ≠ L_z) $$E_{n_x,n_y,n_z} = \frac{\pi^2\hbar^2}{2m}\left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right)$$ *Degeneracy is lifted! (Symmetry breaking)* #### Volume Constraint Problems - **For two boxes with same volume:** - **Cubic:** $V = L^3$ - **Rectangular:** $V = L_x \cdot L_y \cdot L_z$ *If $L^3 = L_x \cdot L_y \cdot L_z$: Compare ground state energies using volume relation.* ### Uncertainty Relations #### Heisenberg Uncertainty Principle - **Position-Momentum:** $$\boxed{\Delta x \cdot \Delta p \geq \frac{\hbar}{2}}$$ *Equality: Gaussian wavepackets (minimum uncertainty)* - **Energy-Time:** $$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$ - **General Form:** $$\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle|$$ #### Calculating Uncertainties - **Definition:** $$\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle^2}$$ - **Position Uncertainty:** $$\Delta x = \sqrt{\int x^2|\psi|^2dx - \left(\int x|\psi|^2dx\right)^2}$$ - **Momentum Uncertainty (Position Space):** $$\langle p^2 \rangle = -\hbar^2\int \psi^*\frac{d^2\psi}{dx^2}dx$$ *Alternative:* $$\langle p^2 \rangle = \hbar^2\int\left|\frac{d\psi}{dx}\right|^2dx$$ #### Real Wavefunction Rule - **If ψ(x) is real:** $$\langle p \rangle = 0$$ *Therefore:* $$\Delta p = \sqrt{\langle p^2 \rangle}$$ #### Momentum Space - **Position Operator:** $$\hat{x} = i\hbar\frac{d}{dp}$$ - **Position Uncertainty from φ(p):** $$\langle x^2 \rangle = -\hbar^2\int \phi(p)\frac{d^2\phi}{dp^2}dp$$ #### Specific Results - **Infinite Square Well (ground state):** $$\Delta x \approx 0.18L, \quad \Delta p \approx \frac{\pi\hbar}{L}$$ $$\Delta x \cdot \Delta p \approx 0.57\hbar > \frac{\hbar}{2}$$ ✓ - **Delta Function Potential (ground state):** $$\Delta x \cdot \Delta p = \frac{\hbar}{\sqrt{2}} \approx 0.707\hbar$$ - **Gaussian Wavepacket:** $$\psi(x) = Ae^{-x^2/(2\sigma^2)}$$ $$\Delta x = \frac{\sigma}{\sqrt{2}}, \quad \Delta p = \frac{\hbar}{\sqrt{2}\sigma}$$ $$\Delta x \cdot \Delta p = \frac{\hbar}{2}$$ (Minimum!) #### Lorentzian in Momentum Space - **Given:** $\phi(p) = \frac{\alpha}{p^2 + \beta^2}$ - **Position uncertainty:** $$\boxed{\Delta x = \frac{\hbar}{\sqrt{2}\beta}}$$ *Fourier pair: Lorentzian ↔ Exponential* ### Universal Formulas #### Normalization $$\int_{-\infty}^{\infty}|\psi(x)|^2dx = 1$$ *For superposition:* $$\sum_n |c_n|^2 = 1$$ #### Probability $$P(x_1 ### Quick Reference Constants #### Physical Constants $$\hbar = 1.055 \times 10^{-34} \text{ J·s}$$ $$h = 2\pi\hbar = 6.626 \times 10^{-34} \text{ J·s}$$ $$m_e = 9.109 \times 10^{-31} \text{ kg}$$ $$m_p = 1.673 \times 10^{-27} \text{ kg}$$ #### Unit Conversions $$1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$$ $$1 \text{ Å} = 10^{-10} \text{ m}$$ $$\hbar c = 197.3 \text{ eV·nm}$$ #### Useful Relations $$E = \hbar\omega = \frac{hc}{\lambda} = \frac{p^2}{2m}$$ $$\lambda = \frac{h}{p}$$ (de Broglie) $$\omega = \frac{E}{\hbar}$$ $$k = \frac{p}{\hbar} = \frac{2\pi}{\lambda}$$ #### Common Integrals $$\int_0^L \sin^2\frac{n\pi x}{L}dx = \frac{L}{2}$$ $$\int_0^L \sin\frac{n\pi x}{L}\sin\frac{m\pi x}{L}dx = \frac{L}{2}\delta_{nm}$$ $$\int_{-\infty}^{\infty}e^{-ax^2}dx = \sqrt{\frac{\pi}{a}}$$ $$\int_{-\infty}^{\infty}x^2e^{-ax^2}dx = \frac{1}{2}\sqrt{\frac{\pi}{a^3}}$$ $$\int_{-\infty}^{\infty}\frac{dp}{p^2+\beta^2} = \frac{\pi}{\beta}$$ #### Trigonometric Identities $$\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$$ $$\cos^2\theta = \frac{1 + \cos(2\theta)}{2}$$ $$\sin^3\theta = \frac{3\sin\theta - \sin(3\theta)}{4}$$ $$e^{i\theta} = \cos\theta + i\sin\theta$$ ### Exam Quick Tips #### Recognition Patterns 1. **Real wavefunction** → ⟨p⟩ = 0 (instant!) 2. **Symmetric potential + superposition** → Use parity selection rules 3. **Equal coefficients** → Simple average: ⟨H⟩ = (ΣEₙ)/N 4. **Sequential measurement** → Use collapsed state 5. **Exponential wavefunction** → Δx·Δp = ℏ/√2 #### Dimensional Checks - **Energy:** ML²T⁻² - **Momentum:** MLT⁻¹ - **Action (ℏ):** ML²T⁻¹ - **Wavefunction (1D):** L⁻¹/² - **Wavefunction (3D):** L⁻³/² #### Common Factor Checks - Ground state ISW: $n = 1$ (not 0!) - Normalization ISW: $\sqrt{2/L}$ (not 1/L) - Energy ratio (1st/ground) in 3D cubic: 2 (not 3) - Uncertainty minimum: ℏ/2 (not ℏ) - Delta potential uncertainty: ℏ/√2