Quantum Mechanics

Cheatsheet Content

### Basic Postulates of Quantum Mechanics 1. **State Vector:** The state of a quantum mechanical system is completely specified by a state vector $|\Psi(t)\rangle$ in a complex Hilbert space. 2. **Observables:** To every classical observable $A$, there corresponds a linear, Hermitian operator $\hat{A}$ whose eigenvalues are the possible results of measurement. 3. **Measurement:** When a measurement of an observable $\hat{A}$ is made on a system in state $|\Psi\rangle$, the probability of obtaining eigenvalue $a_n$ (corresponding to eigenstate $|u_n\rangle$) is $P(a_n) = |\langle u_n | \Psi \rangle|^2$. After measurement, the system collapses to the eigenstate $|u_n\rangle$. 4. **Expectation Value:** The expectation value (average value) of an observable $\hat{A}$ in state $|\Psi\rangle$ is $\langle \hat{A} \rangle = \langle \Psi | \hat{A} | \Psi \rangle$. 5. **Time Evolution:** The time evolution of the state vector is governed by the Schrödinger equation: $$i\hbar \frac{\partial}{\partial t} |\Psi(t)\rangle = \hat{H} |\Psi(t)\rangle$$ where $\hat{H}$ is the Hamiltonian operator. ### Schrödinger Equation - **Time-Dependent Schrödinger Equation (TDSE):** $$i\hbar \frac{\partial}{\partial t} \Psi(\vec{r}, t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\vec{r}, t) \right) \Psi(\vec{r}, t)$$ - **Time-Independent Schrödinger Equation (TISE):** For conservative systems where $V(\vec{r}, t) = V(\vec{r})$, we can separate variables: $\Psi(\vec{r}, t) = \psi(\vec{r}) e^{-iEt/\hbar}$. $$-\frac{\hbar^2}{2m} \nabla^2 \psi(\vec{r}) + V(\vec{r}) \psi(\vec{r}) = E \psi(\vec{r})$$ - **Probability Current Density:** $$\vec{J} = \frac{\hbar}{2mi} (\Psi^* \nabla \Psi - \Psi \nabla \Psi^*)$$ Conservation of probability: $\frac{\partial P}{\partial t} + \nabla \cdot \vec{J} = 0$, where $P = |\Psi|^2$. ### Operators and Commutators - **Position Operator:** $\hat{x} = x$ - **Momentum Operator:** $\hat{p}_x = -i\hbar \frac{\partial}{\partial x}$ - **Angular Momentum Operator:** $\hat{L} = \hat{r} \times \hat{p}$ - $\hat{L}_x = -i\hbar (y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y})$ - $\hat{L}_y = -i\hbar (z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z})$ - $\hat{L}_z = -i\hbar (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x})$ - **Commutation Relations:** - $[\hat{x}, \hat{p}_x] = i\hbar$ - $[\hat{x}, \hat{p}_y] = 0$ - $[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z$ (cyclic permutations apply) - $[\hat{L}^2, \hat{L}_x] = [\hat{L}^2, \hat{L}_y] = [\hat{L}^2, \hat{L}_z] = 0$ - **Uncertainty Principle:** For any two observables $\hat{A}$ and $\hat{B}$: $$\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|$$ Specifically, $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ and $\Delta E \Delta t \ge \frac{\hbar}{2}$. ### Harmonic Oscillator (1D) - **Hamiltonian:** $\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2$ - **Energy Eigenvalues:** $E_n = \left( n + \frac{1}{2} \right) \hbar\omega$, for $n=0, 1, 2, ...$ - **Annihilation Operator:** $\hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega} \hat{p} \right)$ - **Creation Operator:** $\hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} - \frac{i}{m\omega} \hat{p} \right)$ - **Commutation Relation:** $[\hat{a}, \hat{a}^\dagger] = 1$ - **Number Operator:** $\hat{N} = \hat{a}^\dagger \hat{a}$, with $\hat{N}|n\rangle = n|n\rangle$. $\hat{H} = \hbar\omega (\hat{N} + 1/2)$. - **Ladder Operators:** - $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ - $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$ ### Hydrogen Atom - **Reduced Mass:** $\mu = \frac{m_e m_p}{m_e + m_p} \approx m_e$ - **Hamiltonian (spherical coordinates):** $$\hat{H} = -\frac{\hbar^2}{2\mu} \left( \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) - \frac{\hat{L}^2}{\hbar^2 r^2} \right) - \frac{Ze^2}{4\pi\epsilon_0 r}$$ - **Energy Eigenvalues:** $E_n = -\frac{Z^2 R_y}{n^2}$, where $R_y = \frac{\mu e^4}{8\epsilon_0^2 h^3 c} \approx 13.6 \text{ eV}$ (Rydberg constant), $n=1, 2, 3, ...$ (principal quantum number). Energy only depends on $n$. - **Quantum Numbers:** - $n$: principal quantum number ($1, 2, ...$) - $l$: orbital angular momentum quantum number ($0, 1, ..., n-1$) - $m_l$: magnetic quantum number ($-l, -l+1, ..., 0, ..., l-1, l$) - $m_s$: spin magnetic quantum number ($\pm 1/2$) - **Degeneracy:** $n^2$ without spin, $2n^2$ with spin included. - **Eigenfunctions:** $\Psi_{nlm_l}(r, \theta, \phi) = R_{nl}(r) Y_{lm_l}(\theta, \phi)$ - $Y_{lm_l}(\theta, \phi)$ are spherical harmonics. ### Angular Momentum and Spin - **Eigenvalues:** - $\hat{L}^2 |l, m_l\rangle = \hbar^2 l(l+1) |l, m_l\rangle$ - $\hat{L}_z |l, m_l\rangle = \hbar m_l |l, m_l\rangle$ - **Spin Angular Momentum:** - $\hat{S}^2 |s, m_s\rangle = \hbar^2 s(s+1) |s, m_s\rangle$ - $\hat{S}_z |s, m_s\rangle = \hbar m_s |s, m_s\rangle$ - For electron, $s=1/2$, $m_s = \pm 1/2$. - **Pauli Spin Matrices:** For $s=1/2$ systems: $$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ - $\hat{S} = \frac{\hbar}{2} \vec{\sigma}$ - Properties: $\sigma_i^2 = I$, $[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k$, $\{\sigma_i, \sigma_j\} = 2\delta_{ij}I$. - **Total Angular Momentum:** $\hat{J} = \hat{L} + \hat{S}$ - Eigenvalues: $\hat{J}^2 |j, m_j\rangle = \hbar^2 j(j+1) |j, m_j\rangle$ - $\hat{J}_z |j, m_j\rangle = \hbar m_j |j, m_j\rangle$ - Possible $j$ values: $|l-s|, |l-s|+1, ..., l+s$. ### Perturbation Theory - **Time-Independent Perturbation Theory (Non-Degenerate):** - $\hat{H} = \hat{H}_0 + \lambda \hat{H}'$, where $\hat{H}'$ is a small perturbation. - **First-order energy correction:** $E_n^{(1)} = \langle n^{(0)} | \hat{H}' | n^{(0)} \rangle$ - **First-order state correction:** $|n^{(1)}\rangle = \sum_{k \ne n} \frac{\langle k^{(0)} | \hat{H}' | n^{(0)} \rangle}{E_n^{(0)} - E_k^{(0)}} |k^{(0)}\rangle$ - **Time-Dependent Perturbation Theory:** - Transition probability from state $i$ to $f$ under a time-dependent perturbation $\hat{H}'(t)$. - **Fermi's Golden Rule:** For a constant perturbation applied from $t=0$ to $t$: $$P_{i \to f} = \frac{2\pi}{\hbar} |\langle f | \hat{H}' | i \rangle|^2 \rho(E_f) t$$ where $\rho(E_f)$ is the density of final states. - **Transition Rate:** $W_{i \to f} = \frac{2\pi}{\hbar} |\langle f | \hat{H}' | i \rangle|^2 \rho(E_f)$ ### Variational Principle - For any normalized trial wave function $|\tilde{\Psi}\rangle$, the expectation value of the Hamiltonian is an upper bound to the true ground state energy $E_0$: $$E_0 \le \langle \tilde{\Psi} | \hat{H} | \tilde{\Psi} \rangle = \frac{\langle \tilde{\Psi} | \hat{H} | \tilde{\Psi} \rangle}{\langle \tilde{\Psi} | \tilde{\Psi} \rangle}$$ - This principle is used to estimate the ground state energy by minimizing the expectation value with respect to parameters in the trial wave function. ### WKB Approximation - **Condition for validity:** The potential $V(x)$ must vary slowly over a de Broglie wavelength: $\frac{\lambda}{2\pi} \left| \frac{dp}{dx} \right| \ll |p|$. - **Wave function in allowed region ($E > V(x)$):** $$\psi(x) \approx \frac{C}{\sqrt{p(x)}} \exp\left( \pm \frac{i}{\hbar} \int p(x) dx \right)$$ where $p(x) = \sqrt{2m(E-V(x))}$. - **Wave function in forbidden region ($E ### Scattering Theory - **Differential Cross-Section:** $d\sigma/d\Omega = |f(\theta, \phi)|^2$, where $f(\theta, \phi)$ is the scattering amplitude. - **Total Cross-Section:** $\sigma = \int |f(\theta, \phi)|^2 d\Omega$ - **Born Approximation:** For a weak potential $V(\vec{r})$: $$f(\vec{q}) = -\frac{2m}{4\pi\hbar^2} \int V(\vec{r}') e^{-i\vec{q} \cdot \vec{r}'} d^3 r'$$ where $\vec{q} = \vec{k}_f - \vec{k}_i$ is the momentum transfer. - **Partial Wave Analysis:** For spherically symmetric potentials. - $f(\theta) = \frac{1}{k} \sum_{l=0}^\infty (2l+1) e^{i\delta_l} \sin\delta_l P_l(\cos\theta)$ - Phase shift $\delta_l$ depends on the potential and energy. - Total cross-section: $\sigma = \frac{4\pi}{k^2} \sum_{l=0}^\infty (2l+1) \sin^2\delta_l$ ### Identical Particles - **Symmetric Wave Function:** For bosons, $\Psi(1,2) = \Psi(2,1)$. - **Anti-symmetric Wave Function:** For fermions, $\Psi(1,2) = -\Psi(2,1)$. - **Pauli Exclusion Principle:** No two identical fermions can occupy the same quantum state (implies anti-symmetric wave function). - **Exchange Interaction:** Arises due to the symmetry/antisymmetry requirement for identical particles, leading to effective "forces" even without direct potential interaction. ### Relativistic Quantum Mechanics - **Klein-Gordon Equation:** Describes spin-0 particles. $$\frac{1}{c^2} \frac{\partial^2}{\partial t^2} \Psi - \nabla^2 \Psi + \left( \frac{mc}{\hbar} \right)^2 \Psi = 0$$ - Problem: Negative energy solutions, negative probability density. - **Dirac Equation:** Describes spin-1/2 particles (e.g., electrons). $$i\hbar \frac{\partial \Psi}{\partial t} = \left( c\vec{\alpha} \cdot \hat{p} + \beta mc^2 \right) \Psi$$ - $\Psi$ is a 4-component spinor. - $\vec{\alpha}$ and $\beta$ are $4 \times 4$ Dirac matrices. - Successfully predicts spin, antimatter, and accurate fine structure of hydrogen. ### Useful Constants - **Planck's constant:** $\hbar = 1.054 \times 10^{-34} \text{ J s}$ - **Electron mass:** $m_e = 9.109 \times 10^{-31} \text{ kg}$ - **Speed of light:** $c = 3 \times 10^8 \text{ m/s}$ - **Elementary charge:** $e = 1.602 \times 10^{-19} \text{ C}$ - **Boltzmann constant:** $k_B = 1.38 \times 10^{-23} \text{ J/K}$ - **Vacuum permittivity:** $\epsilon_0 = 8.854 \times 10^{-12} \text{ F/m}$ - **1 eV:** $1.602 \times 10^{-19} \text{ J}$