Quantum Mechanics Cheatsheet

Cheatsheet Content

1. Fundamental Concepts Wave Function ($\Psi$): Describes the quantum state of a particle. For a single particle in 1D, $\Psi(x,t)$. Born's Rule: The probability density of finding a particle at position $x$ at time $t$ is $|\Psi(x,t)|^2$. Normalization: $\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$ (for 1D). Operators: Physical observables are represented by Hermitian operators. Position: $\hat{X} = x$ Momentum: $\hat{P} = -i\hbar \frac{\partial}{\partial x}$ Energy (Hamiltonian): $\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$ Commutation Relation: $[\hat{X}, \hat{P}] = \hat{X}\hat{P} - \hat{P}\hat{X} = i\hbar$. 2. Schrödinger Equation Time-Dependent Schrödinger Equation (TDSE) Describes how the wave function evolves over time: $$i\hbar \frac{\partial}{\partial t} \Psi(\vec{r},t) = \hat{H} \Psi(\vec{r},t)$$ Time-Independent Schrödinger Equation (TISE) For systems with a time-independent potential $V(\vec{r})$, solutions are of the form $\Psi(\vec{r},t) = \psi(\vec{r})e^{-iEt/\hbar}$. $$\hat{H} \psi(\vec{r}) = E \psi(\vec{r})$$ Where $E$ are the allowed energy eigenvalues and $\psi(\vec{r})$ are the corresponding stationary states (eigenfunctions). 3. Postulates of Quantum Mechanics The state of a quantum system is completely described by its wave function $\Psi$. To every measurable physical quantity (observable) there corresponds a linear, Hermitian operator. The only possible result of a measurement of an observable $A$ are the eigenvalues $a_n$ of the corresponding operator $\hat{A}$. When a measurement of $A$ is made on a system in state $\Psi$, the probability of obtaining an eigenvalue $a_n$ is given by $P(a_n) = |\langle \phi_n | \Psi \rangle|^2$, where $\phi_n$ are the eigenfunctions corresponding to $a_n$. Immediately after a measurement of $A$ yields $a_n$, the state of the system collapses to the corresponding eigenstate $\phi_n$. The time evolution of the state vector is governed by the TDSE. 4. Expectation Values The average value of an observable $A$ if many measurements are made on identical systems: $$\langle \hat{A} \rangle = \int \Psi^*(\vec{r},t) \hat{A} \Psi(\vec{r},t) d^3r$$ For an operator $\hat{A}$ and eigenstate $\phi_n$, $\langle \hat{A} \rangle = a_n$. 5. Uncertainty Principle Heisenberg Uncertainty Principle: $$\Delta x \Delta p_x \ge \frac{\hbar}{2}$$ Energy-Time Uncertainty Principle: $$\Delta E \Delta t \ge \frac{\hbar}{2}$$ 6. Important Systems & Solutions Particle in a 1D Box (Infinite Potential Well) Potential: $V(x) = 0$ for $0 Energy Levels: $E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}$, for $n=1, 2, 3, \ldots$ Wave Functions: $\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right)$ Harmonic Oscillator (1D) Potential: $V(x) = \frac{1}{2}m\omega^2 x^2$ Energy Levels: $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$, for $n=0, 1, 2, \ldots$ (zero-point energy!) Ground State Wave Function ($n=0$): $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega}{2\hbar}x^2}$ Hydrogen Atom (Briefly) Quantum Numbers: Principal ($n=1,2,3,\ldots$): Determines energy. Angular Momentum ($l=0,1,\ldots,n-1$): Determines shape of orbital. Magnetic ($m_l=-l,\ldots,l$): Determines orientation. Spin ($m_s=\pm 1/2$): Intrinsic angular momentum. Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ 7. Angular Momentum Orbital Angular Momentum Operator: $\hat{\vec{L}} = \hat{\vec{r}} \times \hat{\vec{p}}$ $\hat{L}_z = -i\hbar \frac{\partial}{\partial\phi}$ Eigenvalues of $\hat{L}^2$: $\hbar^2 l(l+1)$ Eigenvalues of $\hat{L}_z$: $\hbar m_l$ Spin Angular Momentum Operator: $\hat{\vec{S}}$ Eigenvalues of $\hat{S}^2$: $\hbar^2 s(s+1)$ Eigenvalues of $\hat{S}_z$: $\hbar m_s$ 8. Perturbation Theory (Time-Independent) Used to find approximate solutions for systems where $H = H_0 + H'$, where $H_0$ is solvable and $H'$ is a small perturbation. First-Order Energy Correction $$E_n^{(1)} = \langle \psi_n^{(0)} | H' | \psi_n^{(0)} \rangle$$ First-Order Wave Function Correction $$\psi_n^{(1)} = \sum_{k \ne n} \frac{\langle \psi_k^{(0)} | H' | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_k^{(0)}} \psi_k^{(0)}$$ 9. Constants & Units Constant Symbol Value (SI units) Reduced Planck Constant $\hbar$ $1.054 \times 10^{-34} \text{ J s}$ Planck Constant $h$ $6.626 \times 10^{-34} \text{ J s}$ Electron Mass $m_e$ $9.109 \times 10^{-31} \text{ kg}$ Elementary Charge $e$ $1.602 \times 10^{-19} \text{ C}$ Speed of Light $c$ $2.998 \times 10^8 \text{ m/s}$ Useful Conversions: $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$ $\hbar c \approx 197 \text{ eV nm}$