Quantum Theory Cheatsheet

Cheatsheet Content

1. Introduction to Quantum Mechanics Quantum Mechanics (QM): Branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant. Classical Mechanics vs. QM: Classical: Deterministic, continuous, macroscopic. Quantum: Probabilistic, quantized, microscopic. Key Concepts: Quantization, wave-particle duality, uncertainty principle, superposition, entanglement, tunneling. 2. Fundamental Constants Planck's Constant: $h \approx 6.626 \times 10^{-34} \text{ J} \cdot \text{s}$ Reduced Planck's Constant: $\hbar = \frac{h}{2\pi} \approx 1.054 \times 10^{-34} \text{ J} \cdot \text{s}$ Speed of Light: $c \approx 2.998 \times 10^8 \text{ m/s}$ Electron Charge: $e \approx 1.602 \times 10^{-19} \text{ C}$ Electron Mass: $m_e \approx 9.109 \times 10^{-31} \text{ kg}$ 3. Wave-Particle Duality De Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$ Photon Energy: $E = hf = \frac{hc}{\lambda}$ Momentum of a Photon: $p = \frac{E}{c} = \frac{h}{\lambda}$ 4. The Wave Function ($\Psi$) Description: A complex-valued probability amplitude for a particle's quantum state. Probability Density: $|\Psi(\mathbf{r},t)|^2$ is the probability density of finding the particle at $\mathbf{r}$ at time $t$. $\int_{\text{all space}} |\Psi(\mathbf{r},t)|^2 dV = 1$ (Normalization) Superposition Principle: If $\Psi_1$ and $\Psi_2$ are valid wave functions, then $c_1\Psi_1 + c_2\Psi_2$ is also a valid wave function. 5. Schrödinger Equation Time-Dependent Schrödinger Equation (TDSE) $i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \hat{H} \Psi(\mathbf{r},t)$ For a single particle, $\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r},t)$. Time-Independent Schrödinger Equation (TISE) For systems with a time-independent potential $V(\mathbf{r})$. $\hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r})$ Solutions are stationary states: $\Psi(\mathbf{r},t) = \psi(\mathbf{r})e^{-iEt/\hbar}$. Hamiltonian Operator: $\hat{H} = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) + V(x,y,z)$ 6. Operators and Observables Observable: A measurable physical quantity. Operator: A mathematical entity associated with an observable. Position: $\hat{x} = x$ Momentum: $\hat{p}_x = -i\hbar \frac{\partial}{\partial x}$ Kinetic Energy: $\hat{T} = -\frac{\hbar^2}{2m} \nabla^2$ Potential Energy: $\hat{V} = V(\mathbf{r})$ Eigenvalue Equation: $\hat{A} \psi = a \psi$. Measurable values are eigenvalues $a$. Expectation Value: $\langle \hat{A} \rangle = \int \Psi^* \hat{A} \Psi dV$. Commutator: $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$. If $[\hat{A}, \hat{B}] = 0$, observables $A$ and $B$ can be simultaneously measured. Commutation Relations: $[\hat{x}, \hat{p}_x] = i\hbar$ $[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z$ 7. Heisenberg Uncertainty Principle For any two observables $A$ and $B$: $\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|$. Position-Momentum: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$. Energy-Time: $\Delta E \Delta t \ge \frac{\hbar}{2}$. 8. Quantum Systems & Solutions Particle in a 1D Box (Infinite Square Well) $V(x) = 0$ for $0 Energy Levels: $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$, for $n=1, 2, 3, \dots$ Wave Functions: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$ Energy is quantized, lowest energy is not zero ($E_1 > 0$). Quantum Harmonic Oscillator (1D) $V(x) = \frac{1}{2} m\omega^2 x^2$. Energy Levels: $E_n = \left(n + \frac{1}{2}\right) \hbar\omega$, for $n=0, 1, 2, \dots$ Zero-Point Energy: $E_0 = \frac{1}{2}\hbar\omega$. Energy levels are equally spaced ($ \hbar\omega $). Quantum Tunneling Probability for a particle to pass through a potential barrier even if its kinetic energy is less than the barrier height. Transmission Probability (approx. for square barrier): $T \approx e^{-2 \kappa L}$ where $\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$. 9. Angular Momentum Orbital Angular Momentum Operator: $\hat{\mathbf{L}} = \hat{\mathbf{R}} \times \hat{\mathbf{P}}$. Eigenvalues of $\hat{L}^2$: $\hbar^2 l(l+1)$, where $l = 0, 1, 2, \dots$ (orbital quantum number). Eigenvalues of $\hat{L}_z$: $\hbar m_l$, where $m_l = -l, -l+1, \dots, l$ (magnetic quantum number). Commutation: $[\hat{L}^2, \hat{L}_z] = 0$. Can measure $L^2$ and $L_z$ simultaneously. 10. Spin Angular Momentum Intrinsic angular momentum of a particle. Not due to spatial motion. Spin Quantum Number ($s$): For electrons, $s = \frac{1}{2}$. Spin Magnetic Quantum Number ($m_s$): $m_s = \pm s = \pm \frac{1}{2}$. Eigenvalues of $\hat{S}^2$: $\hbar^2 s(s+1)$. Eigenvalues of $\hat{S}_z$: $\hbar m_s$. 11. Pauli Exclusion Principle No two identical fermions (half-integer spin particles, like electrons) can occupy the same quantum state simultaneously. Each electron in an atom must have a unique set of quantum numbers ($n, l, m_l, m_s$). 12. Quantum Numbers (for Hydrogen Atom) Principal Quantum Number ($n$): $1, 2, 3, \dots$ (Determines energy level $E_n = -\frac{13.6 \text{ eV}}{n^2}$ and shell). Orbital Angular Momentum Quantum Number ($l$): $0, 1, \dots, n-1$ (Determines orbital shape and subshell: s, p, d, f). Magnetic Quantum Number ($m_l$): $-l, \dots, 0, \dots, l$ (Determines orbital orientation in space). Spin Quantum Number ($m_s$): $\pm \frac{1}{2}$ (Determines intrinsic spin orientation). 13. Dirac Notation (Bra-Ket Notation) Ket: $|\psi\rangle$ represents a quantum state vector. Bra: $\langle\psi|$ represents the dual vector (conjugate transpose of ket). Inner Product (Probability Amplitude): $\langle\phi|\psi\rangle$. If $\langle\phi|\psi\rangle = 0$, states are orthogonal. If $\langle\psi|\psi\rangle = 1$, state is normalized. Outer Product (Operator): $|\phi\rangle\langle\psi|$. Expectation Value: $\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle$. 14. Measurement in QM When an observable $\hat{A}$ is measured, the system collapses to an eigenstate of $\hat{A}$. The outcome of the measurement will be one of the eigenvalues of $\hat{A}$. The probability of measuring eigenvalue $a_n$ for a state $|\psi\rangle$ is $|c_n|^2$, where $|\psi\rangle = \sum_n c_n |a_n\rangle$.