Quantum mechanics Study Guide

📖 Core Concepts Wave function $\psi(\mathbf r,t)$ – a complex‑valued function whose squared magnitude $|\psi|^{2}$ gives the probability density for finding a particle (Born rule). Hamiltonian $\hat H$ – the energy operator; generates time evolution via the Schrödinger equation. Schrödinger equation $$i\hbar\frac{\partial}{\partial t}\psi(\mathbf r,t)=\hat H\psi(\mathbf r,t)$$ Deterministic evolution of the state vector. Observable ↔ Hermitian operator – measurable quantities are represented by self‑adjoint operators; eigenvalues are the possible measurement results. Eigenstate / eigenvalue – $\hat O|\phi\rangle = o|\phi\rangle$; a system in $|\phi\rangle$ yields the definite value $o$ on measuring $\hat O$. Superposition – any linear combination $\sum ci|\phii\rangle$ of eigenstates is also a valid state; measurement probabilities are $|ci|^{2}$. Entanglement – states of composite systems that cannot be written as a simple product $|\psiA\rangle\otimes|\phiB\rangle$; subsystems have no independent pure state. Uncertainty principle – non‑commuting observables $[\hat A,\hat B]\neq0$ satisfy $\sigmaA\sigmaB\ge\frac{1}{2}| \langle[\hat A,\hat B]\rangle|$; for $x$ and $p$: $\sigmax\sigmap\ge\frac{\hbar}{2}$. Measurement postulate (collapse) – a measurement projects the state onto the eigenstate associated with the observed eigenvalue, with probability given by the Born rule. Unitary time‑evolution operator $\hat U(t)=e^{-i\hat H t/\hbar}$ – preserves norm and encodes deterministic dynamics. --- 📌 Must Remember Planck‑Einstein relation: $E=h\nu=\hbar\omega$. De Broglie wavelength: $\lambda=\frac{h}{p}$. Uncertainty: $\sigmax\sigmap\ge\frac{\hbar}{2}$. Particle‑in‑a‑box: $kn=n\pi/L$, $En=\frac{\hbar^{2}kn^{2}}{2m}= \frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}$, $n=1,2,\dots$ $\psin(x)=\sqrt{\frac{2}{L}}\sin(kn x)$. Harmonic oscillator: $En=\hbar\omega\left(n+\tfrac12\right)$, $n=0,1,2,\dots$; eigenfunctions involve Hermite polynomials $Hn(\alpha x)$. Commutation relation: $[\hat x,\hat p]=i\hbar$. Normalization: $\langle\psi|\psi\rangle = 1$ (global phase $e^{i\theta}$ is irrelevant). Conserved observable: $[\hat O,\hat H]=0\;\Rightarrow\;$ $\langle\hat O\rangle$ is constant in time. --- 🔄 Key Processes Solving the stationary Schrödinger equation $\hat H\psi=E\psi$: Identify the potential $V(x)$. Write the differential equation, apply boundary conditions (e.g., $\psi=0$ at infinite walls). Quantize $k$ (or other quantum numbers) → discrete $E$. Time evolution of a known state $|\psi(0)\rangle$: Compute $\hat U(t)=e^{-i\hat H t/\hbar}$. Apply: $|\psi(t)\rangle=\hat U(t)|\psi(0)\rangle$. Measurement of observable $\hat O$: Expand $|\psi\rangle=\sumi ci|oi\rangle$. Probability of outcome $oi$ = $|ci|^{2}$. After measurement, state collapses to $|oi\rangle$. Constructing a composite system $A\otimes B$: Form tensor product $\mathcal HA\otimes\mathcal HB$. Identify product vs entangled states. Checking conservation: compute $[\hat O,\hat H]$. Zero → conserved; non‑zero → expect time dependence and uncertainty relation. --- 🔍 Key Comparisons Wave vs Particle – interference pattern (wave) vs discrete detections (particle) in the double‑slit experiment. Product state vs Entangled state – $|\psiA\rangle\otimes|\phiB\rangle$ (independent) vs $|\Phi\rangle\neq|\psiA\rangle\otimes|\phiB\rangle$ (correlated outcomes). Schrödinger picture vs Heisenberg picture – State evolves in time vs operators evolve; physical predictions identical. Local hidden‑variable theory vs Quantum mechanics – Predicts Bell‑inequality obeying statistics vs experimentally violated correlations. Copenhagen vs Bohmian vs Many‑worlds – Collapse postulate vs deterministic pilot wave vs branching universal wavefunction. --- ⚠️ Common Misunderstandings Wavefunction is a physical wave – it encodes probability amplitudes, not a literal oscillating medium. Uncertainty = measurement error – it reflects intrinsic spread of complementary observables, not instrument precision. Tunnelling means the particle “passes through” the barrier – mathematically it is the exponentially decaying tail of the wavefunction giving non‑zero transmission probability. Entanglement enables faster‑than‑light communication – the no‑communication theorem forbids using entanglement to transmit information. Global phase matters – overall factor $e^{i\theta}$ leaves all physical predictions unchanged. --- 🧠 Mental Models / Intuition Amplitude addition – Think of each possible path as a vector in the complex plane; the total probability comes from the length of the sum (interference). Phase‑space cell – The product $\sigmax\sigmap\sim\hbar/2$ is the minimal “area” a quantum state can occupy; squeezing one variable inflates the other. Barrier as a “leaky wall” – The wavefunction decays inside the barrier like an evanescent wave; the leak probability is the tunnelling rate. Entanglement as a shared script – Two particles follow a correlated script; measuring one reveals the script’s line for the other, without sending a signal. --- 🚩 Exceptions & Edge Cases Degenerate eigenvalues – Collapse projects onto the entire eigenspace, not a single vector. Commuting observables – Simultaneous eigenstates exist; uncertainties can both be zero. Large quantum numbers – Classical limit (correspondence principle) restores Newtonian trajectories. Planck length in LQG – Lengths $<lP$ have no physical meaning; geometry becomes discrete. --- 📍 When to Use Which Bound‑state problems – Use stationary Schrödinger equation with boundary conditions (particle in a box, harmonic oscillator). Scattering / tunnelling – Apply time‑independent equation in regions of constant potential; match wavefunction and derivative at boundaries. Composite systems – Build tensor‑product Hilbert space; use entanglement criteria (e.g., cannot factorize). Estimating measurement limits – Invoke uncertainty relation for $x$–$p$, $E$–$t$, etc. Choosing representation – Momentum space for free particles; position space for potentials; operator algebra (Heisenberg) when symmetry simplifies calculations. --- 👀 Patterns to Recognize Quantization ↔ boundary conditions – Infinite walls → sinusoidal standing waves; periodicity → discrete $k$. Exponential decay inside a barrier – Signals tunnelling; the decay constant $\kappa=\sqrt{2m(V0-E)}/\hbar$. Interference disappearance – Any which‑path information (detector) destroys coherence → no fringes. Commutator ≠ 0 → uncertainty – Whenever $[\hat A,\hat B]\neq0$, expect a non‑zero lower bound on $\sigmaA\sigmaB$. Conserved quantity ↔ symmetry – If Hamiltonian is invariant under translation → momentum conserved; under rotation → angular momentum conserved. --- 🗂️ Exam Traps Confusing $E=h\nu$ with $E=\hbar\omega$ – Remember $\omega=2\pi\nu$, so $E=\hbar\omega$ is the same as $E=h\nu$. Assuming $|\psi|^{2}= \psi$ – The probability density is the square of the absolute value, not the square of the function itself. Mixing up $k$ and $p$ – $p=\hbar k$; $k$ appears in wavefunctions, $p$ in observables. Thinking normalization is optional – Unnormalized states give wrong probabilities; always enforce $\langle\psi|\psi\rangle=1$. Believing a global phase changes outcomes – Any $e^{i\theta}$ factor cancels in $|\psi|^{2}$ and expectation values. Choosing product state when entanglement is required – In Bell‑test questions, the correct answer will involve a non‑factorizable state. Applying uncertainty principle as a statement about measurement disturbance – The principle is about intrinsic spreads, not about how hard we try to measure. ---