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Photons & early quantum, Dirac formalism, spin & angular momentum, time evolution, entanglement, wave mechanics, the oscillator, path integrals, hydrogen, perturbation theory, identical particles, scattering, quantized radiation.

photons & early quantum · states & operators · spin-1/2 & two-state systems · angular momentum · time evolution · composite systems & entanglement · wave mechanics in 1D · harmonic oscillator · path integrals · central potentials & hydrogen · perturbation theory · identical particles · scattering · photons & atoms

Photons & early quantum

Photon. Energy \(E=hf=hc/\lambda\); momentum \(p=h/\lambda=E/c\).

Atomic transitions. \(\Delta E=hf=hc/\lambda\); emission if \(E_i\gt E_f\), absorption if \(E_f\gt E_i\).

Hydrogen spectra. \(\dfrac1\lambda=R_H\left(\dfrac1{n_f^2}-\dfrac1{n_i^2}\right)\); ionization energy from level \(n\): \(13.6/n^2\ \mathrm{eV}\). Levels \(E_n=-13.6\,\mathrm{eV}/n^2\) — derived under central potentials.

Photoelectric effect. \(hf=\Phi+K_{\max}\), \(K_{\max}=eV_s\); threshold \(f_0=\Phi/h\), \(\lambda_0=hc/\Phi\). Intensity controls photon rate/current; frequency controls photon energy and stopping potential.

Matter waves. de Broglie \(\lambda=h/p\); nonrelativistic \(p=mv\), \(K=p^2/(2m)\), \(\lambda=h/\sqrt{2mK}\). Electron through voltage \(V\): \(\lambda=h/\sqrt{2meV}\) nonrel.

Nuclear notation. \(^{A}_{Z}X\): \(Z\) protons, \(N=A-Z\) neutrons, charge \(+Ze\). Isotopes: same \(Z\), different \(N\); ions have electron number \(\ne Z\). Conservation in reactions: nucleon number \(A\), charge \(Z\), energy, momentum, lepton number.

Rutherford. Alpha scattering by small positive nucleus; closest approach head-on: \(K=k_e(2e)(Ze)/r_{\min}\).

Bohr. Angular momentum \(mvr=n\hbar\); \(r_n=a_0n^2\) for H. Historical stepping stone — quantized orbits reproduce the H levels but not the formalism that replaces them.

States & operators

State vector. A physical state is a ket \(|\psi\rangle\) in a complex vector space, normalized \(\langle\psi|\psi\rangle=1\); overall phase \(e^{i\delta}|\psi\rangle\) is unobservable (relative phases are everything).

Amplitudes. In an orthonormal basis \(\{|a_i\rangle\}\): \(|\psi\rangle=\sum_i c_i|a_i\rangle\), \(c_i=\langle a_i|\psi\rangle\) the probability amplitude; \(P(a_i)=|\langle a_i|\psi\rangle|^2\). The bra \(\langle\psi|\) is the adjoint; \(\langle\varphi|\psi\rangle=\langle\psi|\varphi\rangle^*\).

Measurement. Measuring \(A\) yields an eigenvalue \(a_i\) of \(\hat A\); the state collapses to the corresponding eigenstate (projection), so an immediate remeasurement repeats \(a_i\). Amplitudes interfere when paths are indistinguishable; probabilities add when they are distinguished.

Observables. Observables ↔ Hermitian operators \(\hat A=\hat A^\dagger\): real eigenvalues, orthogonal eigenstates, complete eigenbasis. \(\avg{A}=\langle\psi|\hat A|\psi\rangle=\sum_i a_i|c_i|^2\); \((\Delta A)^2=\avg{A^2}-\avg{A}^2\).

Projection & completeness. \(\hat P_i=|a_i\rangle\langle a_i|\), \(\hat P_i^2=\hat P_i\); completeness \(\sum_i|a_i\rangle\langle a_i|=\hat 1\) — insert freely to change basis.

Matrix mechanics. \(A_{ij}=\langle a_i|\hat A|a_j\rangle\); an operator is diagonal in its own eigenbasis with eigenvalues on the diagonal. Adjoint \((\hat A\hat B)^\dagger=\hat B^\dagger\hat A^\dagger\); Hermitian matrix = its conjugate transpose. Basis change is a unitary matrix of overlaps \(\langle b_i|a_j\rangle\).

Unitary operators. Symmetry operations (rotation, translation, time evolution) preserve norms: \(\hat U^\dagger\hat U=\hat 1\), \(\hat U=e^{-i\theta\hat G/\hbar}\) with Hermitian generator \(\hat G\) — angular momentum generates rotations, momentum translations, the Hamiltonian time evolution.

Commutators. \([\hat A,\hat B]=\hat A\hat B-\hat B\hat A\). Commuting observables are compatible: simultaneous eigenbasis, measurements don't disturb each other. Noncommuting observables obey (via the Schwarz inequality) \[\Delta A\,\Delta B\ge\tfrac12\bigl|\avg{[\hat A,\hat B]}\bigr|\]

Spin-1/2 & two-state systems

Stern-Gerlach. Inhomogeneous field: force \(F_z=\mu_z\,\partial B_z/\partial z\) sorts by \(\mu_z\). A silver beam splits into two — \(S_z=\pm\hbar/2\) only; generally \(2j+1\) beams. Sequential SG: measuring \(S_x\) erases prior \(S_z\) information — \(S_x,S_z\) incompatible.

Spin-1/2 states. \(|{\pm}x\rangle=\frac1{\sqrt2}|{+}z\rangle\pm\frac1{\sqrt2}|{-}z\rangle\); \(|{\pm}y\rangle=\frac1{\sqrt2}|{+}z\rangle\pm\frac{i}{\sqrt2}|{-}z\rangle\). The \(i\) is forced by experiment: no real amplitudes reproduce the SG results — quantum mechanics needs complex numbers.

Spin along \(\vv{n}(\theta,\phi)\). \(|{+}\vv{n}\rangle=\cos\frac\theta2\,|{+}z\rangle+e^{i\phi}\sin\frac\theta2\,|{-}z\rangle\) — eigenstate of \(\hat{\vv{S}}\cdot\vv{n}\) with eigenvalue \(+\hbar/2\).

Pauli matrices. \(\hat{\vv{S}}=\frac\hbar2\boldsymbol\sigma\) in the \(S_z\) basis: \(\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}\), \(\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}\), \(\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\); \(\sigma_i^2=1\), \([\sigma_i,\sigma_j]=2i\varepsilon_{ijk}\sigma_k\), \(\{\sigma_i,\sigma_j\}=2\delta_{ij}\).

Rotations. \(\hat R(\phi\uvec{k})=e^{-i\hat J_z\phi/\hbar}\) rotates states counterclockwise about \(z\). A spin-1/2 state picks up \(-1\) under a \(2\pi\) rotation (\(+1\) only after \(4\pi\)) — observed in neutron interferometry.

Magnetic moment. \(\boldsymbol\mu=\dfrac{gq}{2m}\vv{S}\), \(H=-\boldsymbol\mu\cdot\vv{B}\); electron \(g\approx2\).

Photon polarization. Another two-state system: linear \(|x\rangle,|y\rangle\); circular \(|R\rangle,|L\rangle=\frac1{\sqrt2}(|x\rangle\pm i|y\rangle)\) are rotation eigenstates, \(\hat J_z|R\rangle=\hbar|R\rangle\), \(\hat J_z|L\rangle=-\hbar|L\rangle\) (helicity \(\pm1\)). The photon is spin-1 with the \(m=0\) state absent — a signature of masslessness.

Angular momentum

Algebra. Finite rotations about different axes do not commute, so neither do their generators: \[[J_x,J_y]=i\hbar J_z\ \text{(and cyclic)}\] \(J^2=J_x^2+J_y^2+J_z^2\) commutes with each component — diagonalize \(J^2\) and \(J_z\) together.

Eigenvalues. \(J^2|j,m\rangle=j(j+1)\hbar^2|j,m\rangle\), \(J_z|j,m\rangle=m\hbar|j,m\rangle\) with \(j=0,\tfrac12,1,\tfrac32,\dots\) and \(m=j,j-1,\dots,-j\) (\(2j+1\) values) — derived purely from the commutation relations via ladder operators.

Ladder operators. \(J_\pm=J_x\pm iJ_y\); \([J_z,J_\pm]=\pm\hbar J_\pm\); \(J_\pm|j,m\rangle=\sqrt{j(j+1)-m(m\pm1)}\,\hbar\,|j,m\pm1\rangle\) — gives all matrix elements of \(J_x,J_y\).

Matrix representations. Each \(j\) gives a \((2j+1)\)-dimensional block. Eigenstates of \(\hat{\vv{J}}\cdot\vv{n}\) come from diagonalizing the matrix \(J_xn_x+J_yn_y+J_zn_z\); for any axis and any \(j\) the eigenvalues are \(j\hbar,\dots,-j\hbar\) (spin-1: \(\hbar,0,-\hbar\), three SG beams).

Uncertainty. \(\Delta J_x\,\Delta J_y\ge\frac\hbar2|\avg{J_z}|\): no state (except \(j=0\)) is simultaneously sharp in two components — the angular momentum never simply "points" along an axis; \(|m|\le j\lt\sqrt{j(j+1)}\).

Time evolution

Evolution operator. \(|\psi(t)\rangle=\hat U(t)|\psi(0)\rangle\); probability conservation forces \(\hat U\) unitary and its generator — the Hamiltonian, the energy operator — Hermitian. Time-independent \(H\): \(\hat U(t)=e^{-iHt/\hbar}\).

Schrödinger equation. \[i\hbar\frac{\dd}{\dd t}|\psi(t)\rangle=H|\psi(t)\rangle\]

Stationary states. \(H|E_n\rangle=E_n|E_n\rangle\Rightarrow|\psi(t)\rangle=\sum_n c_ne^{-iE_nt/\hbar}|E_n\rangle\). An energy eigenstate evolves only by phase — nothing observable changes; a superposition of two levels oscillates at \(\omega=(E_2-E_1)/\hbar\). To evolve: expand in energy eigenstates, attach phases, re-project.

Expectation values. \[\frac{\dd\avg{A}}{\dd t}=\frac{i}{\hbar}\avg{[H,\hat A]}+\avg{\partial\hat A/\partial t}\] \([\hat A,H]=0\Rightarrow A\) conserved (symmetries ↔ conservation laws). Applied to \(\hat x,\hat p\): \(\dd\avg{x}/\dd t=\avg{p}/m\), \(\dd\avg{p}/\dd t=-\avg{V'(x)}\) — expectation values follow classical equations (Ehrenfest).

Spin precession. \(H=-\boldsymbol\mu\cdot\vv{B}=-\gamma B\hat S_z\) (\(\gamma=gq/2m\)): \(\avg{\vv{S}}\) precesses about \(\vv{B}\) at the Larmor frequency \(\omega_0=\gamma B\) — the evolution operator is a rotation operator about \(\vv{B}\).

Magnetic resonance. Add a transverse field rotating at \(\omega\) (strength \(\omega_1=\gamma B_1\)); spin-flip probability (Rabi's formula) \[P_{+\to-}(t)=\frac{\omega_1^2}{\omega_1^2+(\omega_0-\omega)^2}\,\sin^2\!\left(\frac{\sqrt{\omega_1^2+(\omega_0-\omega)^2}}{2}\,t\right)\] Complete flips only on resonance \(\omega=\omega_0\) — measure \(\omega_0\) sharply (NMR, MRI).

Ammonia molecule. N-above/below-plane states \(|1\rangle,|2\rangle\) tunnel-coupled: \(\langle1|H|2\rangle=-A\). Eigenstates \(\frac1{\sqrt2}(|1\rangle\pm|2\rangle)\) split by \(2A\) (\(f\approx24\,\mathrm{GHz}\), microwave); a molecule prepared on one side oscillates between wells at \(\omega=2A/\hbar\). Maser: energy-sorted molecules + resonant cavity → stimulated emission.

Energy-time uncertainty. \(\Delta E\,\Delta t\ge\hbar/2\), where \(\Delta t=\Delta A/|\dd\avg{A}/\dd t|\) is the time for an observable to change by one standard deviation — not an operator relation. Stationary state: \(\Delta E=0\), \(\Delta t\to\infty\). Finite lifetime → energy spread (natural linewidth).

Composite systems & entanglement

Product states. Two-particle kets live in the direct-product space: \(|\psi\rangle=\sum_{i,j}c_{ij}|a_i\rangle_1\otimes|b_j\rangle_2\); operators for particle 1 act only on its factor. A state that cannot be factored into \(|\varphi\rangle_1|\chi\rangle_2\) is entangled.

Two spin-1/2: triplet & singlet. \(\hat{\vv{S}}=\hat{\vv{S}}_1+\hat{\vv{S}}_2\) obeys the same algebra; \(1/2\otimes1/2=1\oplus0\):

Addition of angular momenta. \(j_1\otimes j_2\) yields \(j=j_1+j_2,\;j_1+j_2-1,\dots,|j_1-j_2|\), once each; state count checks: \(\sum(2j+1)=(2j_1+1)(2j_2+1)\). Clebsch-Gordan coefficients \(\langle j_1m_1;j_2m_2|j,m\rangle\) expand the total-\(j\) basis in the product basis (\(m=m_1+m_2\)).

Hyperfine splitting. H ground state: \(H=\frac{2A}{\hbar^2}\hat{\vv{S}}_1\cdot\hat{\vv{S}}_2\) (e–p moments); \(\hat{\vv{S}}_1\cdot\hat{\vv{S}}_2=\frac12(\hat S^2-\hat S_1^2-\hat S_2^2)\) → \(E=A/2\) (triplet), \(-3A/2\) (singlet); splitting \(2A\): \(f\approx1420\,\mathrm{MHz}\), \(\lambda\approx21\,\mathrm{cm}\) — the radio tracer of interstellar hydrogen.

EPR. Singlet correlations: same-axis measurements on the two particles are perfectly anticorrelated at any separation, for every axis. Einstein's reading: values preexist locally ("elements of reality"); quantum reading: no definite value until measurement.

Bell inequality. Any local hidden-variable model obeys \(P({+}\vv{a};{+}\vv{b})\le P({+}\vv{a};{+}\vv{c})+P({+}\vv{c};{+}\vv{b})\). Quantum singlet: \(P({+}\vv{a};{+}\vv{b})=\frac12\sin^2(\theta_{ab}/2)\) — with \(\theta_{ac}=\theta_{cb}=\theta\), \(\theta_{ab}=2\theta\) the inequality fails for \(\theta\lt\pi/2\). Experiment (Aspect et al.) agrees with quantum mechanics: no local hidden variables.

Teleportation. Four Bell states (maximally entangled pair basis). Alice jointly Bell-measures her unknown \(|\psi\rangle\) and her half of a shared pair, sends 2 classical bits; Bob applies the corresponding unitary and holds \(|\psi\rangle\). The state is never measured or cloned, and the classical channel keeps it slower than light.

Density operator. \(\rho=\sum_kp_k|\psi_k\rangle\langle\psi_k|\) covers ensembles (mixed states); \(\avg{A}=\operatorname{tr}(\rho\hat A)\), \(\operatorname{tr}\rho=1\); pure iff \(\rho^2=\rho\) (\(\operatorname{tr}\rho^2=1\); mixed \(\operatorname{tr}\rho^2\lt1\)); evolution \(i\hbar\,\dd\rho/\dd t=[H,\rho]\). Off-diagonal elements (coherences) distinguish a superposition from a 50-50 mixture.

Reduced density operator. \(\rho_1=\operatorname{tr}_2\,\rho\) predicts all measurements on particle 1 alone. For an entangled pure state the reduced state is mixed — singlet → \(\rho_1=\frac12\hat 1\), completely unpolarized: one particle of an entangled pair carries no information by itself.

Wave mechanics in 1D

Position basis. Continuous spectrum: \(\hat x|x\rangle=x|x\rangle\), \(\langle x|x'\rangle=\delta(x-x')\), \(\int\dd x\,|x\rangle\langle x|=\hat 1\). Wavefunction \(\psi(x)=\langle x|\psi\rangle\); probability density \(|\psi|^2\), normalized \(\int|\psi|^2\dd x=1\).

Translations & momentum. \(\hat T(a)|x\rangle=|x+a\rangle\), \(\hat T(a)=e^{-i\hat p_xa/\hbar}\): momentum generates translations. Hence the canonical commutator and its uncertainty relation \[[\hat x,\hat p_x]=i\hbar\qquad\Longrightarrow\qquad\Delta x\,\Delta p\ge\hbar/2\] In the position basis \(\langle x|\hat p_x|\psi\rangle=\dfrac{\hbar}{i}\dfrac{\partial\psi}{\partial x}\).

Momentum space. \(\langle x|p\rangle=\dfrac{e^{ipx/\hbar}}{\sqrt{2\pi\hbar}}\); \(\varphi(p)=\langle p|\psi\rangle\) and \(\psi(x)\) are a Fourier-transform pair — same state, two representations.

Gaussian wave packet. Minimum uncertainty \(\Delta x\,\Delta p=\hbar/2\); a free packet spreads, \(\Delta x(t)=\Delta x_0\sqrt{1+\hbar^2t^2/(4m^2\Delta x_0^4)}\) — faster the tighter it starts.

Double slit. Amplitudes superpose: \(|\psi_1+\psi_2|^2\) shows single-particle interference; any which-path determination kills the cross term. Wave/particle models chosen by measurement; no simultaneous exact \(x,p\).

Schrödinger equation in position space. \[-\frac{\hbar^2}{2m}\frac{\dd^2\psi}{\dd x^2}+V(x)\psi=E\psi\] (time-dependent version: replace \(E\psi\) by \(i\hbar\,\partial\Psi/\partial t\)).

General properties. \(E\gt V_{\min}\); 1D bound states are nondegenerate with discrete spectrum; the \(n\)th excited state has \(n\) nodes; \(\psi,\psi'\) continuous (\(\psi'\) jumps only at \(\delta\)/infinite walls); symmetric \(V(x)\) → energy eigenstates of definite parity. Curvature \(\psi''/\psi\propto V-E\): oscillatory where classically allowed, decaying where forbidden.

Particle in 1D box. \(\lambda_n=2L/n\), \(p_n=nh/(2L)\), \(E_n=\dfrac{n^2h^2}{8mL^2}=\dfrac{n^2\pi^2\hbar^2}{2mL^2}\); \(\psi_n=\sqrt{2/L}\,\sin(n\pi x/L)\); zero-point energy required by \(\Delta x\,\Delta p\ge\hbar/2\).

Probability current. \(j=\dfrac{\hbar}{m}\operatorname{Im}(\psi^*\psi')\); local conservation \(\partial|\psi|^2/\partial t+\partial j/\partial x=0\). Reflection/transmission coefficients are current ratios.

Step potential. \(E\gt V_0\): \(R=\left(\dfrac{k_1-k_2}{k_1+k_2}\right)^2\), \(T=\dfrac{4k_1k_2}{(k_1+k_2)^2}=1-R\) — partial reflection even above the step. \(E\lt V_0\): total reflection with evanescent penetration \(e^{-\kappa x}\).

Tunnelling. Barrier width \(L\), height \(V_0\gt E\), \(\kappa=\sqrt{2m(V_0-E)}/\hbar\): \(T=\left[1+\dfrac{V_0^2\sinh^2\kappa L}{4E(V_0-E)}\right]^{-1}\approx16\dfrac{E}{V_0}\left(1-\dfrac{E}{V_0}\right)e^{-2\kappa L}\) for \(\kappa L\gg1\) — the \(e^{-2\kappa L}\) dominates. Slowly varying barrier: \(T\approx\exp\left(-2\int\kappa(x)\dd x\right)\) (WKB-style; field emission).

Well transmission resonances. Scattering off a square well: \(T=1\) when an integer number of half de Broglie wavelengths spans the well (\(k_2L=n\pi\)) — Ramsauer-Townsend transparency.

Harmonic oscillator

Why it matters. Any potential near a minimum is \(\approx\frac12m\omega^2x^2\); the oscillator also quantizes the radiation field mode by mode.

Ladder operators. \(a=\sqrt{\dfrac{m\omega}{2\hbar}}\left(\hat x+\dfrac{i\hat p}{m\omega}\right)\), \(a^\dagger\) its adjoint; \([a,a^\dagger]=1\); \(\hat x=\sqrt{\dfrac{\hbar}{2m\omega}}(a+a^\dagger)\), \(\hat p=i\sqrt{\dfrac{m\omega\hbar}{2}}(a^\dagger-a)\).

Spectrum. \[H=\hbar\omega\left(a^\dagger a+\tfrac12\right),\qquad E_n=\left(n+\tfrac12\right)\hbar\omega,\quad n=0,1,2,\dots\] \(a|n\rangle=\sqrt n\,|n-1\rangle\), \(a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle\), \(|n\rangle=\frac{(a^\dagger)^n}{\sqrt{n!}}|0\rangle\).

Matrix elements. \(\langle n'|\hat x|n\rangle\) and \(\langle n'|\hat p|n\rangle\) vanish unless \(n'=n\pm1\) — read off from the ladder relations; basis of selection rules.

Wavefunctions. \(a|0\rangle=0\) → \(\psi_0=(m\omega/\pi\hbar)^{1/4}e^{-m\omega x^2/2\hbar}\); \(\psi_n\) = Gaussian × Hermite polynomial \(H_n\), parity \((-1)^n\). Zero-point energy \(\hbar\omega/2\) saturates the uncertainty bound. Large \(n\): \(|\psi_n|^2\) → classical dwell-time distribution.

Time dependence. \(\avg{x}(t)\) oscillates at \(\omega\) exactly (Ehrenfest is exact here — \(V'\) linear).

Coherent states. \(a|\alpha\rangle=\alpha|\alpha\rangle\); \(|c_n|^2\) Poissonian with \(\avg{n}=|\alpha|^2\), \(\Delta n=|\alpha|\); a minimum-uncertainty Gaussian oscillating without spreading — the closest quantum state to a classical wave of definite amplitude and phase (laser light).

Parity. \(\hat\Pi\psi(x)=\psi(-x)\), eigenvalues \(\pm1\); \([\hat\Pi,H]=0\) for even \(V\) → eigenstates of definite parity, known before solving — the 1D prototype of symmetry arguments.

Path integrals

Sum over paths. Every path from \((x_0,t_0)\) to \((x',t')\) contributes an amplitude of unit modulus and phase = action/\(\hbar\): \[\langle x',t'|x_0,t_0\rangle=\int\mathcal{D}[x(t)]\,e^{iS[x(t)]/\hbar}\] Free particle: \(\langle x',t'|x_0,t_0\rangle=\sqrt{\dfrac{m}{2\pi i\hbar(t'-t_0)}}\,e^{im(x'-x_0)^2/2\hbar(t'-t_0)}\).

Classical limit. For \(S\gg\hbar\), phases cancel except near the stationary path \(\delta S=0\) — particles follow the path of least action because neighboring paths interfere constructively only there. Slits reintroduce interference by restricting which paths exist.

Gravitational interference. Two interferometer arms at different heights accumulate \(\Delta\varphi=m^2g\lambda A\sin\delta/(2\pi\hbar^2)\) for enclosed area \(A\) (tilt \(\delta\)) — observed with neutrons (COW experiment): gravity shifts quantum phases, and mass enters squared.

Central potentials & hydrogen

3D formalism. \([\hat x_i,\hat p_j]=i\hbar\delta_{ij}\) (different axes compatible); \(\hat{\vv{p}}\to\frac{\hbar}{i}\nabla\). Translational invariance ⇒ total \(\vv{P}\) conserved; rotational invariance ⇒ \(\vv{L}\) conserved — symmetry ↔ conservation law.

Two-body reduction. CM and relative coordinates: \(H=\dfrac{P^2}{2M}+\dfrac{p^2}{2\mu}+V(|\vv{r}|)\), \(\mu=\dfrac{m_1m_2}{m_1+m_2}\); in the CM frame solve the one-body relative problem.

Orbital angular momentum. \(\hat{\vv{L}}=\hat{\vv{r}}\times\hat{\vv{p}}\) generates rotations of position states; same algebra as spin. Complete set of commuting observables: \(H,L^2,L_z\) → eigenstates \(|E,l,m\rangle\), \(\psi=R(r)Y_{l,m}(\theta,\phi)\).

Spherical harmonics. \(L_z\to\frac{\hbar}{i}\frac{\partial}{\partial\phi}\); \(Y_{l,m}\propto e^{im\phi}\), single-valuedness forces \(m,l\) integral — half-integral angular momentum cannot be orbital (spin is not "spinning"). \(Y_{0,0}=\frac{1}{\sqrt{4\pi}}\), \(Y_{1,0}=\sqrt{\frac{3}{4\pi}}\cos\theta\), \(Y_{1,\pm1}=\mp\sqrt{\frac{3}{8\pi}}\sin\theta\,e^{\pm i\phi}\).

Radial equation. With \(u=rR(r)\), a 1D problem on \(r\ge0\) with a centrifugal barrier: \[-\frac{\hbar^2}{2\mu}\frac{\dd^2u}{\dd r^2}+\left[V(r)+\frac{l(l+1)\hbar^2}{2\mu r^2}\right]u=Eu\] \(u(0)=0\), \(u\sim r^{l+1}\) near the origin; higher \(l\) pushed out. Quick ground-state estimates: minimize \(E(a)\) with \(p\sim\hbar/a\) (uncertainty principle).

Hydrogen. \(V=-\dfrac{Ze^2}{4\pi\epsilon_0r}\): \[E_n=-\frac{\mu c^2(Z\alpha)^2}{2n^2}=-\frac{13.6\,Z^2}{n^2}\,\mathrm{eV},\qquad\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}\approx\frac{1}{137}\] \(n=n_r+l+1\); Bohr radius \(a_0=\dfrac{4\pi\epsilon_0\hbar^2}{\mu e^2}\approx0.529\,\mathrm{\mathring{A}}\); size scale \(r_n\sim a_0n^2/Z\); \(R_{10}\propto e^{-Zr/a_0}\). Energy depends on \(n\) alone (Coulomb accident) → degeneracy \(n^2\) (\(2n^2\) with spin). Reduced mass: deuterium isotope shift, positronium \(\mu=m_e/2\), muonic atoms.

Diatomic molecules. \(E\approx E_{\text{elec}}+(n+\tfrac12)\hbar\omega+\dfrac{l(l+1)\hbar^2}{2I}\): electronic ≫ vibrational ≫ rotational. Selection rules \(\Delta n=\pm1\), \(\Delta l=\pm1\) → band spectra; rotational line spacing \(\hbar^2/I\) measures bond length.

Finite spherical well. An s-wave bound state exists only if the well is deep/wide enough: \(V_0a^2\ge\pi^2\hbar^2/8\mu\). Deuteron: one barely bound \(l=0\) state (\(\approx2.2\,\mathrm{MeV}\)), no bound excited states.

Infinite spherical well. \(E_{n,l}=\dfrac{\hbar^2\beta_{n,l}^2}{2\mu a^2}\), \(\beta_{n,l}\) the \(n\)th zero of the spherical Bessel function \(j_l\) (\(l=0\): \(\beta=n\pi\)).

3D isotropic oscillator. \(E=(2n_r+l+\tfrac32)\hbar\omega\); level \(n=2n_r+l\) has degeneracy \(\tfrac12(n+1)(n+2)\), alternating parity.

Perturbation theory

Setup. \(H=H_0+H_1\) with \(H_0\) solved: \(H_0|\varphi_n^{(0)}\rangle=E_n^{(0)}|\varphi_n^{(0)}\rangle\).

Nondegenerate results. \[E_n^{(1)}=\langle\varphi_n^{(0)}|H_1|\varphi_n^{(0)}\rangle,\qquad E_n^{(2)}=\sum_{k\ne n}\frac{\bigl|\langle\varphi_k^{(0)}|H_1|\varphi_n^{(0)}\rangle\bigr|^2}{E_n^{(0)}-E_k^{(0)}}\] First-order state mixes in \(|\varphi_k^{(0)}\rangle\) with coefficients \(\langle\varphi_k^{(0)}|H_1|\varphi_n^{(0)}\rangle/(E_n^{(0)}-E_k^{(0)})\); valid when matrix elements ≪ level spacings. Second order always pushes the ground state down.

Degenerate case. Diagonalize \(H_1\) within the degenerate subspace; its eigenvalues are the first-order shifts and its eigenvectors the good zeroth-order states. A symmetry \([H_1,\hat A]=0\) mixes only states with equal \(a\) — use it to block-diagonalize.

Stark effect. Uniform \(\mathcal{E}\): \(H_1=e\mathcal{E}\hat z\). H ground state: no linear shift (definite parity), quadratic shift \(-\frac{\alpha_{\text{pol}}}{2}\mathcal{E}^2\) with \(\alpha_{\text{pol}}=\frac92\,4\pi\epsilon_0a_0^3\). \(n=2\) (degenerate): \(2s\) and \(2p_0\) mix → linear shifts \(\pm3ea_0\mathcal{E}\). Ammonia in \(\mathcal{E}\): two-state \(E_\pm=E_0\pm\sqrt{A^2+(\mu_e\mathcal{E})^2}\) — quadratic when weak, linear when strong; basis of maser state selection.

Fine structure. Internal relativistic perturbations, all \(O(\alpha^2)\) relative to \(E_n\):

Diagonalize spin-orbit in the total-\(j\) basis (\(\hat{\vv{S}}\cdot\hat{\vv{L}}=\frac12(J^2-L^2-S^2)\)); all three combine into \[E_{n,j}=E_n^{(0)}\left[1+\frac{\alpha^2}{n^2}\left(\frac{n}{j+\tfrac12}-\frac34\right)\right]\] Levels \(nL_j\) depend only on \(n,j\): \(2s_{1/2}\)–\(2p_{1/2}\) stay degenerate until the Lamb shift (\(\sim4.4\times10^{-6}\,\mathrm{eV}\), QED vacuum fluctuations).

Zeeman effect. Weak \(B\): \(\Delta E=g_J\mu_Bm_jB\) with the Landé factor \(g_J=1+\dfrac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}\) (\(g_J=2\) for \(s_{1/2}\), \(\tfrac23\) for \(p_{1/2}\)) — "anomalous" patterns from spin. Strong \(B\) decouples \(\vv{L}\) and \(\vv{S}\).

Identical particles

Exchange symmetry. Indistinguishability: \(\hat P_{12}\) commutes with \(H\), physical states are eigenstates. Integer spin → symmetric (bosons); half-integer → antisymmetric (fermions) — spin-statistics. Antisymmetry ⇒ Pauli exclusion: no two identical fermions in the same single-particle state. Many bosons pile into one state (laser, superfluidity, superconductivity).

Two electrons. Total state antisymmetric: symmetric space × singlet spin, or antisymmetric space × triplet spin. Antisymmetric spatial states vanish at \(\vv{r}_1=\vv{r}_2\) — exchange keeps triplet electrons apart, an effective interaction with no classical analogue.

Helium. Zeroth order \(2\times(-54.4)=-108.8\,\mathrm{eV}\); first-order \(e\)–\(e\) repulsion \(\avg{e^2/4\pi\epsilon_0r_{12}}=+34\,\mathrm{eV}\) → \(-74.8\) vs measured \(-79.0\,\mathrm{eV}\). Ground state necessarily a singlet (symmetric \(1s^2\) space). Excited \(1snl\): \(E=E^{(0)}+J\pm K\) (direct ± exchange integral) — triplet (ortho) below singlet (para).

Multielectron atoms. Central-field approximation with screening: energy depends on \(n\) and \(l\); low-\(l\) orbits penetrate the core and bind deeper — filling order \(1s\,2s\,2p\,3s\,3p\,4s\,3d\dots\). Shell closures → noble gases; valence electrons → chemistry; Hund's rule: maximize spin (exchange lowers symmetric-spin configurations).

Covalent bonding. \(\ce{H2+}\): symmetric (bonding) combination piles amplitude between the protons, antisymmetric (antibonding) has a node there. \(\ce{H2}\): two electrons fill the bonding orbital as a spin singlet. \(\ce{He2}\): Pauli forces antibonding occupation — no covalent bond.

Scattering

Cross sections. Asymptotically \(\psi\to e^{ikz}+f(\theta,\phi)\dfrac{e^{ikr}}{r}\); \[\frac{\dd\sigma}{\dd\Omega}=|f(\theta,\phi)|^2,\qquad\sigma=\int|f|^2\dd\Omega\]

Born approximation. First-order (weak potential or high energy): \(f\) = Fourier transform of \(V\) at momentum transfer \(\hbar\vv{q}=\hbar(\vv{k}-\vv{k}')\), \(q=2k\sin(\theta/2)\); central \(V\): \(f(q)=-\dfrac{2\mu}{\hbar^2q}\displaystyle\int_0^\infty rV(r)\sin(qr)\dd r\).

Yukawa → Rutherford. \(V=\dfrac{g^2}{r}e^{-r/a}\): \(f=-\dfrac{2\mu g^2/\hbar^2}{q^2+1/a^2}\). Unscreened limit \(a\to\infty\), \(g^2=\dfrac{Z_1Z_2e^2}{4\pi\epsilon_0}\): \(\dfrac{\dd\sigma}{\dd\Omega}=\left(\dfrac{Z_1Z_2e^2}{16\pi\epsilon_0E}\right)^2\dfrac{1}{\sin^4(\theta/2)}\) — the classical Rutherford formula.

Partial waves. A central potential only shifts the phase of each outgoing \(l\)-wave: \[f(\theta)=\frac1k\sum_l(2l+1)e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta),\qquad\sigma=\frac{4\pi}{k^2}\sum_l(2l+1)\sin^2\delta_l\] Only \(l\lesssim ka\) contribute (classical impact-parameter argument) — low energy is pure s-wave. Optical theorem: \(\sigma=\dfrac{4\pi}{k}\operatorname{Im}f(0)\) (forward interference depletes the beam).

Phase-shift analysis. Attractive \(V\) pulls the wave in (\(\delta_l\gt0\)), repulsive pushes out. Hard sphere radius \(a\): \(\delta_0=-ka\), low energy \(\sigma\to4\pi a^2\) (4× geometric). Resonance: \(\delta_l\) sweeps through \(\pi/2\) → \(\sigma_l\) hits the unitarity bound \(4\pi(2l+1)/k^2\) (quasi-bound state). \(\delta_0=\pi\) → s-wave invisible: Ramsauer-Townsend transparency of noble gases to slow electrons.

Photons & atoms

Charged particle in an EM field. \(H=\dfrac{(\hat{\vv{p}}-q\vv{A})^2}{2m}+q\varphi\) — the potentials, not the fields, enter the Hamiltonian.

Aharonov-Bohm. Paths encircling a solenoid acquire relative phase \(\dfrac{q}{\hbar}\oint\vv{A}\cdot\dd\vv{l}=\dfrac{q\Phi}{\hbar}\): the interference pattern shifts although \(\vv{B}=0\) everywhere on the paths, with flux period \(h/q\). Potentials are physical in quantum mechanics.

Quantized radiation field. Each mode \((\vv{k},s)\) is a harmonic oscillator: \[H=\sum_{\vv{k},s}\hbar\omega_k\left(a^\dagger_{\vv{k},s}a_{\vv{k},s}+\tfrac12\right)\] Photons are the quanta \(n_{\vv{k},s}\); \(a^\dagger,a\) create/destroy them; the vacuum carries zero-point fluctuations.

Time-dependent perturbation theory. First order: \(c_f(t)=-\dfrac{i}{\hbar}\displaystyle\int_0^t\langle f|H_1(t')|i\rangle\,e^{i\omega_{fi}t'}\dd t'\), \(\omega_{fi}=(E_f-E_i)/\hbar\); a harmonic perturbation drives transitions at resonance \(\omega\approx\omega_{fi}\). Transitions into a continuum at density \(\rho(E_f)\) (Fermi's golden rule): \[\Gamma_{i\to f}=\frac{2\pi}{\hbar}\bigl|\langle f|H_1|i\rangle\bigr|^2\rho(E_f)\]

Emission & absorption. Matrix elements give absorption rate \(\propto n\) and emission rate \(\propto n+1\): stimulated emission (photon into the same mode — the laser, with population inversion) plus spontaneous emission from the vacuum term. Dipole approximation (\(\lambda\gg\) atom): \(e^{i\vv{k}\cdot\vv{r}}\approx1\), coupling via \(\langle f|\hat{\vv{r}}|i\rangle\).

Spontaneous emission. \(\Gamma=\dfrac{\omega^3|\langle f|q\hat{\vv{r}}|i\rangle|^2}{3\pi\epsilon_0\hbar c^3}\); lifetime \(\tau=1/\Gamma\) (H \(2p\to1s\): \(\tau\approx1.6\,\mathrm{ns}\)). Dipole selection rules from the matrix element: \(\Delta l=\pm1\), \(\Delta m=0,\pm1\) (photon carries one \(\hbar\)); same rules govern the molecular spectra above.

Cavity QED. A cavity reshapes the vacuum: spontaneous emission enhanced on resonance, suppressed off it. Strong coupling of one atom to one mode: reversible vacuum Rabi oscillations \(|e,0\rangle\leftrightarrow|g,1\rangle\) — emission is not intrinsically irreversible.

Feynman diagrams. Higher-order QED: amplitudes as sums of diagrams of vertices exchanging virtual photons; each vertex contributes amplitude \(\sim\sqrt\alpha\), so each order is suppressed by \(\alpha\approx1/137\) — why perturbation theory works.