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Quantum Mechanics — Equation Sheet

A deep reference for quantum mechanics — from the postulates and Dirac notation through angular momentum, perturbation theory, identical particles, open quantum systems, and a dedicated section on the equations most central to AMO physics.

11 sections · 79 equations

Foundations & Postulates

Time-Dependent Schrödinger Equation
iℏ∂Ψ∂t=H^Ψi\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi
The fundamental equation of quantum dynamics. Governs the time evolution of the wave function Ψ(r, t) under the Hamiltonian operator Ĥ.
Time-Independent Schrödinger Equation
H^ψ=Eψ\hat{H}\psi = E\psi
Eigenvalue equation for stationary states. Solutions ψ_n with energies E_n form a complete, orthonormal basis for the Hilbert space.
Hamiltonian Operator
H^=−ℏ22m∇2+V(r,t)\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)
Kinetic energy operator (−ℏ²∇²/2m) plus potential V. In 1D: −(ℏ²/2m)d²/dx².
Born Rule — Probability Density
P(r,t)=∣Ψ(r,t)∣2=Ψ∗(r,t)Ψ(r,t)P(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2 = \Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t)
|Ψ|² is the probability density of finding the particle at position r at time t. The probability interpretation is a postulate of quantum mechanics.
Normalization Condition
∫−∞∞∣Ψ(r,t)∣2 d3r=1\int_{-\infty}^{\infty}|\Psi(\mathbf{r},t)|^2\,d^3r = 1
The Schrödinger equation preserves normalization for all time, provided Ĥ is Hermitian.
Heisenberg Uncertainty Principle
σx σp≥ℏ2\sigma_x\,\sigma_p \geq \frac{\hbar}{2}
Position and momentum cannot both be known precisely. σ_x and σ_p are the standard deviations of x and p in state Ψ.
Time-Energy Uncertainty Relation
ΔE Δt≥ℏ2\Delta E\,\Delta t \geq \frac{\hbar}{2}
An energy eigenstate has zero energy uncertainty but no characteristic time scale. For an unstable state of lifetime τ, ΔE ≥ ℏ/2τ sets the natural linewidth.
Generalized Uncertainty Principle
σA σB≥12∣⟨[A^,B^]⟩∣\sigma_A\,\sigma_B \geq \frac{1}{2}\bigl|\langle[\hat{A},\hat{B}]\rangle\bigr|
For any two observables  and B̂. The Heisenberg relation is the special case [x̂, p̂] = iℏ. Incompatible observables (non-zero commutator) are fundamentally constrained.

Operators, Commutators & Dirac Notation

Canonical Commutation Relation
[x^i, p^j]=iℏ δij,[x^i,x^j]=[p^i,p^j]=0[\hat{x}_i,\,\hat{p}_j] = i\hbar\,\delta_{ij}, \qquad [\hat{x}_i,\hat{x}_j]=[\hat{p}_i,\hat{p}_j]=0
The fundamental quantum condition. Implies the Heisenberg uncertainty relation and determines the form of all quantum dynamics.
Momentum Operator (position space)
p^=−iℏ∇,⟨p∣r⟩=e−ip⋅r/ℏ(2πℏ)3/2\hat{\mathbf{p}} = -i\hbar\nabla, \qquad \langle\mathbf{p}|\mathbf{r}\rangle = \frac{e^{-i\mathbf{p}\cdot\mathbf{r}/\hbar}}{(2\pi\hbar)^{3/2}}
Momentum acts as a differential operator in position space. Its eigenstates are plane waves.
Expectation Value
⟨A^⟩=⟨ψ∣A^∣ψ⟩=∫ψ∗A^ψ d3r\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle = \int\psi^*\hat{A}\psi\,d^3r
Expectation value of observable  in state |ψ⟩. For a Hermitian operator the result is always real.
Completeness Relation
∑n∣n⟩⟨n∣=1^(discrete),∫∣x⟩⟨x∣ dx=1^(continuous)\sum_n |n\rangle\langle n| = \hat{1} \quad (\text{discrete}), \qquad \int |x\rangle\langle x|\,dx = \hat{1} \quad (\text{continuous})
Any state can be expanded in a complete orthonormal basis. The projection operators |n⟩⟨n| (or |x⟩⟨x| dx) resolve the identity.
Time Evolution Operator
∣ψ(t)⟩=U^(t)∣ψ(0)⟩,U^(t)=e−iH^t/ℏ(time-indep. H^)|\psi(t)\rangle = \hat{U}(t)|\psi(0)\rangle, \qquad \hat{U}(t) = e^{-i\hat{H}t/\hbar}\quad(\text{time-indep. }\hat{H})
Unitary operator Û(t) propagates states forward in time. For a time-independent Hamiltonian it is an exponential; for time-dependent Ĥ use Dyson series.
Heisenberg Equation of Motion
dA^Hdt=iℏ[H^,A^H]+∂A^H∂t\frac{d\hat{A}_H}{dt} = \frac{i}{\hbar}[\hat{H},\hat{A}_H] + \frac{\partial\hat{A}_H}{\partial t}
In the Heisenberg picture states are fixed, operators carry the time dependence. Equivalent to the Schrödinger picture for all measurable quantities.
Ehrenfest's Theorem
ddt⟨p^⟩=−⟨∇V⟩,ddt⟨r^⟩=⟨p^⟩m\frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = -\langle\nabla V\rangle, \qquad \frac{d}{dt}\langle\hat{\mathbf{r}}\rangle = \frac{\langle\hat{\mathbf{p}}\rangle}{m}
Quantum expectation values obey Newton's second law on average. Classical mechanics is recovered when wave-packet spreading is negligible.

Wave Mechanics & 1D Potentials

Infinite Square Well — Energy Levels
En=n2π2ℏ22mL2,n=1,2,3,…E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \quad n = 1,2,3,\ldots
Particle confined to [0, L] with infinite walls. Energy grows as n²; the non-zero ground-state energy E₁ is a direct consequence of the uncertainty principle.
Infinite Square Well — Wave Functions
ψn(x)=2Lsin⁡nπxL\psi_n(x) = \sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}
Normalized stationary states. Mutually orthogonal: ∫ψ_m* ψ_n dx = δ_{mn}. Form a complete basis for L²[0,L].
Free Particle — Plane Wave
ψk(x)=Aeikx,E=ℏ2k22m,p=ℏk\psi_k(x) = Ae^{ikx}, \quad E = \frac{\hbar^2 k^2}{2m}, \quad p = \hbar k
Momentum eigenstate. Not normalizable in isolation; physical states are wavepackets ψ(x,t) = ∫φ(k)e^{i(kx−ωt)} dk.
Transmission Coefficient — Rectangular Barrier
T≈e−2κa,κ=2m(V0−E)ℏ2(E<V0)T \approx e^{-2\kappa a}, \qquad \kappa = \sqrt{\frac{2m(V_0-E)}{\hbar^2}} \quad (E < V_0)
Exponential suppression of tunneling through a barrier of width a and height V₀ > E. Underlies alpha decay, STM, and tunnel junctions.
Reflection & Transmission Amplitudes (step)
r=k1−k2k1+k2,t=2k1k1+k2,∣r∣2+k2k1∣t∣2=1r = \frac{k_1-k_2}{k_1+k_2}, \quad t = \frac{2k_1}{k_1+k_2}, \qquad |r|^2 + \frac{k_2}{k_1}|t|^2 = 1
Amplitude coefficients for a particle incident on a potential step from region 1 (wavevector k₁) to region 2 (k₂). Probability current is conserved.
Delta Function Potential — Bound State
V(x)=−α δ(x)  ⟹  E0=−mα22ℏ2,ψ0=mαℏ2 e−mα∣x∣/ℏ2V(x) = -\alpha\,\delta(x) \implies E_0 = -\frac{m\alpha^2}{2\hbar^2}, \quad \psi_0 = \sqrt{\frac{m\alpha}{\hbar^2}}\,e^{-m\alpha|x|/\hbar^2}
A delta function attractive potential always supports exactly one bound state. Energy is negative; ψ decays exponentially away from the origin.

The Quantum Harmonic Oscillator

Harmonic Oscillator Hamiltonian
H^=p^22m+12mω2x^2\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2
Exactly solvable potential appearing throughout quantum physics: vibrating molecules, phonons, quantum optics, and QFT field modes.
Energy Eigenvalues
En=ℏω ⁣(n+12),n=0,1,2,…E_n = \hbar\omega\!\left(n + \frac{1}{2}\right), \quad n = 0,1,2,\ldots
Equally spaced energy levels separated by ℏω. The non-zero ground-state energy ½ℏω is the quantum zero-point energy.
Ladder (Creation & Annihilation) Operators
a^=mω2ℏ ⁣(x^+ip^mω),a^†=mω2ℏ ⁣(x^−ip^mω)\hat{a} = \sqrt{\frac{m\omega}{2\hbar}}\!\left(\hat{x}+\frac{i\hat{p}}{m\omega}\right), \quad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\!\left(\hat{x}-\frac{i\hat{p}}{m\omega}\right)
â lowers and ↠raises the quantum number by 1. They factor the Hamiltonian and give the algebraic solution without solving a differential equation.
Ladder Operator Commutator
[a^, a^†]=1,H^=ℏω ⁣(a^†a^+12)[\hat{a},\,\hat{a}^\dagger] = 1, \qquad \hat{H} = \hbar\omega\!\left(\hat{a}^\dagger\hat{a}+\tfrac{1}{2}\right)
Bosonic commutation relation. The number operator N̂ = â†â counts excitation quanta.
Action on Number States
a^∣n⟩=n ∣n−1⟩,a^†∣n⟩=n+1 ∣n+1⟩,a^∣0⟩=0\hat{a}|n\rangle = \sqrt{n}\,|n-1\rangle, \quad \hat{a}^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle, \quad \hat{a}|0\rangle = 0
Recursion relations for Fock states. Repeated application of ↠on the vacuum |0⟩ generates all number states.
Position & Momentum in Terms of Ladder Operators
x^=ℏ2mω(a^+a^†),p^=imωℏ2(a^†−a^)\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a}+\hat{a}^\dagger), \qquad \hat{p} = i\sqrt{\frac{m\omega\hbar}{2}}(\hat{a}^\dagger-\hat{a})
Express x̂ and p̂ as sums of ladder operators. Directly gives matrix elements ⟨m|x̂|n⟩ ∝ (√n δ_{m,n−1} + √(n+1) δ_{m,n+1}).
Ground State Wave Function
ψ0(x)=(mωπℏ) ⁣1/4 ⁣exp⁡ ⁣(−mωx22ℏ)\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{\!1/4}\!\exp\!\left(-\frac{m\omega x^2}{2\hbar}\right)
Gaussian ground state with width x₀ = √(ℏ/mω). The oscillator length x₀ sets the scale for quantum fluctuations.

Angular Momentum & Spin

Angular Momentum Commutation Relations
[L^i,L^j]=iℏ ϵijkL^k,[L^2,L^i]=0[\hat{L}_i,\hat{L}_j] = i\hbar\,\epsilon_{ijk}\hat{L}_k, \qquad [\hat{L}^2,\hat{L}_i] = 0
Fundamental algebra of angular momentum. L² commutes with all components so L² and L_z share eigenstates.
L² and L_z Eigenvalues
L^2∣ℓ,m⟩=ℏ2ℓ(ℓ+1)∣ℓ,m⟩,L^z∣ℓ,m⟩=ℏm∣ℓ,m⟩\hat{L}^2|\ell,m\rangle = \hbar^2\ell(\ell+1)|\ell,m\rangle, \quad \hat{L}_z|\ell,m\rangle = \hbar m|\ell,m\rangle
Quantum numbers: ℓ = 0, 1, 2, … (orbital); m = −ℓ, −ℓ+1, …, ℓ. Same algebra holds for spin S and total angular momentum J.
Raising & Lowering Operators
L^±=L^x±iL^y,L^±∣ℓ,m⟩=ℏℓ(ℓ+1)−m(m±1) ∣ℓ,m±1⟩\hat{L}_{\pm} = \hat{L}_x \pm i\hat{L}_y, \qquad \hat{L}_\pm|\ell,m\rangle = \hbar\sqrt{\ell(\ell+1)-m(m\pm1)}\,|\ell,m\pm1\rangle
L₊ raises m by 1; L₋ lowers it. The square-root factor vanishes at the extremes m = ±ℓ, terminating the ladder.
Pauli Matrices (spin-½)
σx=(0110),σy=(0−ii0),σz=(100−1)\boldsymbol{\sigma}_x = \begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \boldsymbol{\sigma}_y = \begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad \boldsymbol{\sigma}_z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}
Spin-½ matrices satisfying {σ_i, σ_j} = 2δ_{ij} and [σ_i, σ_j] = 2iε_{ijk}σ_k. The spin operator is Ŝ = (ℏ/2)σ.
Spinor Rotation
R^(n^,ϕ)=e−iϕJ^⋅n^/ℏ=cos⁡ϕ2 1−isin⁡ϕ2 (n^⋅σ)(spin-12)\hat{R}(\hat{\mathbf{n}},\phi) = e^{-i\phi\hat{\mathbf{J}}\cdot\hat{\mathbf{n}}/\hbar} = \cos\frac{\phi}{2}\,\mathbf{1} - i\sin\frac{\phi}{2}\,(\hat{\mathbf{n}}\cdot\boldsymbol{\sigma})\quad(\text{spin-}\tfrac{1}{2})
A spin-½ state requires a rotation of 4π (not 2π) to return to itself. The Bloch-sphere rotation angle is twice the physical rotation.
Addition of Angular Momenta
∣j,m⟩=∑m1+m2=m⟨j1,m1;j2,m2 ∣ j,m⟩ ∣j1,m1⟩∣j2,m2⟩|j,m\rangle = \sum_{m_1+m_2=m}\langle j_1,m_1;j_2,m_2\,|\,j,m\rangle\,|j_1,m_1\rangle|j_2,m_2\rangle
Clebsch-Gordan expansion. The combined quantum number runs |j₁−j₂| ≤ j ≤ j₁+j₂.
Wigner-Eckart Theorem
⟨α′,j′,m′∣Tq(k)∣α,j,m⟩=⟨j,m;k,q ∣ j′,m′⟩ ⟨α′,j′∥T(k)∥α,j⟩2j′+1\langle\alpha^\prime,j^\prime,m^\prime|T_q^{(k)}|\alpha,j,m\rangle = \langle j,m;k,q\,|\,j^\prime,m^\prime\rangle\,\frac{\langle\alpha^\prime,j^\prime\|T^{(k)}\|\alpha,j\rangle}{\sqrt{2j^\prime+1}}
The matrix element of any rank-k tensor operator T factors into a Clebsch-Gordan coefficient (geometry) and a reduced matrix element (dynamics). Explains and unifies selection rules.

The Hydrogen Atom

Bohr Energy Levels
En=−mee42(4πϵ0)2ℏ21n2=−13.6 eVn2,n=1,2,3,…E_n = -\frac{m_e e^4}{2(4\pi\epsilon_0)^2\hbar^2}\frac{1}{n^2} = -\frac{13.6\text{ eV}}{n^2}, \quad n = 1,2,3,\ldots
Exact energy eigenvalues of the hydrogen atom. Degeneracy is n² (ignoring spin); the accidental SO(4) symmetry causes ℓ-independence.
Bohr Radius
a0=4πϵ0ℏ2mee2=ℏmecα≈0.529 A˚a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_e e^2} = \frac{\hbar}{m_e c\alpha} \approx 0.529\text{ Å}
Scale of atomic physics. The ground-state wave function peaks at r = a₀; ⟨r⟩_{1s} = 3a₀/2.
Rydberg Formula (spectral lines)
1λ=R∞ ⁣(1n12−1n22),R∞=mee48ϵ02h3c≈1.097×107 m−1\frac{1}{\lambda} = R_\infty\!\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \quad R_\infty = \frac{m_e e^4}{8\epsilon_0^2 h^3 c} \approx 1.097\times10^7\text{ m}^{-1}
Wavelength of photons emitted or absorbed in transitions n₂ → n₁. Lyman: n₁=1; Balmer: n₁=2; Paschen: n₁=3.
Hydrogen Wave Function
ψnℓm(r,θ,ϕ)=Rnℓ(r) Yℓm(θ,ϕ)\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\,Y_\ell^m(\theta,\phi)
Factorizes into radial function R_{nℓ}(r) (Laguerre polynomials × e^{−r/na₀}) and spherical harmonics Y_ℓ^m(θ,φ).
Electric Dipole Selection Rules
Δn=any,Δℓ=±1,Δm=0,±1\Delta n = \text{any}, \quad \Delta\ell = \pm1, \quad \Delta m = 0,\pm1
Conditions for non-zero electric dipole matrix element ⟨n′ℓ′m′|r|nℓm⟩. Arise from parity and Wigner-Eckart theorem. Magnetic dipole/electric quadrupole transitions are weaker by α².
Fine Structure Energy Shift
Enjfs=−Enα2n2 ⁣(nj+12−34)E_{nj}^{\rm fs} = -\frac{E_n\alpha^2}{n^2}\!\left(\frac{n}{j+\tfrac{1}{2}} - \frac{3}{4}\right)
Combined spin-orbit and relativistic kinetic corrections, order α² relative to the Bohr levels. Splits levels by j = ℓ ± ½.
Lamb Shift
ELamb(2s1/2)−ELamb(2p1/2)≈1058 MHzE_{\rm Lamb}(2s_{1/2}) - E_{\rm Lamb}(2p_{1/2}) \approx 1058\text{ MHz}
QED correction lifting the degeneracy of 2s₁/₂ and 2p₁/₂ levels (which are degenerate in the Dirac equation). First measured by Lamb & Retherford (1947); a triumph of QED.

Perturbation Theory

First-Order Energy Correction
En(1)=⟨n(0)∣H^′∣n(0)⟩E_n^{(1)} = \langle n^{(0)}|\hat{H}^\prime|n^{(0)}\rangle
First-order shift is just the expectation value of the perturbation in the unperturbed state. Works whenever E_n^{(0)} is non-degenerate.
First-Order State Correction
∣n(1)⟩=∑m≠n⟨m(0)∣H^′∣n(0)⟩En(0)−Em(0) ∣m(0)⟩|n^{(1)}\rangle = \sum_{m\neq n}\frac{\langle m^{(0)}|\hat{H}^\prime|n^{(0)}\rangle}{E_n^{(0)}-E_m^{(0)}}\,|m^{(0)}\rangle
The perturbed state acquires admixtures from other unperturbed states. Diverges when there is degeneracy — use degenerate perturbation theory instead.
Second-Order Energy Correction
En(2)=∑m≠n∣⟨m(0)∣H^′∣n(0)⟩∣2En(0)−Em(0)E_n^{(2)} = \sum_{m\neq n}\frac{|\langle m^{(0)}|\hat{H}^\prime|n^{(0)}\rangle|^2}{E_n^{(0)}-E_m^{(0)}}
Always lowers the ground-state energy (negative denominator for m ≠ ground state). Essential for computing van der Waals forces, polarizability, and the Stark effect.
Degenerate Perturbation Theory
det⁡[H^′−E(1)1^]degen. subspace=0\det\bigl[\hat{H}^\prime - E^{(1)}\hat{1}\bigr]_{\text{degen. subspace}} = 0
When multiple unperturbed states share the same energy E_n^{(0)}, diagonalize H′ within the degenerate subspace. The "good" basis states are the eigenvectors.
Fermi's Golden Rule
Γi→f=2πℏ∣⟨f∣H^′∣i⟩∣2ρ(Ef)\Gamma_{i\to f} = \frac{2\pi}{\hbar}\bigl|\langle f|\hat{H}^\prime|i\rangle\bigr|^2\rho(E_f)
Transition rate to a continuum of final states at energy E_f. ρ(E_f) is the density of states. Valid when the perturbation is weak and the coupling to the continuum is irreversible.
Variational Principle
E0≤⟨ψ~∣H^∣ψ~⟩≡⟨H^⟩ψ~E_0 \leq \langle\tilde{\psi}|\hat{H}|\tilde{\psi}\rangle \equiv \langle\hat{H}\rangle_{\tilde{\psi}}
For any normalized trial state |ψ̃⟩, ⟨Ĥ⟩ is an upper bound on the ground-state energy. The bound is exact iff |ψ̃⟩ is the true ground state.
Adiabatic Theorem
∣ψ(t)⟩≈eiγn(t)eiαn(t)∣n(t)⟩if∣⟨m∣H˙∣n⟩(Em−En)2∣≪1|\psi(t)\rangle \approx e^{i\gamma_n(t)}e^{i\alpha_n(t)}|n(t)\rangle \quad \text{if} \quad \left|\frac{\langle m|\dot{H}|n\rangle}{(E_m-E_n)^2}\right| \ll 1
If the Hamiltonian changes slowly compared to the energy gaps, a system in eigenstate |n⟩ remains in the instantaneous eigenstate. α_n is the dynamical phase; γ_n is the Berry (geometric) phase.

Identical Particles & Second Quantization

Symmetrization Postulate
P^ij∣ψ⟩={+∣ψ⟩bosons (integer spin)−∣ψ⟩fermions (half-integer spin)\hat{P}_{ij}|\psi\rangle = \begin{cases}+|\psi\rangle & \text{bosons (integer spin)}\\-|\psi\rangle & \text{fermions (half-integer spin)}\end{cases}
Exchange of identical particles leaves the state unchanged (bosons) or changes its sign (fermions). The spin-statistics theorem guarantees this connection.
Slater Determinant
ψ(x1,…,xN)=1N!det⁡[ϕi(xj)]i,j=1N\psi(\mathbf{x}_1,\ldots,\mathbf{x}_N) = \frac{1}{\sqrt{N!}}\det\bigl[\phi_i(\mathbf{x}_j)\bigr]_{i,j=1}^N
Antisymmetric N-fermion state. The determinant structure ensures the Pauli exclusion principle: two rows equal ⟹ determinant = 0.
Bosonic Ladder Operator Algebra
[a^i, a^j†]=δij,[a^i,a^j]=[a^i†,a^j†]=0[\hat{a}_i,\,\hat{a}_j^\dagger] = \delta_{ij}, \qquad [\hat{a}_i,\hat{a}_j] = [\hat{a}_i^\dagger,\hat{a}_j^\dagger] = 0
Creation/annihilation operators for bosons in mode i. Fock states |n₁, n₂, …⟩ are built from the vacuum by (â†_i)^{n_i}/√(n_i!).
Fermionic Ladder Operator Algebra
{c^i, c^j†}=δij,{c^i,c^j}={c^i†,c^j†}=0\{\hat{c}_i,\,\hat{c}_j^\dagger\} = \delta_{ij}, \qquad \{\hat{c}_i,\hat{c}_j\} = \{\hat{c}_i^\dagger,\hat{c}_j^\dagger\} = 0
Anticommutation relations for fermionic operators. (ĉ†_i)² = 0 enforces the Pauli exclusion principle: at most one fermion per mode.
Many-Body Hamiltonian (second quantization)
H^=∑ijhijc^i†c^j+12∑ijklVijklc^i†c^j†c^lc^k\hat{H} = \sum_{ij}h_{ij}\hat{c}_i^\dagger\hat{c}_j + \frac{1}{2}\sum_{ijkl}V_{ijkl}\hat{c}_i^\dagger\hat{c}_j^\dagger\hat{c}_l\hat{c}_k
One-body (kinetic + single-particle potential) and two-body (interaction) terms in the occupation-number representation.
Field Operator
Ψ^(r)=∑kϕk(r) a^k,[Ψ^(r),Ψ^†(r′)]=δ(3)(r−r′)\hat{\Psi}(\mathbf{r}) = \sum_k\phi_k(\mathbf{r})\,\hat{a}_k, \qquad [\hat{\Psi}(\mathbf{r}),\hat{\Psi}^\dagger(\mathbf{r}^\prime)] = \delta^{(3)}(\mathbf{r}-\mathbf{r}^\prime)
Field operator annihilates a particle at position r. The commutation relation (bosons) / anticommutation relation (fermions) encodes quantum statistics in position space.

Density Matrix & Open Quantum Systems

Density Matrix (mixed state)
ρ^=∑kpk∣ψk⟩⟨ψk∣,pk≥0,Tr(ρ^)=1\hat{\rho} = \sum_k p_k|\psi_k\rangle\langle\psi_k|, \qquad p_k \geq 0, \quad \mathrm{Tr}(\hat{\rho}) = 1
Statistical ensemble of pure states |ψ_k⟩ with probabilities p_k. For a pure state ρ = |ψ⟩⟨ψ|; for a mixed state, ρ² ≠ ρ.
Expectation Value from Density Matrix
⟨A^⟩=Tr(ρ^A^)=∑n⟨n∣ρ^A^∣n⟩\langle\hat{A}\rangle = \mathrm{Tr}(\hat{\rho}\hat{A}) = \sum_{n}\langle n|\hat{\rho}\hat{A}|n\rangle
The trace is basis-independent. This formalism handles both quantum superpositions (off-diagonal ρ elements) and classical uncertainty (diagonal ρ).
Von Neumann Equation
iℏdρ^dt=[H^,ρ^]i\hbar\frac{d\hat{\rho}}{dt} = [\hat{H},\hat{\rho}]
Quantum analogue of Liouville's equation. Preserves purity: Tr(ρ²) is constant. Applies to closed systems only.
Von Neumann Entropy
S=−kB Tr(ρ^ln⁡ρ^)=−kB∑kλkln⁡λkS = -k_B\,\mathrm{Tr}(\hat{\rho}\ln\hat{\rho}) = -k_B\sum_k\lambda_k\ln\lambda_k
S = 0 for a pure state; S = k_B ln N for a maximally mixed N-level system. Equals the entanglement entropy for a bipartite pure state.
Bloch Sphere Representation
ρ^=12(1+r⋅σ),∣r∣≤1\hat{\rho} = \frac{1}{2}\bigl(\mathbf{1} + \mathbf{r}\cdot\boldsymbol{\sigma}\bigr), \qquad |\mathbf{r}|\leq 1
Any qubit state is a point inside (mixed) or on the surface of (pure) the unit Bloch sphere. Unitary evolution → rotations; decoherence → the vector shrinks toward the origin.
Lindblad Master Equation
dρ^dt=−iℏ[H^,ρ^]+∑kγk ⁣(L^kρ^L^k†−12{L^k†L^k,ρ^})\frac{d\hat{\rho}}{dt} = -\frac{i}{\hbar}[\hat{H},\hat{\rho}] + \sum_k\gamma_k\!\left(\hat{L}_k\hat{\rho}\hat{L}_k^\dagger - \frac{1}{2}\{\hat{L}_k^\dagger\hat{L}_k,\hat{\rho}\}\right)
Most general Markovian, trace-preserving, completely positive master equation. The jump operators L̂_k describe dissipation channels (spontaneous emission, dephasing, etc.).

WKB & Semiclassical Methods

WKB Wave Function
ψ(x)≈Cp(x)exp⁡ ⁣(±iℏ∫x ⁣p(x′) dx′),p(x)=2m(E−V(x))\psi(x) \approx \frac{C}{\sqrt{p(x)}}\exp\!\left(\pm\frac{i}{\hbar}\int^x\!p(x^\prime)\,dx^\prime\right), \quad p(x) = \sqrt{2m(E-V(x))}
Valid when the potential changes slowly on the scale of the de Broglie wavelength: |dp/dx| ≪ p²/ℏ. Breaks down near classical turning points.
WKB Tunneling Probability
T≈exp⁡ ⁣(−2ℏ∫ab ⁣2m(V(x)−E) dx)T \approx \exp\!\left(-\frac{2}{\hbar}\int_a^b\!\sqrt{2m(V(x)-E)}\,dx\right)
Probability of tunneling through a classically forbidden barrier between turning points a and b. The exponent is the imaginary action (instanton action in QFT).
Bohr-Sommerfeld Quantization
∮p(x) dx=2πℏ ⁣(n+12),n=0,1,2,…\oint p(x)\,dx = 2\pi\hbar\!\left(n+\frac{1}{2}\right), \quad n = 0,1,2,\ldots
Quantization condition for a periodic orbit with two soft classical turning points. The ½ arises from the Maslov index at each turning point.
Connection Formulas (linear turning point)
2Apcos⁡ ⁣(1ℏ∫xap dx′−π4)⟷A∣p∣e−1ℏ∫ax∣p∣dx′\frac{2A}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\int_x^a p\,dx^\prime - \frac{\pi}{4}\right) \longleftrightarrow \frac{A}{\sqrt{|p|}}e^{-\frac{1}{\hbar}\int_a^x|p|dx^\prime}
Patch between the oscillatory (classically allowed) and exponentially decaying (forbidden) WKB solutions across a turning point at x = a.
Berry Phase
γn(C)=i∮C⟨n(R)∣∇R∣n(R)⟩⋅dR\gamma_n(C) = i\oint_C\langle n(\mathbf{R})|\nabla_{\mathbf{R}}|n(\mathbf{R})\rangle\cdot d\mathbf{R}
Geometric phase accumulated by state |n⟩ as the Hamiltonian parameters R traverse a closed loop C in parameter space. Observable in interference experiments and central to topological physics.

AMO Physics — Key Equations

Electric Dipole Interaction Hamiltonian
H^dip=−d^⋅E(t)=−er^⋅E(t)\hat{H}_{\rm dip} = -\hat{\mathbf{d}}\cdot\mathbf{E}(t) = -e\hat{\mathbf{r}}\cdot\mathbf{E}(t)
Dominant atom-light interaction in the long-wavelength (dipole) approximation λ ≫ a₀. The full multipole expansion adds magnetic dipole, electric quadrupole, … terms.
Rabi Frequency
Ω=∣⟨e∣d^⋅ϵ^∣g⟩∣ E0ℏ=∣deg∣E0ℏ\Omega = \frac{|\langle e|\hat{\mathbf{d}}\cdot\hat{\boldsymbol{\epsilon}}|g\rangle|\,E_0}{\hbar} = \frac{|d_{eg}|E_0}{\hbar}
Coupling strength between ground |g⟩ and excited |e⟩ states driven by a field of amplitude E₀ and polarization ε̂. Sets the speed of coherent population transfer.
Two-Level Hamiltonian (Rotating Wave Approximation)
H^RWA=ℏ2(ΔΩΩ−Δ),Δ=ωL−ω0\hat{H}_{\rm RWA} = \frac{\hbar}{2}\begin{pmatrix}\Delta & \Omega \\ \Omega & -\Delta\end{pmatrix}, \qquad \Delta = \omega_L - \omega_0
In the rotating frame, slow terms survive (RWA drops counter-rotating terms ∼ e^{±2iω_L t}). Δ is the laser detuning; on resonance (Δ = 0) the Hamiltonian is purely off-diagonal.
Rabi Oscillation (population)
Pe(t)=Ω2Ωeff2sin⁡2 ⁣(Ωeff t2),Ωeff=Ω2+Δ2P_e(t) = \frac{\Omega^2}{\Omega_{\rm eff}^2}\sin^2\!\left(\frac{\Omega_{\rm eff}\,t}{2}\right), \qquad \Omega_{\rm eff} = \sqrt{\Omega^2+\Delta^2}
Probability of finding the atom in |e⟩ oscillates at the generalized Rabi frequency Ω_eff. On resonance (Δ = 0): P_e reaches 1 at t = π/Ω (π-pulse).
Optical Bloch Equations
ρ˙ee=iΩ2(ρge−ρeg)−Γρee,ρ˙ge=(iΔ−Γ2) ⁣ρge−iΩ2(ρee−ρgg)\dot{\rho}_{ee} = \frac{i\Omega}{2}(\rho_{ge}-\rho_{eg}) - \Gamma\rho_{ee}, \qquad \dot{\rho}_{ge} = \left(i\Delta - \frac{\Gamma}{2}\right)\!\rho_{ge} - \frac{i\Omega}{2}(\rho_{ee}-\rho_{gg})
Lindblad master equation for a two-level atom with spontaneous emission rate Γ. ρ_{ee} is the excited-state population; ρ_{ge} is the coherence (off-diagonal density matrix element).
Steady-State Excited Population
ρeess=Ω2/4Δ2+Γ2/4+Ω2/2=s/21+s+(2Δ/Γ)2\rho_{ee}^{\rm ss} = \frac{\Omega^2/4}{\Delta^2 + \Gamma^2/4 + \Omega^2/2} = \frac{s/2}{1+s+(2\Delta/\Gamma)^2}
Lorentzian in detuning Δ with power-broadened linewidth √(1 + s) Γ. Saturation parameter s = 2Ω²/Γ² = I/I_sat; maximum excited population is ½ (inversion impossible for two-level system).
Einstein A Coefficient (Spontaneous Emission Rate)
A21=Γ=ω03∣deg∣23πϵ0ℏc3=e2ω033πϵ0me2c3ω02meω02ℏ∣⟨e∣x^∣g⟩∣2A_{21} = \Gamma = \frac{\omega_0^3|d_{eg}|^2}{3\pi\epsilon_0\hbar c^3} = \frac{e^2\omega_0^3}{3\pi\epsilon_0 m_e^2 c^3\omega_0^2}\frac{m_e\omega_0}{2\hbar}|\langle e|\hat{x}|g\rangle|^2
Spontaneous emission rate; equals the natural linewidth Γ. The lifetime τ = 1/A₂₁. Scales as ω³|d|², so UV transitions decay much faster than microwave transitions.
Einstein B Coefficients & Detailed Balance
B12=π2c3ℏω3A21,g1B12=g2B21B_{12} = \frac{\pi^2 c^3}{\hbar\omega^3}A_{21}, \qquad g_1 B_{12} = g_2 B_{21}
B_{12} (absorption) and B_{21} (stimulated emission) are related to A₂₁ by the photon mode density. g₁, g₂ are degeneracies. Together they reproduce the Planck distribution at thermal equilibrium.
Hyperfine Hamiltonian
H^hf=Ahf I^⋅J^,⟨I^⋅J^⟩=ℏ22[F(F+1)−I(I+1)−J(J+1)]\hat{H}_{\rm hf} = A_{\rm hf}\,\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}, \qquad \langle\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}\rangle = \frac{\hbar^2}{2}[F(F+1)-I(I+1)-J(J+1)]
Magnetic interaction between nuclear spin I and electron angular momentum J. F = I + J is the total angular momentum quantum number, |I−J| ≤ F ≤ I+J.
Anomalous Zeeman Effect (Landé g-factor)
ΔE=gJmJμBB,gJ=1+J(J+1)+S(S+1)−L(L+1)2J(J+1)\Delta E = g_J m_J \mu_B B, \qquad g_J = 1 + \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}
Splitting of a level with total angular momentum J in a magnetic field B. g_J = 1 for pure orbital; g_J = 2 for pure spin. The anomalous factor arises from the electron's spin g ≈ 2.
AC Stark Shift (Light Shift)
Uac=−3πc22ω03 ΓΔ I(r)=ℏΩ24Δ(∣Δ∣≫Γ)U_{\rm ac} = -\frac{3\pi c^2}{2\omega_0^3}\,\frac{\Gamma}{\Delta}\,I(\mathbf{r}) = \frac{\hbar\Omega^2}{4\Delta} \quad (|\Delta| \gg \Gamma)
Energy shift of an atomic level in an off-resonant laser field of intensity I and detuning Δ = ω_L − ω_0. Red-detuned (Δ < 0): atom is attracted to field maxima (optical dipole trap). Blue-detuned: repelled.
Doppler & Recoil Temperature Limits
TD=ℏΓ2kB(Doppler),TR=ℏ2k2mkB=2ERkB(recoil)T_D = \frac{\hbar\Gamma}{2k_B} \quad (\text{Doppler}), \qquad T_R = \frac{\hbar^2 k^2}{m k_B} = \frac{2E_R}{k_B} \quad (\text{recoil})
Fundamental laser cooling limits. T_D (∼ μK) is the Doppler cooling limit; sub-Doppler cooling (Sisyphus, EIT) reaches T_R (∼ 100 nK). E_R = ℏ²k²/2m is the single-photon recoil energy.
Jaynes-Cummings Hamiltonian
H^JC=ℏωca^†a^+ℏω02σ^z+ℏg(a^σ^++a^†σ^−)\hat{H}_{\rm JC} = \hbar\omega_c\hat{a}^\dagger\hat{a} + \frac{\hbar\omega_0}{2}\hat{\sigma}_z + \hbar g\bigl(\hat{a}\hat{\sigma}_+ + \hat{a}^\dagger\hat{\sigma}_-\bigr)
Exactly solvable model for a two-level atom coupled to a single cavity mode with coupling g. Predicts vacuum Rabi splitting 2g, photon blockade, and collapses & revivals of Rabi oscillations.