Formula sheet · PU-401

Advanced Quantum Theory — formula sheet

Every boxed result from the 16 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.

D-350

Interaction Picture and the Dyson Series

\[ \hat{U}_I(t,t_0)=\mathcal{T}\exp\!\left[-\frac{i}{\hbar}\int_{t_0}^{t}\hat{V}_I(t')\,dt'\right]=\sum_{n=0}^{\infty}\frac{1}{n!}\Big(\tfrac{-i}{\hbar}\Big)^{n}\!\int_{t_0}^{t}\!\!\cdots\!\!\int_{t_0}^{t}\!\mathcal{T}\big[\hat{V}_I(t_1)\cdots\hat{V}_I(t_n)\big]\,dt_1\cdots dt_n \]

The full-picture propagator factorises as \( \hat{U}(t,t_0)=\hat{U}_0(t,t_0)\,\hat{U}_I(t,t_0) \): free evolution times a correction built entirely from the perturbation. The \( n \)-th term is the amplitude for the system to feel the perturbation \( n \) times, at times \( t_1>t_2>\cdots>t_n \), the earliest interaction acting first on the state — time-ordering enforces this causal sequence. Setting \( \hat{V}=0 \) collapses the series to \( \mathbb{1} \), recovering pure free evolution.

D-227

Fermi's Golden Rule

\[ \boxed{\;\Gamma_{i\to f} \;=\; \frac{2\pi}{\hbar}\,\bigl|\langle f|\hat{V}|i\rangle\bigr|^{2}\,\rho(E_f)\Big|_{E_f=E_i}\;} \]

The transition rate is constant in time (hence a well-defined lifetime \(\tau=1/\Gamma\)). It is proportional to the squared coupling strength \(|\langle f|\hat V|i\rangle|^2\) — so a weak perturbation gives a slow rate — and to how many final states are available at the conserved energy, \(\rho(E_i)\). Energy is conserved between initial and final states, enforced by the delta function that emerged in the long-time limit.

D-351

Semiclassical Radiation and Dipole Selection Rules

\[ \boxed{\ \Delta\ell=\pm1,\qquad \Delta m=0,\pm1,\qquad \pi_i\pi_f=-1\ (\text{parity change}),\qquad \Delta s=0\ } \]

An electric-dipole photon carries one unit of angular momentum and odd parity. The atom must therefore change its orbital angular momentum by exactly one unit (\(\Delta\ell=\pm1\)), change parity, and change its magnetic quantum number by at most one unit (\(\Delta m=0\) for light linearly polarized along \(z\), \(\Delta m=\pm1\) for the two circular polarizations in the \(xy\)-plane). The spin is a spectator, \(\Delta s=0\). Any pair of hydrogen levels not meeting all of these is E1-forbidden and produces no first-order line.

D-352

Einstein A and B Coefficients

\[ g_1 B_{12} = g_2 B_{21}, \qquad \frac{A_{21}}{B_{21}} = \frac{\hbar\,\omega^3}{\pi^2 c^3} \]

Absorption and stimulated emission are the same process run in opposite directions: their rate constants are equal up to the degeneracy weighting. Spontaneous emission is tied to stimulated emission by a universal factor that scales as the cube of the transition frequency — so a single atomic matrix element controls all three, and high-frequency transitions radiate spontaneously far faster than low-frequency ones.

D-354

The Wigner-Eckart Theorem

\[ \boxed{\;\langle \alpha'\, j'\, m'\lvert \hat{T}^{(k)}_q\rvert \alpha\, j\, m\rangle = \langle j\, m;\, k\, q \,\vert\, j'\, m'\rangle\;\frac{\langle \alpha'\, j'\Vert \hat{T}^{(k)}\Vert \alpha\, j\rangle}{\sqrt{2j'+1}}\;} \]

Every dependence on the magnetic quantum numbers \(m,q,m'\) — that is, on the orientation of the states and operator in space — lives entirely in the Clebsch–Gordan coefficient \(\langle jm;kq\vert j'm'\rangle\), a pure number fixed by group theory. All the physics of the specific operator and states is compressed into one orientation-independent scalar, the reduced matrix element \(\langle\alpha'j'\Vert\hat T^{(k)}\Vert\alpha j\rangle\). Two selection rules are immediate: \(m'=m+q\) and the triangle condition \(\lvert j-k\rvert\le j'\le j+k\).

D-356

WKB Approximation and Connection Formulas

\[ \psi_{\text{WKB}}(x)=\frac{C_{\pm}}{\sqrt{|p(x)|}}\exp\!\left(\pm\frac{i}{\hbar}\!\int^{x}\! p\,dx'\right),\qquad p=\sqrt{2m(E-V)} \] \[ \frac{2}{\sqrt{p}}\cos\!\Big(\tfrac{1}{\hbar}\!\int_x^a p\,dx'-\tfrac{\pi}{4}\Big)\ \longleftrightarrow\ \frac{1}{\sqrt{|p|}}e^{-\frac{1}{\hbar}\int_a^x |p|\,dx'} \]

The semiclassical wavefunction is a wave whose phase accumulates the classical action \(\tfrac{1}{\hbar}\int p\,dx\) and whose amplitude \(\propto 1/\sqrt{p}\) piles up probability where the particle moves slowly — exactly the classical dwell-time distribution. Across a turning point the connection formula ties the decaying barrier tail to an oscillation lagging by \(\pi/4\). Applied between two turning points, the two \(\pi/4\) shifts add to the half-integer in Bohr–Sommerfeld quantisation \(\int_a^b p\,dx=(n+\tfrac12)\pi\hbar\).

D-357

Gamow Tunneling and Alpha Decay

\[ T\approx e^{-G},\quad G=\frac{2k}{\hbar}\sqrt{\frac{2m}{E}}\Big[\arccos\!\sqrt{\tfrac{R}{b}}-\sqrt{\tfrac{R}{b}(1-\tfrac{R}{b})}\Big]\ \xrightarrow{R\ll b}\ 2\pi\eta,\quad \lambda\approx\frac{v_{\mathrm{in}}}{2R}\,e^{-G} \]

The escape probability per wall-collision is the exponential of a negative action accumulated across the classically forbidden region between the nuclear surface \(R\) and the Coulomb outer turning point \(b=2Z_d e^2/4\pi\epsilon_0 E\). Because that action is huge (\(G\sim30\)–\(90\)) and enters exponentially, a factor-of-two change in \(E\) swings the half-life by dozens of decades. The leading piece \(2\pi\eta\) depends only on daughter charge and alpha speed; the negative correction rewards a wider nucleus (larger \(R\)), which thins the barrier it must cross.

D-358

The Adiabatic Theorem

\[ \boxed{\;\; \frac{\hbar\,\big|\langle n(t)|\dot H(t)|m(t)\rangle\big|}{\big(E_n(t)-E_m(t)\big)^2}\;\ll\;1 \quad (\forall\, n\neq m) \;\;\Longrightarrow\;\; |\psi(t)\rangle \simeq e^{i\theta_m(t)}\,e^{i\gamma_m(t)}\,|m(t)\rangle \;\;} \]

If the Hamiltonian is stirred slowly compared with the intrinsic oscillation timescale \(\hbar/|E_n-E_m|\) set by the gap, a system launched in eigenstate \(m\) is carried along the instantaneous eigenstate \(|m(t)\rangle\), acquiring only two phases: the dynamical phase \(\theta_m=-\frac1\hbar\!\int E_m\,dt'\) (the naive "energy times time") and the geometric phase \(\gamma_m=i\!\int\langle m|\dot m\rangle\,dt'\) that depends only on the path traced in parameter space, not on how fast it is traversed.

D-359

Berry's Geometric Phase

\[ \gamma_n(C) = \oint_C i\,\langle n|\nabla_{\mathbf{R}} n\rangle\cdot d\mathbf{R} = \iint_S \mathbf{F}_n\cdot d\mathbf{S} \]

The extra phase left on an adiabatically transported eigenstate after one closed loop equals the circulation of the Berry connection around the loop, equivalently the flux of the Berry curvature through any capping surface. It is geometric — set entirely by the shape of \(C\) in parameter space and the local structure of the eigenstates — not by the clock. The curvature \(\mathbf{F}_n\) acts as a "magnetic field" living in parameter space whose sources are the degeneracies of the spectrum.

D-360

The Lippmann-Schwinger Equation

\[ \psi_{\vec{k}}^{+}(\vec{r}\,) = e^{i\vec{k}\cdot\vec{r}} - \frac{m}{2\pi\hbar^2}\int \frac{e^{ik|\vec{r}-\vec{r}\,{}'|}}{|\vec{r}-\vec{r}\,{}'|}\,V(\vec{r}\,{}')\,\psi_{\vec{k}}^{+}(\vec{r}\,{}')\,d^3r' \]

The full scattering wavefunction equals the incident plane wave plus a coherent superposition of outgoing spherical wavelets \( e^{ikR}/R \), each emitted from a volume element at \( \vec{r}\,{}' \) with strength proportional to the local potential \( V(\vec{r}\,{}') \) and the local wavefunction \( \psi^{+}(\vec{r}\,{}') \) there. It is Huygens' principle for quantum scattering. The equation is implicit — \( \psi^{+} \) appears on both sides — which is exactly what encodes multiple scattering.

D-361

The Born Approximation

\[ f^{(1)}(\mathbf q)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\mathbf q\cdot\mathbf r'}\,V(\mathbf r')\,d^{3}r',\qquad \frac{d\sigma}{d\Omega}=\bigl|f^{(1)}(\mathbf q)\bigr|^{2},\qquad \mathbf q=\mathbf k-\mathbf k',\; q=2k\sin\frac{\theta}{2} \]

The scattering amplitude at scattering angle \(\theta\) is minus a constant times the spatial Fourier transform of the potential, sampled at the momentum transfer \(\hbar\mathbf q\). Large-angle scattering probes short-wavelength (small-scale) features of \(V\); forward scattering (\(q\to 0\)) sees only the volume integral \(\int V\,d^{3}r\). The cross section, being the square, loses the phase — so scattering measures the modulus of the Fourier transform, and reconstructing \(V\) requires a phase assumption or additional data.

D-362

Partial-Wave Analysis and Phase Shifts

\[ f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)\,e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta),\qquad f_\ell=\frac{e^{2i\delta_\ell}-1}{2ik} \]

Every effect of a central, finite-range potential on elastic scattering is stored in the set of real phase shifts \(\{\delta_\ell(k)\}\). The \(\ell\)-th outgoing partial wave is the free wave advanced (or retarded) in phase by \(\delta_\ell\); an attractive potential pulls the wave inward, \(\delta_\ell>0\), a repulsive one pushes it out, \(\delta_\ell<0\). Integrating cross-terms gives the total elastic cross section \(\sigma=\frac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell\), and the forward direction obeys the optical theorem \(\sigma=\frac{4\pi}{k}\,\mathrm{Im}\,f(0)\).

D-363

The Optical Theorem

\[ \boxed{\;\sigma_{\text{tot}} = \dfrac{4\pi}{k}\,\operatorname{Im} f(0)\;}\qquad \operatorname{Im} f(0)=\frac{1}{k}\sum_{l}(2l+1)\sin^2\delta_l \]

The total probability scattered out of the incident beam is set entirely by how much the forward-going wave is attenuated — quantified by the imaginary (out-of-phase) part of \(f(0)\). A purely real forward amplitude would remove nothing from the beam and hence describe zero total cross section; scattering necessarily gives the forward amplitude a positive imaginary part. This is the shadow behind every scatterer.

D-229

Identical Particles and Exchange Symmetry

\[ \boxed{\;\Psi_{\text{F}}=\frac{1}{\sqrt{N!}}\det[\varphi_i(x_j)],\qquad E_\pm=\varepsilon_a+\varepsilon_b+J\pm K,\qquad \Delta E_{\text{S–T}}=2K\;} \]

Nature admits only the fully symmetric (boson) or fully antisymmetric (fermion) sector; fermions are described by Slater determinants, which vanish when two orbitals coincide (Pauli). Even though the Coulomb interaction knows nothing about spin, antisymmetry forces the spatial part to depend on the spin state, so the two-electron energy carries a spin-dependent piece \(\pm K\). The exchange integral \(K\) — a positive electrostatic quantity — pushes the parallel-spin (triplet) configuration below the antiparallel (singlet) one by \(2K\).

D-364

The Density Matrix and von Neumann Equation

\[ \hat{\rho}=\sum_i p_i|\psi_i\rangle\langle\psi_i|,\qquad \langle\hat{A}\rangle=\operatorname{Tr}(\hat{\rho}\hat{A}),\qquad i\hbar\,\partial_t\hat{\rho}=[\hat{H},\hat{\rho}],\qquad \hat{\rho}_A=\operatorname{Tr}_B\hat{\rho}. \]

The density operator is the state of any quantum system for which we lack a wavefunction. Its three defining properties — Hermitian, positive, unit trace — encode "these are genuine probabilities." Expectation values are traces against it; it evolves by the commutator with \(\hat{H}\) (the operator analogue of the classical Liouville equation, with an extra minus sign relative to Heisenberg observables); and the accessible state of a part of a larger system is obtained by tracing out the rest. Purity is measured by \(\operatorname{Tr}\hat{\rho}^2\le 1\), with equality iff the state is pure.

D-365

The Feynman Path Integral

\[ K(x_f,t_f;x_i,t_i) = \int_{x_i}^{x_f} \mathcal{D}x\; \exp\!\left( \frac{i}{\hbar}\int_{t_i}^{t_f} L(x,\dot{x})\,dt \right) \;\xrightarrow[\hbar\to 0]{}\; A\,e^{\,iS_{\text{cl}}/\hbar} \]

The amplitude to go from \( (x_i,t_i) \) to \( (x_f,t_f) \) is obtained by giving every conceivable trajectory the same modulus but a phase equal to its action in units of \( \hbar \), then adding. Neighbouring paths near an extremum of \( S \) share almost the same phase and add constructively; far from an extremum they cancel. As \( \hbar\to 0 \) only the classical path survives, and Hamilton's principle \( \delta S = 0 \) is the statement that this path is where the phases line up.

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