Formula sheet · PU-301

Quantum Mechanics II — formula sheet

Every boxed result from the 17 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-213

Spectral Theorem for Hermitian Observables

\[ \hat{A}=\sum_n a_n\,\hat{P}_n,\qquad a_n\in\mathbb{R},\qquad \hat{P}_n\hat{P}_m=\delta_{nm}\hat{P}_n,\qquad \sum_n\hat{P}_n=\hat{\mathbb{1}} \]

A Hermitian observable is fully specified by a set of real eigenvalues (the measurable values) and mutually orthogonal projectors onto their eigenspaces. Any state expands as \(|\psi\rangle=\sum_n\hat{P}_n|\psi\rangle\), and the Born rule assigns outcome \(a_n\) the probability \(\langle\psi|\hat{P}_n|\psi\rangle=\lVert\hat{P}_n\psi\rVert^{2}\). Because the projectors resolve the identity, these probabilities sum to \(\langle\psi|\psi\rangle=1\).

D-215

Angular Momentum Spectrum from Commutators

\[ \hat{J}^2\lvert j,m\rangle=\hbar^2\,j(j+1)\,\lvert j,m\rangle,\qquad \hat{J}_z\lvert j,m\rangle=\hbar\,m\,\lvert j,m\rangle \]\[ j\in\Bigl\{0,\tfrac12,1,\tfrac32,\dots\Bigr\},\qquad m\in\{-j,-j+1,\dots,j-1,j\} \]\[ \hat{J}_\pm\lvert j,m\rangle=\hbar\sqrt{j(j+1)-m(m\pm1)}\;\lvert j,m\pm1\rangle \]

The magnitude of angular momentum is quantised as \(\lvert\vec{J}\rvert=\hbar\sqrt{j(j+1)}\), and its projection on any chosen axis takes the \(2j+1\) equally spaced values \(\hbar m\). The strict inequality \(j(j+1)>j^2\) means the vector can never point exactly along \(z\) — a purely algebraic statement of the uncertainty between components. The half-integer branch, forbidden for orbital motion, is what the algebra permits and what spin realises.

D-216

Spherical Harmonics as Angular Momentum Eigenstates

\[ \boxed{\ \hat{L}^2\,Y_\ell^{m}=\hbar^2\,\ell(\ell+1)\,Y_\ell^{m},\qquad \hat{L}_z\,Y_\ell^{m}=\hbar m\,Y_\ell^{m},\qquad Y_\ell^{m}(\theta,\phi)=(-1)^m\sqrt{\tfrac{2\ell+1}{4\pi}\tfrac{(\ell-m)!}{(\ell+m)!}}\;P_\ell^{m}(\cos\theta)\,e^{im\phi}\ } \]

The angular part of any central-force wavefunction is one of a discrete grid of functions labelled by two integers: \(\ell\ge0\) sets the total angular momentum magnitude \(\hbar\sqrt{\ell(\ell+1)}\), and \(m\) (with \(|m|\le\ell\)) sets its projection \(\hbar m\) on the chosen axis. The \(e^{im\phi}\) factor carries the azimuthal circulation; \(P_\ell^m(\cos\theta)\) shapes the polar profile and has exactly \(\ell-|m|\) nodes in \(\theta\).

D-217

Separation of the Central-Potential Schrodinger Equation

\[-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2}+V_{\text{eff}}(r)\,u=E\,u,\qquad V_{\text{eff}}(r)=V(r)+\frac{\hbar^2\ell(\ell+1)}{2mr^2},\qquad u=rR,\ \ u(0)=0\]

The full 3D problem for a central potential collapses to a single one-dimensional Schrodinger equation for \(u(r)\) on the half-line, moving in the effective potential \(V_{\text{eff}}\). The only trace of three-dimensionality is (i) the boundary condition \(u(0)=0\), and (ii) the repulsive centrifugal term \(\hbar^2\ell(\ell+1)/2mr^2\), which is added to the true potential and grows with angular momentum. The energy \(E\) and the quantum numbers \(\ell,m\) are inherited unchanged; \(m\) does not appear in the radial equation, giving the \((2\ell+1)\)-fold degeneracy.

D-218

Hydrogen Atom Bound-State Spectrum

\[ E_n=-\frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2}\,\frac{1}{n^2}=-\frac{E_{\mathrm R}}{n^2},\qquad E_{\mathrm R}=\tfrac12\alpha^2\mu c^2\approx 13.606\ \mathrm{eV},\qquad n=1,2,3,\dots \]

D-219

Spin-1/2 and the Pauli Matrices

\[ \hat{S}_i=\frac{\hbar}{2}\sigma_i, \qquad \sigma_i\sigma_j=\delta_{ij}\mathbf{1}+i\varepsilon_{ijk}\sigma_k, \qquad \hat{U}(\theta,\hat{\mathbf{n}})=\cos\tfrac{\theta}{2}\,\mathbf{1}-i\sin\tfrac{\theta}{2}\,\hat{\mathbf{n}}\cdot\vec{\sigma}. \]

The three Pauli matrices are the generators of \( SU(2) \) in its fundamental representation; scaled by \( \hbar/2 \) they are the physical spin components of a spin-\( \tfrac12 \) particle. Their product rule packages orthonormality (the \( \delta_{ij}\mathbf{1} \) piece, from squaring) and the rotation algebra (the \( i\varepsilon_{ijk}\sigma_k \) piece, from commuting) into one line. The rotation operator rotates a spinor by turning the "spin vector" \( \langle\vec\sigma\rangle \) through the ordinary angle \( \theta \) on the Bloch sphere, while the state itself only advances by \( \theta/2 \) — so a \( 2\pi \) physical rotation yields \( \hat{U}=-\mathbf{1} \).

D-220

Addition of Angular Momenta

\[ \mathcal{H}_{j_1}\otimes\mathcal{H}_{j_2} \;=\; \bigoplus_{j=|j_1-j_2|}^{j_1+j_2}\mathcal{H}_j, \qquad |j\,m\rangle=\!\!\sum_{m_1+m_2=m}\!\!\langle j_1 m_1;j_2 m_2|j\,m\rangle\,|j_1 m_1\rangle|j_2 m_2\rangle. \]

Two angular momenta of sharp magnitude combine into a ladder of total-spin multiplets running from the difference to the sum of the individual quantum numbers, each occurring exactly once. The Clebsch-Gordan coefficients are the fixed, real amplitudes (given Condon-Shortley phases) that write each coupled state as a superposition of uncoupled product states sharing the same total \( m=m_1+m_2 \). The dimensions balance: \( \sum_j(2j+1)=(2j_1+1)(2j_2+1) \).

D-221

Non-Degenerate Perturbation Theory

\[ E_n = E_n^{(0)} + \langle n^{(0)} \lvert \hat{V} \rvert n^{(0)} \rangle + \sum_{m \neq n} \frac{\lvert \langle m^{(0)} \lvert \hat{V} \rvert n^{(0)} \rangle \rvert^2}{E_n^{(0)} - E_m^{(0)}} + \cdots \]

D-222

Degenerate Perturbation Theory

\[ \det\!\big(W - E^{(1)}\mathbf{1}\big) = 0, \qquad W_{ab} = {}^{(0)}\langle n,a|\,\hat{H}'\,|n,b\rangle^{(0)}, \qquad |n,\alpha\rangle^{(0)} = \sum_b c_b^{(\alpha)}\,|n,b\rangle^{(0)} \]

To split a degenerate level, build the matrix of the perturbation in any orthonormal basis of the degenerate shell and diagonalize it. Its \(g\) eigenvalues are the first-order energy corrections \(E_{n,\alpha}^{(1)}\); its eigenvectors are the unique "good" combinations of the degenerate states that acquire definite energies. If the eigenvalues are distinct, the perturbation has fully lifted the degeneracy at first order.

D-223

Fine Structure of Hydrogen

\[ E_{nj}=-\frac{mc^2\alpha^2}{2n^2}\left[\,1+\frac{\alpha^2}{n^2}\left(\frac{n}{\,j+\tfrac12\,}-\frac34\right)\right],\qquad E^{\text{weak}}_{Z}=\mu_B\,g_J\,B\,m_j \]

The Bohr energy acquires a fractional correction of order \( \alpha^2\approx 5.3\times10^{-5} \) that depends on \( n \) and \( j \) but not on \( \ell \): states of equal \( n \) and \( j \) (e.g. \( 2S_{1/2} \) and \( 2P_{1/2} \)) remain degenerate at this order. Larger \( j \) sits higher. Switching on \( B \) splits each \( j \)-level into \( 2j+1 \) uniformly spaced lines whose spacing \( \mu_B g_J B \) carries the Landé factor, so different terms split at visibly different rates — the anomalous Zeeman effect.

D-224

Rayleigh-Ritz Variational Principle

\[ E[\psi] = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle} \ge E_0, \qquad \text{equality} \iff \hat{H}|\psi\rangle = E_0 |\psi\rangle \]

The average energy measured in any state you can write down is at least the true ground-state energy. A trial wavefunction can never fool the Hamiltonian into reporting an energy below the floor; the best you can do is hit the floor exactly, which happens precisely when the trial state is the ground state. Minimising \( E[\psi] \) over a family therefore squeezes the estimate down towards \( E_0 \) from above.

D-225

WKB Approximation and Bohr-Sommerfeld Quantization

\[ \boxed{\;\displaystyle\int_a^b \sqrt{2m\bigl(E_n-V(x)\bigr)}\,dx=\left(n+\tfrac12\right)\pi\hbar\;\;\Longleftrightarrow\;\;\oint p\,dx=\left(n+\tfrac12\right)h\;}\]

The area enclosed by the classical orbit in phase space \((x,p)\) is quantised in half-integer units of Planck's constant \(h\). Each energy level "owns" a phase-space cell of area \(h\); the extra half-cell \(\tfrac12 h\) is the semiclassical remnant of zero-point motion, arising because each smooth turning point costs a \(\pi/4\) reflection phase. The integer \(n\) counts the nodes of the interior wavefunction.

D-226

Time-Dependent Perturbation Theory

\[ c_f^{(1)}(t) = -\frac{i}{\hbar}\int_{t_0}^{t}\langle f|\hat V(t')|i\rangle\, e^{i\omega_{fi}t'}\,dt', \qquad \omega_{fi}=\frac{E_f-E_i}{\hbar} \]

The amplitude to end up in \(|f\rangle\) is the coupling matrix element \(\langle f|\hat V|i\rangle\) accumulated over time, but each instant is weighted by the phase \(e^{i\omega_{fi}t'}\) at the Bohr transition frequency \(\omega_{fi}\). Fourier-like, the transition is efficient only when \(\hat V(t)\) contains frequency components near \(\omega_{fi}\) — this is the origin of resonance. The transition probability is \(P_{i\to f}(t)=|c_f^{(1)}(t)|^2\).

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-228

Electric Dipole Transitions and Selection Rules

\[ \Gamma_{fi} = \frac{e^2\,\omega_{fi}^3}{3\pi\varepsilon_0\hbar c^3}\,\big|\langle f|\vec r|i\rangle\big|^2, \qquad \boxed{\;\Delta\ell = \pm1,\quad \Delta m = 0,\pm1,\quad \text{parity flips}\;} \]

An excited atom radiates at a rate set by the cube of the transition frequency times the squared dipole matrix element. The transition is dead unless the orbital angular momentum changes by exactly one unit and the parity reverses; the magnetic quantum number may change by at most one, the deficit carried off as the photon's spin projection. In full \(jj\)/fine-structure notation the companion rules are \(\Delta J = 0,\pm1\) (but \(J=0\nrightarrow J=0\)), \(\Delta m_J = 0,\pm1\), and \(\Delta S = 0\) in LS coupling.

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-230

Born Approximation and Partial Waves

\[ \frac{d\sigma}{d\Omega}=|f(\theta,\phi)|^2,\qquad f_{\mathrm B}(\mathbf q)=-\frac{m}{2\pi\hbar^2}\int e^{i\mathbf q\cdot\mathbf r'}V(\mathbf r')\,d^3r',\qquad \sigma_{\text{tot}}=\frac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell=\frac{4\pi}{k}\,\mathrm{Im}\,f(0) \]

The scattered intensity in a direction is the squared amplitude \(f\). In the Born (weak, fast) regime that amplitude is simply the Fourier transform of the potential evaluated at the momentum transferred to the particle: forward scattering (\(q\to0\)) probes the volume integral of \(V\), while large-angle scattering (\(q\) large) probes short-distance structure. The exact, complementary statement is the partial-wave sum, where all the physics of channel \(\ell\) is compressed into one real number \(\delta_\ell\); the optical theorem then fixes the total cross-section by the forward amplitude alone, a bookkeeping of the flux removed from the incident beam.

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