Perturbation Theory and Transitions

Perturbation Theory and Quantum Transitions

When Is an Exact Solution Impossible?

Exact solutions exist only for a few problems: the harmonic oscillator, the hydrogen atom, a particle in a rectangular well. Real-world problems—atoms in fields, molecules, interacting particles—do not have analytical solutions. Perturbation theory is a systematic method for finding approximate solutions when a problem is "similar" to an exactly solvable one.

Idea: if we know how to solve a problem with Hamiltonian $\hat{H}_0$, and the real Hamiltonian is $\hat{H} = \hat{H}_0 + \lambda\hat{H}'$ ($\lambda$ is small), then solutions are sought in the form of series expansions in $\lambda$.

Stationary Perturbation Theory

Formulation: $\hat{H} = \hat{H}_0 + \lambda\hat{H}'$, where $\lambda\hat{H}'$ is a small perturbation. The eigenvectors $|n^0\rangle$ and eigenvalues $E_n^0$ of $\hat{H}_0$ are known. To find: $E_n(\lambda) = E_n^0 + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \ldots$

First-order correction to the energy:

$ E_n^{(1)} = \langle n^0|\hat{H}'|n^0\rangle $

This is the matrix element of the perturbation evaluated on the unperturbed state. Physically: "the average correction to the energy due to the perturbation."

Correction to the wavefunction:

$ |n^{1}\rangle = \sum_{k \neq n} \frac{\langle k^0|\hat{H}'|n^0\rangle}{E_n^0 - E_k^0} \cdot |k^0\rangle $

Denominator: $E_n^0 - E_k^0$. If levels are close—the correction is large ("strong mixing"). If levels are far apart—the mixing is small.

Second-order correction to the energy:

$ E_n^{(2)} = \sum_{k \neq n} \frac{|\langle k^0|\hat{H}'|n^0\rangle|^2}{E_n^0 - E_k^0} $

The second order always lowers the ground state (denominator $E_0^0 - E_k^0 < 0$ for $k > 0$, numerator

gt; 0$)—polarizability is always positive.

Numerical Example: Stark Effect

Hydrogen atom in an electric field $Ez$: perturbation $\hat{H}' = eEz$.

For the ground state $\psi_{100}$: $E_1^{(1)} = \langle 1s|eEz|1s\rangle = eE\langle z\rangle_{100} = 0$ (since $z$ is odd and $|\psi_{100}|^2$ is even in $z \rightarrow$ the integral is zero).

Second order: $E_1^{(2)} = -\alpha E^2 / 2$, where $\alpha = e^2 \sum_{k \neq 1s} |\langle k|z|1s\rangle|^2 / (E_k - E_1)$. This is the electric polarizability: $\alpha(\mathrm{H}) \approx 4.5a_0^3 \approx 0.667 \times 10^{-30}~\text{m}^3$.

Stark effect: energy shift $\Delta E = -\alpha E^2 / 2$. For $E = 10^7~\text{V/m}$: $\Delta E \approx 3.3 \times 10^{-4}~\text{eV}$—small but measurable.

Time-Dependent Perturbation Theory and Fermi's Golden Rule

Problem: Hamiltonian depends on time: $\hat{H}(t) = \hat{H}_0 + V(t)$ for $t > 0$. Initial state: $|\psi(0)\rangle = |i\rangle$. Find the probability of transition to $|f\rangle$.

Transition amplitude in first order:

$ c_{i \rightarrow f}(t) = \frac{1}{i\hbar} \int_0^t \langle f|V(t')|i\rangle e^{i\omega_{fi} t'} dt' $

where $\omega_{fi} = (E_f - E_i)/\hbar$—the difference of Bohr frequencies.

For a constant perturbation $V$: as $t \rightarrow \infty$, the transition probability per unit time:

$ \Gamma_{i \rightarrow f} = \frac{2\pi}{\hbar} |\langle f|V|i\rangle|^2 \rho(E_i) $

This is Fermi's golden rule. Transition rate: proportional to the square of the matrix element and the density of final states $\rho(E_i)$.

Analysis: $|\langle f|V|i\rangle|^2$—the "matrix element" of the perturbation between initial and final states. The square: interference contribution (amplitude × conjugate). $\rho(E_i)$—the density of final states at energy $E_i$: the more "places" the system can transition to, the faster the transition.

Applications of the Golden Rule

Optical transitions: An atom absorbs a photon of frequency $\omega$. $V = -er \cdot E_0$. The matrix element $\langle f|er|i\rangle$ is the dipole matrix element. Selection rules: $\Delta l = \pm1$ (since $\langle f|r|i\rangle \neq 0$ only if $l$ changes by 1). This explains which transitions are "allowed" in atomic spectra.

Nuclear $\beta$-decay: neutron $\rightarrow$ proton + electron + antineutrino. The weak interaction is the perturbation. Decay rate by the golden rule determines the half-life.

Semiconductors: transitions between bands during photon absorption. The absorption threshold is the band gap width. In silicon $E_g = 1.1~\text{eV} \rightarrow \lambda_{thresh} = hc/E_g \approx 1100~\text{nm}$ (infrared)—therefore silicon solar cells absorb the entire visible light spectrum and part of the infrared.

Fine Structure and the Real Spectrum of Hydrogen

In addition to the main spectrum, the hydrogen atom exhibits fine structure—level splitting due to three effects:

1. Spin-orbit interaction: $\hat{H}_{SO} \propto \hat{L} \cdot \hat{S}$. Shift $\sim \alpha^4 m_e c^2$ ($\alpha = 1/137$—the fine-structure constant)
2. Relativistic correction to kinetic energy
3. Lamb shift (quantum electrodynamic)

Total splitting of the $2p$ levels ($j=1/2$ and $j=3/2$): $\Delta E \approx 4.5 \times 10^{-5}~\text{eV}$—this is the fine structure of the H$\alpha$ line.

Perturbation Theory in Atomic Physics and Quantum Electrodynamics

The perturbation method is the main computational tool in real quantum physics. The fine structure of the hydrogen spectrum is explained by relativistic corrections and spin-orbit interaction, both calculated by first-order perturbation theory. Quantum Electrodynamics (QED)—the most accurate theory in science—is based on a perturbative expansion in powers of the fine-structure constant $\alpha \approx 1/137$. The anomalous magnetic moment of the electron has been computed up to the 12th order of the expansion and agrees with experiment to 12 significant digits—an unprecedented level of precision in physics. In atomic clocks, frequency shifts from external fields (Stark shift from radiation, Zeeman shift from magnetic field) are calculated using perturbation theory and accounted for in calibrating the time standard. In molecular physics, the Møller–Plesset method (MP2) accounts for electron correlation over Hartree–Fock as a perturbation: this allows prediction of molecular interaction energies with an accuracy of 1–5 kcal/mol. Computational chemistry and drug design rely on these methods for screening candidate molecules. In nuclear physics, the perturbation method is used to calculate nuclear levels, taking into account interactions beyond the superposition principle.

Assignment: (a) Calculate $E^{(1)}$ and $E^{(2)}$ for the anharmonic oscillator $\hat{H} = \hat{H}{HO} + \lambda \hat{x}^3$ ($E^{(1)} = 0$ by symmetry; for $E^{(2)}$ use matrix elements via $\hat{a}$). (b) Atom in a time-dependent field $E(t) = E_0 \cos(\omega t)$. Calculate $P{i \rightarrow f}(t \rightarrow \infty)$. For which $\omega$ is the probability maximal? (Resonant absorption: $\omega = \omega_{fi}$.) (c) Why does $H' = x^3$ not give a first-order correction to the ground state of the oscillator? Use parity.