Perturbation Theory

Chem 3240 · Lecture 6.2

Davit Potoyan

The Big Idea

  • Most real problems are not exactly solvable.

  • Split the Hamiltonian into a solvable part plus a small correction.

  • Quantum analog of a Taylor expansion.

  • The art is finding the small parameter.

Splitting the Hamiltonian

  • Start from an exactly solvable problem (box, oscillator, …):

\[\hat{H}^0 \mid n^0\rangle = E^0_n \mid n^0\rangle\]

  • Add a small perturbation \(\hat{H}^1\), switched by \(\lambda\): \[\hat{H} = \hat{H}^0 + \lambda \hat{H}^1\]
  • “Small” means level shifts stay below the level spacing.

Expand Like a Taylor Series

  • Energies and states become power series in \(\lambda\):

\[E_n = E^0_n + \lambda E^1_n + \lambda^2 E^2_n + \dots\]

\[\mid n\rangle = \mid n^0\rangle + \lambda \mid n^1\rangle + \lambda^2 \mid n^2\rangle + \dots\]

  • Plug into \(\hat{H}\mid n\rangle = E_n\mid n\rangle\) and collect powers of \(\lambda\).

Order by Order

  • Each power of \(\lambda\) gives one equation:

\[\hat{H}^0\mid n^0\rangle = E^0_n\mid n^0 \rangle\]

\[\hat{H}^0\mid n^1\rangle + \hat{H}^1\mid n^0\rangle = E^0_n\mid n^1 \rangle + E^1_n\mid n^0 \rangle\]

  • Zeroth order is just the exact solution.
  • The sum of superscripts sets the order.

The Master Formula

\[E_n = E^0_n + H_{nn} + \sum_{k \neq n} \frac{\mid H_{nk}\mid^2}{E^0_n - E^0_k}\]

  • Everything is built from the solved eigenstates and eigenvalues.
  • Matrix elements of the perturbation: \[H_{nk} = \langle n^0\mid \hat{H}^1\mid k^0\rangle\]

First Order: a Diagonal Average

  • The first-order shift is the diagonal matrix element:

\[E_n^1 = \langle n^0 \mid \hat{H}^1\mid n^0 \rangle\]

  • Looks like an expectation value, but the solved states sandwich the perturbation.
  • \(\hat{H}^0\) and \(\hat{H}^1\) in general do not share eigenfunctions.

Second Order: Mixing in Other States

\[E_n^2 = \sum_{k \neq n} \frac{\mid H_{nk}\mid^2}{E^0_n - E^0_k}\]

  • Built from off-diagonal elements \(H_{nk}\).

  • Nearby levels dominate: small denominators amplify their terms.

  • For the ground state every denominator is negative, so \(E_0^2 < 0\): mixing pushes it down.

Particle in a Box, Shifted Up

  • Add a constant potential \(V_0\) across the whole box.

\[E_n^1 = \langle n \mid V_0 \mid n \rangle = V_0 \cdot \frac{2}{L}\int_0^L \sin^2\frac{n\pi x}{L}\,dx = V_0\]

  • Every level rises by the same \(V_0\).
  • Spacings are unchanged to first order.

Anharmonic Oscillator

  • Oscillator plus a cubic kick: \(\hat{H} = \hat{H}_0 + \gamma x^3\).

\[E_n^1 = \langle n\mid \gamma x^3 \mid n\rangle = 0\]

  • First order vanishes: \(x^3\) is odd, \(|\psi_n|^2\) is even.
  • The shift appears only at second order, from odd \(k\) terms.

Where It Shows Up

  • Hydrogen in a magnetic field: \(\hat{H}^1 = \frac{e}{2m_e}B\,\hat{L}_z\), shift \(m_l\,\beta_B B\).

  • Spin-orbit coupling: \(\hat{H}^1 = A_{SO}\,\hat{L}\hat{S}\).

  • Anharmonic vibrations, a shifted box, and a perturbative view of bonding.

Takeaway

Perturbation theory turns a hard problem into a solvable one plus a small correction, expanding the energy as \(E_n = E^0_n + H_{nn} + \sum_{k\neq n}\frac{\mid H_{nk}\mid^2}{E^0_n - E^0_k}\). The first-order shift is a diagonal average; the second-order shift mixes in other states, with nearby levels dominating.