Higher Orders and Applications
General formulas, Brillouin-Wigner theory, and connections to modern physics
Having established the first- and second-order formulas and worked through concrete examples, we now examine the general structure of perturbation theory to all orders, compare different formulations, and explore connections to quantum field theory.
General Formula to All Orders
The energy correction at arbitrary order $k$ can be written systematically. The third-order energy correction is:
The pattern generalizes: the $k$-th order correction involves $k$ matrix elements and $(k-1)$ energy denominators, with subtractions to enforce normalization.
At higher orders, the expressions become combinatorially complex. The general structure is captured elegantly by the resolvent operator formalism.
The Resolvent Operator
Define the resolvent (Green's function) of the unperturbed Hamiltonian:
where $\hat{Q}_n$ projects out of the state $|n^{(0)}\rangle$. The perturbation series can then be written compactly:
This is the Rayleigh-Schrodinger perturbation series, where the resolvent is always evaluated at the unperturbed energy.
Brillouin-Wigner vs Rayleigh-Schrodinger
There are two main formulations of perturbation theory, each with distinct advantages:
Rayleigh-Schrodinger (RS) Perturbation Theory
- Energy denominators use unperturbed energies $E_n^{(0)}$
- Each order is independent of higher orders
- Size-extensive: corrections scale properly with system size (crucial for many-body physics)
- Standard choice for most applications
Brillouin-Wigner (BW) Perturbation Theory
- Energy denominators use the exact energy $E_n$ (which appears on both sides)
- Self-consistent: must solve implicitly for $E_n$
- Often converges faster than RS for single states
- Not size-extensive: problematic for large systems
When to Use Which?
For most applications in atomic and molecular physics, Rayleigh-Schrodinger perturbation theory is preferred because of its size-extensivity property. The BW formulation can be useful when you need good results for a single state and can solve the implicit equation numerically.
Connection to Feynman Diagrams
The perturbation series has a beautiful diagrammatic representation. Each term in the expansion corresponds to a diagram:
- First order: A single vertex connecting the initial state to the perturbation. Corresponds to the matrix element $\langle n|\hat{H}'|n\rangle$.
- Second order: Two vertices connected by a propagator (intermediate state line). The propagator is $1/(E_n^{(0)} - E_m^{(0)})$. This represents a virtual transition to state $|m\rangle$ and back.
- Higher orders: More vertices and propagators, with summation over all intermediate states.
In quantum field theory, these diagrams become Feynman diagrams, where the vertices represent interactions (e.g., photon emission/absorption) and the propagators represent particle propagation. The rules for computing Feynman diagrams are a direct generalization of the perturbation theory rules you have learned here.
Selection Rules from Matrix Elements
Many matrix elements $\langle m|\hat{H}'|n\rangle$ vanish due to symmetry, greatly simplifying the perturbation sums. Key selection rules include:
For central potentials with angular momentum eigenstates $|n,l,m_l\rangle$:
- Electric dipole ($\hat{H}' \propto r\cos\theta$): $\Delta l = \pm 1$, $\Delta m_l = 0$
- Electric quadrupole ($\hat{H}' \propto r^2 Y_2^q$): $\Delta l = 0, \pm 2$, $\Delta m_l = 0, \pm 1, \pm 2$
- Magnetic dipole ($\hat{H}' \propto \hat{L}_z$): $\Delta l = 0$, $\Delta m_l = 0$
These selection rules determine which terms survive in the perturbation sums and which transitions are allowed or forbidden in spectroscopy.
Wigner's $2n+1$ Rule
An elegant result relates the accuracy of energy calculations to the accuracy of wave functions:
For example:
- Zeroth-order wave function ($n=0$) gives energy correct to first order
- First-order wave function ($n=1$) gives energy correct to third order
- Second-order wave function ($n=2$) gives energy correct to fifth order
This is why perturbation theory is so efficient for energies: you get "more than you paid for" when computing the wave function corrections.
Key Concepts Summary
- Perturbation setup: $\hat{H} = \hat{H}_0 + \lambda\hat{H}'$ where $\hat{H}_0$ is exactly solvable
- First-order energy: $E_n^{(1)} = \langle n^{(0)}|\hat{H}'|n^{(0)}\rangle$ (expectation value of perturbation)
- First-order state: $|n^{(1)}\rangle = \sum_{m\neq n} \frac{\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle}{E_n^{(0)}-E_m^{(0)}}|m^{(0)}\rangle$ (mixing of other states)
- Second-order energy: $E_n^{(2)} = \sum_{m\neq n} \frac{|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle|^2}{E_n^{(0)}-E_m^{(0)}}$ (always lowers ground state)
- Validity: Perturbation matrix elements must be small compared to level spacings
- Rayleigh-Schrodinger: Size-extensive, uses unperturbed energy denominators (standard choice)
- Brillouin-Wigner: Self-consistent, uses exact energy in denominators (faster convergence for single states)
- Selection rules: Symmetry determines which matrix elements vanish, simplifying calculations
- Wigner 2n+1 rule: Wave function to order $n$ gives energy to order $2n+1$
- Feynman diagrams: Diagrammatic representation generalizes to quantum field theory
- Failure: Degeneracy, near-degeneracy, strong coupling, and asymptotic divergence limit applicability
Related Topics: Degenerate Perturbation Theory - Essential when unperturbed levels share the same energy | Variational Method - Alternative approximation that gives energy upper bounds | Time-Dependent Perturbation Theory - For time-varying perturbations and transitions