perturbation theory

Perturbation theory is a technique in Quantum Mechanics for finding approximate solutions to real-world problems. It has a wide variety of applications throughout physics and applied mathematics.

In Quantum Mechanics we often need to solve the Schrodinger equation that corresponds to a given physical potential. Even for a simple potential, the equation is extremely difficult to solve. For a real-world potential finding a solution analytically is almost impossible. In perturbation theory, we break down a complicated potential into a solvable potential, plus a small complicated remainder. We solve the first part, and then consider how the second part influences our answer.

For the mathematics below we assume that the energy levels are all distinct. If some of the energy levels are repeated, then this analysis breaks down at the first order wavefunction corrections. In that case we need to use an extension to these ideas known as degenerate perturbation theory.

See also the example of perturbation theory to find out how to put these results into practice.

Outline

We use the Dirac formalism of Quantum Mechanics. Let H₀ be a soluble Hamiltonian, with corresponding orthogonal wavefunctions |n> and energy eigenvalues ε_n. We consider a small perturbation to H₀, represented by λH', where λ is small. Hence we look at

H(λ) = H0 + λH'.

We therefore want to solve

H(λ) |ψn(λ)> = En(λ) |ψn(λ)>.

For perturbation theory to work, we assume that |ψ_n(λ)> → |n> and hence also E_n(λ) → ε_n, as λ → 0. We therefore assume Taylor Series for |ψ_n(λ)> and E_n(λ) of the form

En(λ) = εn + λFn + λ²Gn + ...
|ψn(λ)> = N (|n> + λ|φn> + λ²|χn> + ...)

where N is a normalisation constant. Substituting into the Schrodinger equation gives

(H0 + λH')(|n> + λ|φn(λ)> + ...) = (εn + λFn + ...)(|n> + λ|φn(λ)> + ...).

We work out the terms in the series by equating powers of λ in this expansion.

First order energy corrections

We equate the terms with a coefficient of λ in the Schrodinger equation.

H'|n> + H0|φn> = Fn|n> + εn|φn>      (*)

Since adding on a scalar multiple of |n> to |φ_n> leaves this equation invariant, we demand the condition <n|φ_n>=0. Applying <n| to the above equation then gives

Fn = <n|H'|n>.

This is the first order shift in energy.

First order wavefunction corrections

We apply <r| to equation (*). By orthogonality, <r|n> = 0, so

<r|H'|n> + εr <r|φn> = εn <r|φn>
<r|φn> = - <r|H'|n> / (εr - εn)

Since the {|r>} form a complete set we get the first order shift to the wavefunction to be

|φn> = sum(-|r><r|H'|n>/(εr - εn), r ≠ n).

Second order energy corrections

Returning to the original Schrodinger equation, we equate terms with a coefficient of λ².

H'|ψn> + H0|χn> = Gn|n> + Fn|φn> + εn|χn>

Similar to as before we impose <n|χ_n>=0. Applying <n| to the equation gives

Gn = <n|H'|φn>
   = -sum(<n|H'|r><r|H'|n>/(εr - εn), r ≠ n)
   = -sum(|<r|H'|n>|²/(εr - εn), r ≠ n)

which is the second order shift in energy.