Momentum Perturbation

A particle of mass m moves in one dimension according to the Hamiltonian

(1)

(2)

All eigenfunctions ѱ_n­(x) and eigenvalues E_n are known. Suppose we add a term to the Hamiltonian, where λ and m are constants and p is the momentum operator:

(3)

Derive an expression for the eigenvalues and eigenstates of the new Hamiltonian H.

SOLUTION

The first step is to rewrite the Hamiltonian by completing the square on the momentum operator:

(1)

The constant just shifts the zero of the momentum operator. The rewritten Hamiltonian in (1) suggests the perturbed eigenstates:

(2)

The action of the displaced momentum operator p + λ on the new eigenstates is

(3)

so the Hamiltonian gives

(4)

and the eigenvalues are simply

(5)