Momentum Perturbation
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 66
19-8-2016
1262
Momentum Perturbation
A particle of mass m moves in one dimension according to the Hamiltonian
(1)
(2)
All eigenfunctions ѱn(x) and eigenvalues En 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)
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