wang
(Wang)
#1
10 Bound State Perturbation Theory
We have now solved exactly a number of quantum mechanical problems. Systems of physical
interest in Nature, however, are often more complex than these problems, and cannot be
solved exactly. Perturbation theory is the approximation technique that allows one to study
systems that may be viewed as small deviations (i.e. perturbations)away from exactly solved
ones. In principle, successively higher orders in perturbation theory will yield results closer
and closer to the solution of the full system. In practice, however, perturbation theory has
many limitations that we shall illustrate below.
One distinguishes the following broad classes of perturbation theory, in order of generally
increasing difficulty,
1. time independent perturbation theory
• bound state spectrum
• continuous spectrum (scattering theory)
2. time dependent perturbation theory
These distinctions are mostly of a technical nature, with bound state perturbation theory
resembling most closely the perturbation theory of finite systems.
The general problem may be posed as follows. Let the HamiltonianH 0 be solvable
and denote its energy eigenvalues byEn^0 and its eigenstates by |En^0 〉. For any givenEn^0 ,
there may be just a single state (regular perturbation theory) orseveral states (degenerate
perturbation theory). The question now is how to calculate the energy eigenvaluesEn(λ)
and the eigenstates|En(λ)〉for the family of Hamiltonians
H=H 0 +λH 1 (10.1)
whereλis a real parameter,H 0 andH 1 are independent ofλ, and we assume thatH 1 †=H 1.
The parameterλmay correspond to an adjustable quantity in an experiment, such as an
external electric field (such as in the Stark effect) or a magnetic field (such as in the Zeeman
effect). It may also be a fixed quantity in which we decide to expand, such as for example
the coupling of a certain interaction. Thus, we have
H 0 |E^0 n〉 = E^0 n|En^0 〉
(H 0 +λH 1 )|En(λ)〉 = En(λ)|En(λ)〉 (10.2)
We seek a solution of the type
En(λ) = En^0 +λEn^1 +λ^2 E^2 n+O(λ^3 )
|En(λ)〉 = |En^0 〉+λ|E^1 n〉+λ^2 |En^2 〉+O(λ^3 ) (10.3)
