Perturbation Theory
Studying physics, one may get in the beginning the impression that there are closed analytical solutions for all problems.
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Perturbation Theory
Due to the lack of analytic solutions of physical problems, several perturbative methods have been developed. The time-independent perturbation theory discussed in this chapter is an important and much-used technique by which we can calculate the fine structure of the spectrum of the hydrogen atom.
Studying physics, one may get in the beginning the impression that there are closed analytical solutions for all problems. That impression is deceptive, as is well known.1 All in all, in physics, the set of explicitly and exactly solvable problems is of measure zero; and this is particularly relevant to quantum mechanics. There are a handful of potentials for which one can specify an explicit analytic solution of the SEq, but that’s about the end of it. If we pick at random any more or less physically reasonable model potential of an appropriate function space, the chance that we know an explicit analytic solution is practically zero. For this reason, one either depends on numerical calculations or, if one wants to have more or less analytic results, on some form of approximation. There are various methods2 ; here, we address the so-called perturbation theory.3 This method can be applied when the interaction being considered can be decomposed into a part V which covers the essential physical effects, and another relatively small part W (the ‘perturbation’) which describes more detailed structures. Of course it is an especially favorable case when there exist closed analytic solutions for V . One distinguishes between time-independent (=stationary) and time-dependent perturbation theory. Since we consider only time-independent potentials in this text, we restrict ourselves in the following to stationary perturbation theory.
1 Physics is regarded as an exact science. That does not mean that physical models are always exact or can be solved exactly. Physical models are inherently approximations, and only a few can indeed be solved exactly. A main characteristic of physics is that it deliberately keeps track of the inaccuracies of its approaches. To repeat a quote from Bertrand Russell: “Although this may be seen as a paradox, all exact science is dominated by the idea of approximation”. 2 In Chap. 23, we will discuss the Ritz variational principle, a different approximation procedure. 3 Also called Rayleigh-Schrödinger perturbation theory.
J. Pade, Quantum Mechanics for Pedestrians 2: Applications and Extensions, Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-319-00813-4_19, © Springer International Publishing Switzerland 2014
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19 Perturbation Theory
19.1 Stationary Perturbation Theory, Nondegenerate We start with a Hamiltonian H : H =−
2 2 ∇ + V + W = H (0) + W. 2m
(19.1)
The prerequisite for the following considerations is that W be sufficiently ‘small’, i.e. that(0)for all states occurring in the calculation, it holds that: |ϕ| W |ψ| ϕ| H |ψ. In this case, we can write H = H (0) + W = H (0) + εWˆ
(19.2) with the smallness parameter ε 1, where the ma
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