Real-life applications of WKB approximation for bound states?

Question

A typical quantum mechanics lecture contains a discussion of the WKB (or semiclassical) approximation. Within this framework, two results are usually derived:

1. WKB approximation for bound states (i.e. Griffiths Ch. 8.1)

2. WKB approximation for tunneling (i.e. Griffiths Ch 8.2)

For the latter case (2) I can think of very many cases where the WKB approximation may be useful even in current research. However, I am not really aware of any "real-life" applications of the WKB approximation (1) for calculating energies of bound states. Teaching this subject, I feel however that it would be more motivating for students to have such examples.

Does anyone know an interesting example, where the WKB approximation was used to calculate bound state energies? Preferably in a condensed matter context (but any other context is fine, too!), and maybe even from recent times (say last 30 years)?

Comments

In this link: http://arxiv.org/pdf/1305.3330.pdf you will find an application of WKB approximation to calculate the probability of excited states in a model of molecular quantum dynamics (with a numerical computation); it is fairly recent too. In general, I think that WKB approximation is still commonly used to do semiclassical approximations of stationary states of linear (and also nonlinear) Schrödinger equations, but I do not know the subject well so I could not provide more context

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Thanks for all the comments so far! Let me clarify my question a little bit: I am aware that there is a huge field of using semiclassical approximations, that can be seen as a generalization of the one-dimensional WKB formula.

I was more specifically wondering, if anyone knew of a neat example/application of the classical one-dimensional WKB formula for bound states, where the quantization condition reads

\[\int_{x_1}^{x_2} p(x) dx = (n+\nu) \pi \hbar\]

where \(\nu=0, 1/4, 1/2\) depending on the turning points (hard wall or smooth slope).

The curious thing is that for the corresponding formula for tunneling a lot of examples come to mind immediately (alpha decay, Fowler-Nordheim tunneling, ...) - for the above equation for the bound states, I failed to come up with a nice "real-life" example ...

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There is a simple, but interesting phenomenon of quantization of the neutron energy in the gravitational field.

Answer 1

I only have a book in Russian. I do not know whether it was translated in English or not. It contains WKB and some other calculations and applications. I know that WKB of the "second order" is rather precise.

EDIT: The pre-history is the following: In about 1987 I met my old professor who taught us additional chapters of QED while we were students (1979) and we had a conversation about Pade approximants used to sum up asymptotic series like here. My professor told me that there was a book about integral methods where the WKB method was developed to some higher orders and it turned out to be rather precise and useful. The convergence of the WKB is much better (quicker) than that of the Pade method.

Comments

Maybe you can paraphrase what topics are WKB applied to in this book?

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I did not read this book, I only heard of it. It develops "integral" methods applicable in the physics of condensed sate, zone theory, defects, new materials, doped materials, disordered media, critical phenomena of localized quasi-particles, boundary effects, resonance effects, etc. In fact, the book contains some unknown but effective methods of QM. It may have many real-life applications. The book is called "Integral Methods in Quantum Mechanics".

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Ok, this already contains nontrivial information, +1 from me. Not sure if this addresses OP's interest since he/she specifically asked for applications to bound state problems.

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Of course, it contains applications to the bound states.

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@VladimirKalitvianski can you explain (or at least mention) one or two of the specific WKB bound state examples contained in the book in some more detail in your answer?

Are you talking about this book, and if so do you know a link where the OP can get it in English, in case he is interested in more details?

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@Dilaton: Yes, it is this book. I cannot read this book, sorry. I am not in good shape.

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Thanks for the pointer - unfortunately I cannot read Russian ... it might be interesting though to ask a Russian colleague to have a look at the contents.