Harmonic Oscillator energy = WKB approximate energy why?

[maverick280857]

Hi,

Why is it that the WKB approximation produces the correct eigenvalues for the Harmonic Oscillator problem, but the wrong wavefunctions, whereas for the square well, we get correct wavefunctions and the wrong eigenvalues?

I've been trying to dig through the approximations we make in deriving the WKB expressions. The quantization rule that gives the correct eigenvalue for the harmonic oscillator is derived under the condition that near the turning points, the potential can be expanded as a linear function. And V(x) for the SHO is a quadratic function, the "next best" to a linear function.

Can we give a more rigorous reason for this?

Thanks.

 

[Bob_for_short]

Hi,

Why is it that the WKB approximation produces the correct eigenvalues for the Harmonic Oscillator problem, but the wrong wavefunctions, whereas for the square well, we get correct wavefunctions and the wrong eigenvalues?

I've been trying to dig through the approximations we make in deriving the WKB expressions. The quantization rule that gives the correct eigenvalue for the harmonic oscillator is derived under the condition that near the turning points, the potential can be expanded as a linear function. And V(x) for the SHO is a quadratic function, the "next best" to a linear function.

Can we give a more rigorous reason for this?

Thanks.

In fact, the WKB approximation is not obliged to reproduce the exact eigenvalues and wave functions. It is by chance that some of that coincides with the exact ones. But WKB solutions converge quickly to the exact solutions when n increases.

In case of oscillator, the wave functions are always approximate since they are never valid at the turning points.

In case of a square well with infinitely high walls, the wave function is valid (=0) at the boundary points, so it may coincide with the exact ones.

Bob.

 

[maverick280857]

Thanks for your reply Bob.

In fact, the WKB approximation is not obliged to reproduce the exact eigenvalues and wave functions. It is by chance that some of that coincides with the exact ones. But WKB solutions converge quickly to the exact solutions when n increases.

And that is precisely why I want to know why it should "work" for the SHO. I wanted to understand if this chance coincidence can be rigorized.

 

[Bob_for_short]

Thanks for your reply Bob. And that is precisely why I want to know why it should "work" for the SHO. I wanted to understand if this chance coincidence can be rigorized.

This is simple: the initial approximation is sufficiently precise. In other words, the higher order WKB corrections to the energy level are very small (1/(pi^2 or so). Here I speak of corrections at a given n. So anyway the initial approximation of E_n is not so far from the exact level. That is why it may occasionally coincide with the exact one.

Bob.