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I'm reading a book which says: (HO stands for harmonic oscillator):

The spectrum of the HO has equidistant energy eigenvalues. A potential that increases quicker than the HO has states which become less dense as the quantum number increases, that is, $$\Delta E = E(n+1) - E(n)$$ increases as $n$ increases. A potential that increases slower than the harmonic oscillator or stops and decreases at some point, has a denser spectrum as n increases. Can you think why?

I understand how to show this using WKB approximation and showing that $\Delta E(n)$ increases with $n$ in the first case and decreases in the second case, for potentials of the form $x^k$.

However, I do not understand the physics behind it. Why is this expected? What is it saying about the particle?

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  • $\begingroup$ Linked. $\endgroup$
    – Cosmas Zachos
    Commented Jun 7 at 13:03
  • $\begingroup$ Thanks that makes sense. Is there though a more physical explanation why the energy spectrum will become less dense as n increases? $\endgroup$
    – MTYS
    Commented Jun 7 at 16:16
  • $\begingroup$ Here's my attempt at physical explanation: An extremely slowly changing potential is going to be approximately free space, which has a continuous energy spectrum because there are basically no restrictions on what the particle can be doing. i.e. maximum density, on the other hand if a potential is strongly confining that particle is going be going bananas! So, the energy gaps are huge. $\endgroup$
    – mike1994
    Commented Jun 7 at 17:36

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I think this naturally follows from the way the energy gap changes. The way I think about this is this.
Let's first stick only to HO potentials, but increase the frequency $(\omega)$, to vary the rate at which the potential changes. Here, clearly in the energy gap increases as you increase the frequency, which lowers the density of the energy spectrum. Now if you look at potentials of the form $x^k$, you can guess (either variationally or using perturbation theory) that the shift in energy states would be proportional and follow the same trend as before.
Hope it helps!

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