Dependence of energy $E$ with principal quantum number $n$

Question

Is there any way to tell how the energy levels of a potential well depend upon the quantum number $n$, looking at the potential, before solving?

I mean, for harmonic oscillator, $E\sim n$, for infinite square well, $E\sim n^2$, so, can it be guessed how it would be for a power law potential $\sim x^p$, where $p$ is some power?

Qmechanic · Accepted Answer · 2017-12-22T13:56:40.247

3

One may show, using semiclassical WKB methods explained in e.g. my Phys.SE answer here, that the discrete energy levels go as $$ E_n ~~\propto~~{\rm sgn}(p)~ n^{\frac{1}{1/p+1/2}}, \qquad n\in\mathbb{N}, $$ for a 1D potential $$ \Phi(x)~~\propto~~{\rm sgn}(p)~|x|^p, \qquad -2~<~p~\leq~\infty .$$

edited Dec 22 '17 at 13:56

answered Dec 19 '17 at 17:02

Qmechanic

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Is there no method to show it directly from schrodinger equation?i am afraid i am not yet familiar with wkb methods. – user157588 Dec 23 '17 at 05:11
FWIW, WKB is a semiclassical approximation to the Schrödinger equation. – Qmechanic Dec 23 '17 at 13:38

Dependence of energy $E$ with principal quantum number $n$

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