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

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?

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 $$\Phi(x)~~\propto~~{\rm sgn}(p)~|x|^p, \qquad -2~<~p~\leq~\infty .$$