2
$\begingroup$

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?
$\endgroup$
3
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Is there no method to show it directly from schrodinger equation?i am afraid i am not yet familiar with wkb methods. $\endgroup$ – user157588 Dec 23 '17 at 5:11
  • $\begingroup$ FWIW, WKB is a semiclassical approximation to the Schrödinger equation. $\endgroup$ – Qmechanic Dec 23 '17 at 13:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.