# Particle on a potential well

let be a particle in a potential well with mass m=1/2 so we have the equation:

$$(p^{2}+V(x))\phi=E_{n}\phi$$

we don,t know if V is real or complex but we have that if En is an energy,its complex conjugate En^*=Ek is also another energy of the system,my question is if the potential is real...

Proof?:taking normalized Eigenfunctions of the Hamiltonian....with $$<\phi|\phi>=1$$ then we would have:

$$(<\phi_{n}|T+V|\phi_{n}>)^{*}=(<\phi_{k}|T+V|\phi_{k}>)$$

so in the end separating and knowing that $$<\phi|p^{2}|\phi>$$ is always real then we would have that:

$$\int_{-\infty}^{\infty}|\phi_{n}|^{2}V^*(x)-int_{-\infty}^{\infty}|\phi_{k}|^{2}V(x)=r$$ with r a real number....

so we would have for every k and n and complex part of the potential b(x) that:

$$} (|\phi_{n}|^{2}+|\phi_{k}|^{2})b(x)dx=0$$

so the complex part of the potential is 0...is that true?

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lightgrav
Homework Helper
I'm not able to load your LaTeX graphics ... so I can't comment.
(maybe that's why nobody *else* is responding, either.)

lightgrav
Homework Helper
It looks like you're trying to end [ tex ] code with [ / tex ],
instead of [ \tex ] . Should be an easy edit fix!