- #1

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[tex](p^{2}+V(x))\phi=E_{n}\phi [/tex]

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 [tex]<\phi|\phi>=1 [/tex] then we would have:

[tex](<\phi_{n}|T+V|\phi_{n}>)^{*}=(<\phi_{k}|T+V|\phi_{k}>)[/tex]

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

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

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

[tex]} (|\phi_{n}|^{2}+|\phi_{k}|^{2})b(x)dx=0 [/tex]

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