- #1
eljose
- 484
- 0
Let be a Hamiltonian in the form H=T+V we don,t know if V is real or complex..all we know is that if E_n is an energy also E*_n=E_k will be another energy, my question is if this would imply V is real...
my proof is taking normalized Eigenfunctions we would have that:
(<\phi_{n}|H|\phi_{n}>)*=<\phi_{k}|H|\phi_{k}>
so the expected value of T is always real,then we would have the identity with the complex part b(x) of the potential:
\int_{-\infty}^{\infty}dx(|\phi_{n}|^{2}+|\phi_{k}|^{2})b(x)=0
for every k,and n so necessarily b=0 so the potential is real and all the eigenfunctions would be real.
my proof is taking normalized Eigenfunctions we would have that:
(<\phi_{n}|H|\phi_{n}>)*=<\phi_{k}|H|\phi_{k}>
so the expected value of T is always real,then we would have the identity with the complex part b(x) of the potential:
\int_{-\infty}^{\infty}dx(|\phi_{n}|^{2}+|\phi_{k}|^{2})b(x)=0
for every k,and n so necessarily b=0 so the potential is real and all the eigenfunctions would be real.
