Real Solutions in Time Independent Schrodinger's Equation?

[spaghetti3451]

This question is about the time independent schrodinger's equation, but is best posted here.

In the TISE, all the terms in H-hat are real, so it is possible, and not uncommon, for the solutions u(x) to also be purely real.

I don't understand why H-hat is real implies that u (x) is real.

 

[dextercioby]

One must understand 2 things under the following (admittedly simplifying) 2 assumptions:

i) The Hamiltonian is a compact self-adjoint operator on a complex separable Hilbert space.
ii) Its spectral equation is a PDE/ODE with real coefficients (one coefficient is the spectral value which is always real as the operator is s-adj).

1. A basis of the space of solutions of the PDE/ODE in ii) can always be chosen to be formed by real functions.
2. Even in the conditions of 1, generally, due to the fact that the Hilbert space where solutions of the spectral equations are sought is complex, a solution to the spectral equation is a complex function.

 

[spaghetti3451]

Boy oh boy. this is too complicated for my level!