Formal properties of eigenfunctions

[black_hole]

Homework Statement

Give a physicist's proof of the following statements regarding energy eigenfunctions:
(a) We can always choose the energy eigenstates E(x) we work with to be purely real
functions (unlike the physical wavefunction, which is necessarily complex). Note: This does not mean that every energy eigenfunction is real, rather if you find an eigenfunction that is not real, it can always be written as a complex linear combination
of two real eigenstates with the same energy.

Hint: If E(x) is an energy eigenstate with energy eigenvalue E, what can be said about E(x)
?

Homework Equations

The Attempt at a Solution

I'm having a hard to deciphering here what is warranted I show in this problem. Does anyone know what this is trying to get at. Maybe my mathematical experience with proofs is getting in the way because I'm trying to do it more generally.

 

[Dick]

Homework Statement

Give a physicist's proof of the following statements regarding energy eigenfunctions:
(a) We can always choose the energy eigenstates E(x) we work with to be purely real
functions (unlike the physical wavefunction, which is necessarily complex). Note: This does not mean that every energy eigenfunction is real, rather if you find an eigenfunction that is not real, it can always be written as a complex linear combination
of two real eigenstates with the same energy.

Hint: If E(x) is an energy eigenstate with energy eigenvalue E, what can be said about E(x)
?

Homework Equations

The Attempt at a Solution

I'm having a hard to deciphering here what is warranted I show in this problem. Does anyone know what this is trying to get at. Maybe my mathematical experience with proofs is getting in the way because I'm trying to do it more generally.

The hint is probably that it satisfies the Schrödinger equation. Can you show that the real and imaginary parts satisfy the equation separately? Show this wouldn't be true if the eigenvalue were not real.