Time independent wave function (prove theorem)

[Cogswell]

Homework Statement

Prove the following theorum:
The time-independent wave function $\psi (x)$ can always be taken to be real (unlike $\Psi (x,t)$, which is necessarily complex). This doesn't mean that every solution to the time independent Schrödinger equation is real; what it says is that if you've got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So you might as well stick to $\psi$ 's that are real.
Hint: If $\psi(x)$ satisfies equation [1.0], for a give E, so too does its complex conjugate, and hence also the real linear combinations $( \psi + \psi ^*)$ and $i( \psi - \psi ^*)$

Homework Equations

Equation [1.0]:

$$E \psi = -\dfrac{\hbar ^2}{2m} \dfrac{d^2 \psi}{dx^2} + V \psi$$

The Attempt at a Solution

I don't fully get what it. So it's saying that if I have a $\psi (x)$ that is a complex function of x, then that can somehow be expressed in the form:

$\psi_{complex} (x) = \psi_1 (x) + \psi_2 (x) + \psi_3 (x) ...$

How does that work? I also don't know how to incorporate the hint into it.

(Sorry quantum mechanics isn't my strong point so it would help if you explained it to me in simple terms)

 

[Fredrik]

Homework Statement

Prove the following theorum:
The time-independent wave function $\psi (x)$ can always be taken to be real (unlike $\Psi (x,t)$, which is necessarily complex). This doesn't mean that every solution to the time independent Schrödinger equation is real; what it says is that if you've got one that is not, it can always be expressed as a linear combination of solutions (with the same energy) that are. So you might as well stick to $\psi$ 's that are real.
Hint: If $\psi(x)$ satisfies equation [1.0], for a give E, so too does its complex conjugate, and hence also the real linear combinations $( \psi + \psi ^*)$ and $i( \psi - \psi ^*)$

Homework Equations

Equation [1.0]:

$$E \psi = -\dfrac{\hbar ^2}{2m} \dfrac{d^2 \psi}{dx^2} + V \psi$$

The Attempt at a Solution

I don't fully get what it. So it's saying that if I have a $\psi (x)$ that is a complex function of x, then that can somehow be expressed in the form:

$\psi_{complex} (x) = \psi_1 (x) + \psi_2 (x) + \psi_3 (x) ...$

How does that work? I also don't know how to incorporate the hint into it.

(Sorry quantum mechanics isn't my strong point so it would help if you explained it to me in simple terms)

Do you know how to express the real and imaginary parts of a complex number z in terms of z and z*? (If not, write down the definition of z* and figure it out).

Note that the eigenfunctions of H are supposed to span the entire space of functions that we're dealing with. If $f_1,f_2,\dots$ are eigenfunctions of H with eigenvalues $E_1,E_2,\dots$, then the set $\{f_n\}_{n=1}^\infty$ is a basis for that space. Your job is to take an arbitrary basis like that and find a way to replace it with one that consists only of real-valued functions.