Combination of wavefunctions in oscillator

[cfwoods]

Homework Statement

The equation $\psi(x) = \frac{1}{sqrt(2)}\psi_0 (x) + \frac{i}{sqrt(5)}\psi_1 (x) + \gamma\psi_2 (x)$

is a combination of the first three eigenfunctions in the 1D harmonic oscillator. So, $\psi_0 = Ae^{-mωx^2 /2\hbar}$ and so on for the first and second excited states. If $\psi_0$, $\psi_1$ and $\psi_2$ are normalised, and $\psi(x)$ is also normalised, determine $|\gamma|$

The Attempt at a Solution

You can obtain the normalised functions for the ground state and first two excited states from a variety of methods, and you can then expand out $\psi(x)$. I tried plugging that back into the time dependent Schrödinger equation, but that didn't help (and it also gave a messy derivative) so I am at a loss as to how i can proceed.

 

[DrClaude]

You can obtain the normalised functions for the ground state and first two excited states from a variety of methods, and you can then expand out $\psi(x)$.

It is stated that $\psi_0$, $\psi_1$, and $\psi_2$ are taken to be normalized.

I tried plugging that back into the time dependent Schrödinger equation

How is that realted to normalization?

Let's start from the beginning: what equation does $\psi(x)$ statisfy if it is normalized?

 

[cfwoods]

Hmm it satisfies $\int^{∞}_{-∞} \psi^{*}\psi dx = 1$ I think

 

[DrClaude]

Hmm it satisfies $\int^{∞}_{-∞} \psi^{*}\psi dx = 1$ I think

Correct. So plug in there the $\psi(x)$ of the problem. Keep the notation $\psi_0$ and so on (i.e., do not write the explicit functions of $x$) and use the properties of the harmonic oscillator wave functions to simplify the result.