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 Schrodinger 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 Schrodinger 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.

 

[paranormal]

I would first clarify the goal of this exercise. Are we trying to determine the value of |\gamma| or understand the physical significance of this combination of wavefunctions?

If the goal is to determine |\gamma|, we can use the fact that \psi(x) is also normalized to set up an integral and solve for |\gamma|. Alternatively, we can use the fact that the wavefunction must be continuous and differentiable to set up a system of equations and solve for |\gamma|.

If the goal is to understand the physical significance of this combination of wavefunctions, we can look at the form of the wavefunction and see that it contains contributions from the ground state, first excited state, and second excited state. This suggests that the resulting wavefunction may exhibit characteristics of all three states, which could have implications for the system being modeled. Further analysis and exploration of this wavefunction may provide insights into the system's behavior.