Corresponding Energy Levels for Normalised Eigenfunctions

[Ryomega]

Homework Statement

Particle of mass m in a 1D infinite square well is confined between 0 ≤ x ≤ a

Given that the normalised energy eigenfunction of the system is:

U_n(x) = ($\frac{2}{a}$)^{$\frac{1}{2}$} sin ($\frac{nx\pi}{a}$)

where n = 1, 2, 3...

what are the corresponding energy levels?

Homework Equations

Normalisation: 1 = $\int$ lψl² dx (from negative to positive infinity)

Energy Level for infinite well: E_n = $\frac{h^2}{2m}$ ($\frac{n}{2a}$)²

Where h = reduced Planck constant (sorry couldn't find it)

The Attempt at a Solution

At first I thought, I could just plug in 1, 2, 3 in U_n(x)
But then I realized the question said that U_n(x) is normalised.
Does this mean I have to reverse normalise U_n(x)? If so, is this order right?

Make U_n(x) = 1
Differentiate ($\frac{2}{a}$)^{$\frac{1}{2}$} sin ($\frac{nx\pi}{a}$)
Then square root whatever the result.

Am I going in the right direction or am I missing something out?

Thank you!

 

[Muphrid]

You don't have to unnormalize the wavefunction. You already have the answer in the $E_n$. The intent of the problem is for you to compute $E_n$ from just the wavefunction and the Hamiltonian, instead of just taking the formula from your notes or book.

In other words, remember that
$$\hat H |U_n \rangle = E_n |U_n \rangle$$

Because, well, that's the basic content of the time-independent Schrödinger equation.

 

[Ryomega]

oh... lol. That's simple enough!

So basically apply the hamiltonian to Un(x)