# Corresponding Energy Levels for Normalised Eigenfunctions

1. Jun 9, 2012

### Ryomega

1. The problem statement, all variables and given/known data

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:

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

where n = 1, 2, 3...

what are the corresponding energy levels?

2. Relevant equations

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

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

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

3. The attempt at a solution

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

Make Un(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!

2. Jun 9, 2012

### 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 Schrodinger equation.

3. Jun 9, 2012

### Ryomega

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

So basically apply the hamiltonian to Un(x)