Corresponding Energy Levels for Normalised Eigenfunctions

  • Thread starter Ryomega
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  • #1
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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:

Un(x) = ([itex]\frac{2}{a}[/itex])[itex]\frac{1}{2}[/itex] sin ([itex]\frac{nx\pi}{a}[/itex])

where n = 1, 2, 3...

what are the corresponding energy levels?

Homework Equations



Normalisation: 1 = [itex]\int[/itex] lψl2 dx (from negative to positive infinity)

Energy Level for infinite well: En = [itex]\frac{h^2}{2m}[/itex] ([itex]\frac{n}{2a}[/itex])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 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 ([itex]\frac{2}{a}[/itex])[itex]\frac{1}{2}[/itex] sin ([itex]\frac{nx\pi}{a}[/itex])
Then square root whatever the result.

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

Thank you!
 

Answers and Replies

  • #2
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You don't have to unnormalize the wavefunction. You already have the answer in the [itex]E_n[/itex]. The intent of the problem is for you to compute [itex]E_n[/itex] from just the wavefunction and the Hamiltonian, instead of just taking the formula from your notes or book.

In other words, remember that
[tex]\hat H |U_n \rangle = E_n |U_n \rangle[/tex]

Because, well, that's the basic content of the time-independent Schrodinger equation.
 
  • #3
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oh... lol. That's simple enough!

So basically apply the hamiltonian to Un(x)
 

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