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Corresponding Energy Levels for Normalised Eigenfunctions

  1. Jun 9, 2012 #1
    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) = ([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?

    2. Relevant 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)

    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 ([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!
     
  2. jcsd
  3. Jun 9, 2012 #2
    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.
     
  4. Jun 9, 2012 #3
    oh... lol. That's simple enough!

    So basically apply the hamiltonian to Un(x)
     
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