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Homework Help: Quantum mechanics - angular momentum problem

  1. Mar 21, 2005 #1
    Can someone help me with this please? If anyone does give me any kind of *small* hints, I'd be very grateful.

    Show that the eigenfunctions of the L_{z} operator with different eigenvalues are orthogonal. I don't have a clue how to start this, I'm not even sure what an eigenfunction is. I know how to do eigenvalues with matrices, but not QM. My notes have this:

    Φ(φ) = e^(im_{l}φ)

    L_{z}Φ = -ihbar dΦ/dφ = m_{l}hΦ(φ)

    I don't understand.

    An atomic electron is in a state where measurement of L_{z} may yield the values -3hbar and 3hbar with equal probabilities. Write down a normalised wavefunction describing this state. Again I'm stuck, do I need to use Φ(φ) = e^(im_{l}φ) in some way?

    The recommended book for our course (Eisberg & Resnick) is confusing me more. Grr at stupid quantum mechanics :mad:.
     
    Last edited: Mar 21, 2005
  2. jcsd
  3. Mar 21, 2005 #2

    Doc Al

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    Staff: Mentor

    That second equation is the eigenvalue equation for angular momentum operator; the first equation describes the eigenfunctions, which are the solutions of the eigenvalue equation. You'll need to do some reading.

    Hint: Two functions (F & G, say) are orthogonal if the integral of F*G over the range of the functions is zero. For this problem, let [itex]F = e^{i m_1 \phi}[/itex] and [itex]G = e^{i m_2 \phi}[/itex]. (These are eigenfunctions for the eigenvalues [itex]m_1[/itex] and [itex]m_2[/itex].) Now show that the integral of F*G is zero unless [itex]m_1 = m_2[/itex].

    The wavefunction can be expressed as a sum of eigenfunctions with appropriate coefficients ([itex]C_m[/itex]). Hint: The wavefunction in this case is the sum of only two component eigenfunctions: one corresponds to m = 3, the other to m = -3. What are those eigenfunctions? What must the coefficients of these be if the probability of measuring either eigenvalue is equal? Hit those books!
     
  4. Mar 22, 2005 #3
    For the orthogonality thing, I picked m_{l} = 1 and m_{l} = 2 and then just did that integral like you said. I had to integrate between 0 and 2pi (cos of the angle) and it came out alright.

    The rest of the question I just gave up on. Thanks :smile:.

    The book for QM really makes no sense to me.. I can just about understand stuff on potential steps/barriers and doing the reflection/transmission coefficient stuff. Apart from that, I just get really lost. If I didn't have to do QM this year, I really wouldn't.
     
  5. Mar 22, 2005 #4

    dextercioby

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    Homework Helper

    You considered a particular case.It's not enough...You need to prove for arbitrary integers "m" & "n".

    As for the second,i'm sure you can solve it,it's just that u need to open the book at the right page...

    What's normalization...?

    Daniel.
     
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