Quantum mechanics - angular momentum problem

[Nylex]

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 .

 

[Doc Al]

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)

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 F = e^{i m_1 \phi} and G = e^{i m_2 \phi}. (These are eigenfunctions for the eigenvalues m_1 and m_2.) Now show that the integral of F*G is zero unless m_1 = m_2.

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 wavefunction can be expressed as a sum of eigenfunctions with appropriate coefficients (C_m). 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!

 

[Nylex]

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 .

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.

 

[dextercioby]

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.