What operators are involved in angular momentum states?

[leonmate]

Hoping this is in the right section! The module is nuclear and atomic physics but it crosses over into quantum occasionally.

I've attached an image of the bit I'm trying to work out.

I've got an exam on this topic in just over a week, so sorry if these posts get annoying, I have a feeling I'm going to posting a few.. Physics is infuriating when you get stuck and can't find where to look for solutions!

The issue I have right now is I'm looking at a number of these operators acting on states like the one below and while I know a few operators: position, momentum, hamiltonian etc I keep getting these odd ones thrown at me and I don't know how to work these through... What's written on the sheet, am I supposed to just accept that's a result or is there a relatively simple way of working it through (if I can work it through I tend to understand and remember it more!)

Thanks,
Leon

 

[blue_leaf77]

I don't know how to work these through

From this statement, I can tell you haven't read enough material from your lecture notes or textbook.

is there a relatively simple way of working it through

It's a relatively simple matter to work with 1 particle angular momentum operators, but of course only after you get sufficient knowledge on it. So, trying to explain the whole thing from the beginning is the same as retyping the existing chapter in your textbook. I suggest Quantum Mechanics by Griffith or Modern Quantum Mechanics by Sakurai, or eventually the textbook your lecturer is using.

 

[leonmate]

Ok, fair enough, I'm finding this a really tough subject

Perhaps my lecture notes aren't great, ill try one of those textbooks

thanks

 

[leonmate]

Ok, found a handy website with an article that's explained this well:

http://www.physicspages.com/2011/07/20/angular-momentum-eigenvalues/

I was curious about this statement:

'We can assume that the eigenvalue of

for

is

for some number

. That is, for this eigenfunction' (above eq 19)

Why is it that we can assume this? I get that we have quantized states - is that why? so we can have hbar, 2hbar, 3hbar etc.
Also, is this the just the case for angular momentum or is it more than that?

 

[blue_leaf77]

$L_z$ is an angular momentum operator, since $\hbar$ has the same unit as angular momentum, it makes sense saying that any eigenvalue of $L_z$ be a multiple of $\hbar$. However at this point, nothing is specified about $l$, it can be any real number. Later it will be proven that $l$ must be integer.

 

[ChrisVer]

If you find it difficult to work with bras and kets to get your result, you can as well work with normal wavefunctions and the angular momentum operator written in the position representation...with the Y being your eigenfunctions and calculating Lz, L^2 (eg in spherical coordinates).
If I recall well the difference is that with the brackets you will reach to half-integer angular momenta too, while with the one I'm proposing you won't...