How Do You Approach Angular Momentum Operator Algebra in Quantum Mechanics?

[gfd43tg]

Homework Statement

Homework Equations

The Attempt at a Solution

This whole thing about angular momentum has me totally confused and stumped, but I am trying this problem given in a youtube video lecture I watched.

I know of this equation
$L^{2} = L_{\pm}L_{\mp} + L_{z}^{2} \mp \hbar L_{z}$
$L^{2}f = \lambda f$
$L_{z}f = \mu f$

In the case $L^{2}f = 2 \hbar^{2}f$, the eigenvalue $\lambda = 2 \hbar^{2}$

So I expand
$$L^{2}f = (L_{-}L_{+} + L_{z}^{2} + \hbar L_{z})f$$
But, I don't know what $L_{z}$ should be. Also, how do I know if I should pick $L_{-}L_{+}$ or $L_{+}L_{-}$?

 

[stevendaryl]

I'm a little puzzled by the examples, because normally $L^2$ has eigenvalues $\hbar^2 l(l+1)$, where $l$ is a nonnegative integer. For your second example, $l$ would have to be fractional. It's possible for the total angular momentum (which includes both spin angular momentum and orbital angular momentum) to be fractional, but usually $L$ refers to orbital angular momentum.

But in any case, it's overkill to consider raising and lowering operators. The fact that's relevant is that if $L^2$ has the value $\hbar^2 l(l+1)$, then $L_z$ can take on any of the following values:

• $l$
• $l-1$
• $l-2$
• ...
• $-l$

 

[gfd43tg]

Hi stevedaryl,

sorry for my late response. I was able to go to my prof's office and get some help on this. Using your post alone, I was not equipped with the necessary understanding to answer this question. I needed to understand the sphere with cones inside to get some intuition for what $l$ and $m$ mean. After rereading your response, it makes sense what you were saying, I just needed to know about $m$