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I think there's a typo on the Quiz 4 practice sheet. On problem 4 second line, it reads: <math>f_l^{m+1}=AL_+f_l^m</math>.

According to equation [4.120] it should read: <math>Af_l^{m+1}=L_+f_l^m</math>.

Can someone verify?

At this point, it is an undetermined constant. Whether it is on the left or right doesn't really matter.

Your fix does correspond to the formula in the book, though (eq. 4.120).

I guess the A in the quiz is equal to 1/A in the textbook.

On the first problem for the practice quiz, how is this normalized? Is there one factor that goes in front of the whole expression for chi? Or should there be a normalization factor in front of each component?

Edit: I think we find a specific value of theta. Does anybody know for sure?

My intention was that there should be an overall factor in front of chi.

Here is where I am and I'm a bit confused because my normalization constant is in terms of <math>\theta

\chi=A\begin{pmatrix}1+\cos{\theta}
\sin{\theta}\end{pmatrix}

normalizing…

\frac{1}{A^2}=\begin{pmatrix}1+\cos{\theta} & \sin{\theta}\end{pmatrix}\begin{pmatrix}1+\cos{\theta}
\sin{\theta}\end{pmatrix}

\frac{1}{A^2}=1+2\cos{\theta}+\cos^2{\theta}+\sin^2{\theta}=2+2\cos{\theta}

A=\frac{1}{\sqrt{2(1+\cos{\theta})}}

using

\cos{\frac{\theta}{2}}=\sqrt{\frac{1+\cos{\theta}}{2}}

A=\frac{1}{2\cos{\frac{\theta}{2}}}\\</math>

I think it is okay to have the normalization include <math>\theta</math>. The idea was to not have a square root, which you accomplished.

So do the probabilities have theta in them as well? We got <math>

\\P_{+\frac{\hbar}{2}}=\frac{1+2\cos{\theta}+\cos^2{\theta}}{4\cos^2{\frac{\theta}{2}}}
P_{-\frac{\hbar}{2}}=\frac{\sin^2{\theta}}{4\cos^2{\frac{\theta}{2}}}</math>

Yes, the probability should have theta in it. I think you can consider theta to be a value given in the problem, or something that could be experimentally determined.

are the relations <math>S{x}=[{S+} +{S-}]/2 </math>and<math> S{y}=[{S+} -{S-}]/2i </math>correct for all spin particles or only spin 1/2 particles?

That relationship should be good for any angular momentum. It comes from the definition of L+ and L- (eq. 4.105)

I just had a general question from section 4.3 on angular momentum. When applying the raising operator to the ladder of angular momentum states, why is it that the “process cannot go on forever”? I guess I don't see why we would eventually reach a state for which the z-component exceeds the total.

If the z-component keeps increasing, then it could definitely exceed the total angular momentum. Take the l=1 case. Here we can have <math>m_l</math> = -1,0,or 1. If you apply <math>L_+</math> to the -1 state, you return the zero state. If you apply it to the 0 state, you return the +1 state. If you apply it to the +1 state, you would end up with <math>L_z</math> being +2. This doesn't make any sense. The angular momentum in the z-direction would be greater than the particle's total angular momentum.

The z-component cannot exceed the total because <math>|L|^2=|L_x|^2+|L_y|^2+|L_z|^2 \ge 0+0+|L_z|^2 =|L_z|^2</math>.

Question 3a:

<math>J_+|\Psi>=\hbar\sqrt{2}|1,0>+\hbar|\frac{1}{2},\frac{1}{2}></math>

Or is there a different notation I should be using? And is this what other people are getting?

I get <math>J_+|\Psi>=\hbar\sqrt{2}|1,0>|\frac{1}{2},-\frac{1}{2}>+\hbar|1,-1>|\frac{1}{2},\frac{1}{2}></math>

I have <math>J_+|\Psi>=\hbar\sqrt{2}|1,0>|\frac{1}{2},-\frac{1}{2}>+\hbar|1,-1>|\frac{1}{2},\frac{1}{2}></math> too.

That is what I get also.

I got the same as Liux, Andromeda, and Devlin.

That's what I was getting, but is there more work to show for this? It seems, …..too basic???

Where does the root2 come from? It seems like this whole problem can be done just by reasoning except for the root2.

equation 4.121: <math>A_lm=\hbar\sqrt{(l+-m)(l-+m+1)}</math>

This is a normal length test, right (i.e., 50 min.)? I'm wondering why the practice test has 5 questions. Are other people expecting 5 questions? Do you think the idea is that we'll have 5 really short questions, or that there will be a very lenient curve?

I don't think it will be 5 questions long, I think he just wants us to know how to do these things. I expect it to be as long as the previous tests (well, hopefully not the first one since that one did take too long). I don't think he wants to create curves, he would rather we just do well. Good luck everyone!

Just like for the last test, this isn't actually a practice or previous test…It's a list of the types of problems we should know how to do for the test, and I don't think should be comparable in length.

I would agree with the previous two comments; it's just a sample of what we should know how to do for the exam. This last section since the last exam has covered quite a bit of material, and there's a lot that we can do with it, so the example problems are meant to be a supplement to studying: if you can sit down and do the example problems without outside reference material then you'll do well on the exam. If you get stuck on a certain type of problem then you know that's what you need to work on before the exam time. Also, each exam in the past has always had a problem that's very similar to one on the practice exams…

Also, I don't see a question in number 4, just a bunch of statements.

Yeah, this puzzled me as well. I think we're being subtly asked to find the normalization factor like in problem 4.18 from HW 10.

I didn't see one either, maybe it was just useful information??

I just assumed it meant to show how to find it using the given relations.

Hey, how do you do the first exam problem from today?

Well, it was difficult for me and I have no idea what I have to say about the test.

For first order theory, infinite square well, how do we know when to integrate over entire well or just parts of well?

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