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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>

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>

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