What is the solution for the quantum homework questions?

[noospace]

Homework Statement

1. Consider a beam of z-oriented electrons, 80 % up, 20 % down which is passed through an x-oriented Stern-Gerlach machine. What percentage of electrons are measured in the +/- x-directions?

2. Consider deuterium. Nuclear spin = 1 with 1 electron orbiting in the n =1 state. Write down the ket for the total angular momentum $|\frac{3}{2} \frac{1}{2}\rangle$ as a linear combination of composite states.

The Attempt at a Solution

1. I write the eigenstates of $S_z$ in a superposition $\sqrt{0.8} (1,0)^T + \sqrt{0.2} (1,0)^T$ (where T denotes transpose) and set it equal to an linear combination of the $S_x$ eigenstates $a(1,1)^T + b(1,-1)^T$. Solving for a and b I get 45 % and 5 %. Interestingly they don't add to 100 % which was what I was expecting. Is this physically reasonable?

2. Do I just write $|3/2,3/2\rangle = |1,1\rangle |1/2,1/2 \rangle$ and apply lowering operators? I can do this because l = 0 so there is no orbital contribution to the angular momentum right?

 

[Meir Achuz]

"Consider a beam of z-oriented electrons, 80 % up, 20 % down"

This does not mean the WF you wrote. Each electron would be either pure up or pure down, and 1/2 would go each way in the z direction, just as for an umpolarized beam.
A problem here is that the usual SG experiment does not work for charged particles.

 

[Meir Achuz]

You are correct for 2.