- #1
Wolvesloveme
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Homework Statement
The problem is a series of problems all relating to each other. I have gotten most of this done but I will add my answers and continue until the last part which is my main question: We choose a magnetic field in the z direction. Use your knowledge of 2-dimensional representations of 1. Sx, Sy, and Sz to express the Hamiltonian as a matrix:
My answer to this:
H_z = -γBcosθ1/2\hbar (1 0, 0 -1)
H_x = -γBsinθ1/2\hbar (0 1, 1 0 )
H_y= 0
Part 2: Find the eigenvalues and eigenspinors of the Hamiltonian matrix:
My answer to this:
Eigenvalues of H_z= \lambda=\pmγBcosθ1/2\hbar
Eigenvalues of H_x= \lambda=\pmγBsinθ1/2\hbar
Eigenvalues of H_y= 0
__________________________
Eigenspinor of Hz using those eigenvalues: [0 1] and [1 0]
Eigenspinor of Hx using those eigenvalues: [1 -1] and [1 1]
Eigenspinor of Hy using those eigenvalues: [0 0] <---what is up with all my "y" answers??
Part 3: Normalize the eigenspinors
My answer: Everything that I am doing is saying that these are already normalized? Is that not correct?
Part 4: Get back to experimental measurements. Suppose we collect the atoms that are in the state after going through the region of magnetic field described above. With what probability would a subsequent measurement of the spin in each of the following directions yield ? (The measurements are not consecutive.)
a. z-axis
b. x-axis
c. y-axis
MY answer: Here is where I am stuck.
Homework Equations
I know I have my eigenspinors for my 1/2 spin particles, I have the eigenvalues also. I know the probabilities having read multitude of experiment data on what is going on, and I know I am supposed to measure starting from ONE state, to get the result to the WANTED state.
The Attempt at a Solution
I know this is not in line with the forum guidelines but I really do mean it when I say I am dead stuck. I only know that I have to start with one state to get the other, or perhaps I am confused in this sense?
