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Finding Quantum Probabilities

  1. Mar 13, 2014 #1
    1. The problem statement, all variables and given/known data
    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[itex]\hbar[/itex] (1 0, 0 -1)
    H_x = -γBsinθ1/2[itex]\hbar[/itex] (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= [itex]\lambda[/itex]=[itex]\pm[/itex]γBcosθ1/2[itex]\hbar[/itex]
    Eigenvalues of H_x= [itex]\lambda[/itex]=[itex]\pm[/itex]γBsinθ1/2[itex]\hbar[/itex]
    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.

    2. Relevant 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 Im supposed to measure starting from ONE state, to get the result to the WANTED state.

    3. 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?
     
  2. jcsd
  3. Mar 14, 2014 #2

    vela

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    Please tell us the original problem statement as originally given to you. I have no idea what you mean by H_x, H_y, and H_z.
     
  4. Mar 14, 2014 #3

    strangerep

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    Science Advisor

    Whew! I'm glad you said that. I had no idea either. :uhh:
     
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