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Spin system, quantum mechanics

  1. Mar 28, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider a spin system with noninteracting spin 1/2 particles. The magnetic moment of the system is written as:
    μ = (ħq/2mc)σ
    Where σ = (σx, σy, σz) is the Pauli spin operator of the particle. A magnetic field of strength Bz is applied along the z direction and a second field Bx is applied along the x direction. The Hamiltonian of the particles is:
    H = H0 + V
    H0 = -μzBz
    V = -μxBx
    a) Find the eigenvalues and eigenkets of H0
    b) Express V in terms of σ+ and σ-
    c) Find the eigenvalues and eigenkets of H

    2. Relevant equations
    σ+ = σx + iσy
    σ- = σx - iσy
    σ+ = |+><-|
    σ- = |-><+|
    σz|+> = 1|+>
    σz|-> = -1|->

    3. The attempt at a solution
    For part a) I'm pretty sure I did it right
    H0 = (-qBzħ/2mc)σz or H0 = ε0σz if ε0 = -qBzħ/2mc.
    The operators H0 and σz commute so they have the same eigenkets |+> and |->
    Using the expression for H0 and the eigenvalue equations for σz given above the eigenkets of H0 are ε0 and -ε0.
    Part b I'm not so sure. I wrote the expression for V in the same way that I did for H0:
    V = (-qBxħ/2mc)σx
    Then using equations given above I worked out that σx = (σ+ + σ-)/2, so I subbed that into the the expression to get:
    V = (-qBxħ/4mc)(σ+ + σ-)
    When I try to do part c I start running into problems and I think it is because I did something in part a or b wrong. Can anyone tell me if I've made any mistakes in part a or b?
     
  2. jcsd
  3. Mar 28, 2016 #2

    blue_leaf77

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    That looks right.
    What are your problems?
     
  4. Mar 28, 2016 #3
    When I try to find the eigenvalues of H I get the expression:

    (-ε0σz - (qBxħ/4mc)(σ+ + σ-))|+> = ε|+>

    and I don't know how to evaluate it properly. I tried plugging these in: σ+ = |+><-|, σ- = |-><+| but I have no idea what to do after that. Are you supposed to expand it so that you're taking the eigenvalue of the first term and the eigenvalues of the second term?
     
  5. Mar 28, 2016 #4

    blue_leaf77

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    Write the matrix form of ##H## using the knowledge of the matrix form of Pauli matrices. Then solve the eigenvalue problem in resulting matrix equation.
     
  6. Mar 29, 2016 #5
    I'm not very familiar with the matrix form of Pauli matrices, it wasn't covered in this course... Do you know of a source I can read that would help?
     
  7. Mar 29, 2016 #6

    DrClaude

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    Staff: Mentor

    There is an alternative route to the solution. You know that the |+> and |-> kets for a complete basis, therefore the eigenstates of H can be written as a|+> + b|->. Try solving H (a|+> + b|->) = E (a|+> + b|->) for a and b (along with proper normalization).
     
  8. Mar 30, 2016 #7

    blue_leaf77

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    Pauli matrices are related to the spin matrices of spin 1/2 particles. But if you are not yet familiar with those matrices, DrClaude's suggestion above will also do the job.
     
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