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Calculate the eigenfunctions for a spin half particle

  1. Mar 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Spin can be represented by matrices. For example, a spin half particle can be described by the following Pauli spin matrices
    [tex] s_x = \frac{\hbar} {2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , s_y = \frac{\hbar} {2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} , s_z = \frac{\hbar} {2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} [/tex]

    Calculate the corresponding eigenfunctions which we will denote as ɑ- and β-eigenfunctions corresponding to spin 1/2 particles. Further show that sj can be determined by the commutation of the other two matrices sn and sm , n,m≠j.
    2. Relevant equations


    3. The attempt at a solution
    I have calculated the eigenvalues (first section of the question) of all three matrices to be ±ħ/2

    I think the diagonal matrix sz has eigenfunctions |α> = (1; 0) and |β> = (0; 1)
    From that I found the eigenfunctions of sx and sx to be
    |x+> = |α> + |β> & |x-> = -|α> + |β> and
    |y+> = -i|α> + |β> & |y-> = i|α> + |β> respectively

    But I'm not entirely sure I'm correct...
     
  2. jcsd
  3. Mar 9, 2015 #2

    DrClaude

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    Staff: Mentor

    Apart from normalization, these are correct.
     
  4. Mar 9, 2015 #3
    Thanks @DrClaude

    Any ideas as to how I show the commutation part of the question?
     
  5. Mar 9, 2015 #4

    DrClaude

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    Staff: Mentor

    Calculate the commutator for each pair of operators and see what you get.
     
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