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Rotations of spins and of wavefunctions

  1. Dec 15, 2015 #1
    • Thread moved from a technical forum, so homework template missing
    This is a question regarding the intrinsic angular momentum S of a particle of spin 1.
    Assuming S = s(s+1)I = 2I and I is the identity operator. In our case s = 1.
    Let |z> be a ket of norm 1 such that Sz |z> = 0, and let |x> and |y> be the ket vectors
    obtained from it by rotations of + 1/2 Pi about the y-axis and − 1/2 Pi about the x-axis
    respectively. Prove the following relations, as well as those resulting from circular
    permutation of x, y, and z:
    Sx |x> = 0 , Sx |y> = i|z> , Sxˆ2 |y> = |y> , Sx |z> = −i|y> , Sx2|z> = |z>
    Use these to show that |x>, |y>, |y> form an orthonormal basis and that the matrices
    representing Sx , Sy and Sz in that basis are:
    Sx = \begin{bmatrix}
    0 & 0 & 0 \\
    0& 0 & -i\\
    0&i &0
    \end{bmatrix}
    Sy = \begin{bmatrix}
    0 & 0 & i \\
    0& 0 & 0\\
    -i&0 &0
    \end{bmatrix}
    Sz = \begin{bmatrix}
    0 & -i & 0 \\
    i& 0 & 0\\
    0&0 &0
    \end{bmatrix}



    Show that the eigenvalues of Sz are the same as those of Jz , where Jz is the matrix
    representation of the z component of the angular momentum in the |j m> basis, for
    j = 1. Then find the most general unitary matrix that transforms
    Sz into Jz : U Sz U† = Jz .Choose the arbitrary parameters in U so that it also
    transforms Sx into Jx and Sy into Jy .
    This problem can be found in Quantum Mechanics by Albert Messiah volume 1 chapter 8 problem 10 part ii and iii.
    see attachment.
    My humble attempt at the problem,
    Used a rotation 3-d matrix for x,y,z or:
    Rx = \begin{bmatrix}
    1 & 0 & 0 \\
    0& cos(a) & -sin(a)\\
    0&sin(a) &cos(a)
    \end{bmatrix}
    Ry = \begin{bmatrix}
    cos(a) & 0 & sin(a) \\
    0& 1 & 0\\
    -sin(a)&0 &cos(a)
    \end{bmatrix}
    Rz = \begin{bmatrix}
    cos(a) & -sin(a & 0 \\
    sin(a) &cos(a) & 0\\
    0&0 &1
    \end{bmatrix}
    After that substituting angle for x and y does not give me any meaning to go further. Probably I am starting in a wrong way. Help appreciated.
     

    Attached Files:

  2. jcsd
  3. Dec 17, 2015 #2

    vela

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    Those rotation matrices aren't the correct ones for this problem. You need the ones that work on the angular momentum space.
     
  4. Dec 18, 2015 #3

    DrClaude

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    Staff: Mentor

    Why not? I seem to get the right answer by the unitary transform ##S_x = R_y S_z R_y^{\dagger}##.
     
  5. Dec 18, 2015 #4

    blue_leaf77

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    Homework Helper

    I think one could also do this problem matrix-less and hence saving lots of space, instead only works with ket notations. For example for the first task to prove ##S_x|x\rangle = 0##, one can use the substitution ##|x\rangle = \exp(-iS_y \pi/2) |z\rangle##. Have tried following this path, unfortunately stuck with the involved algebra. By the way the OP seems to be long absent anyway.
     
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