1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sakurai, Chapter 1 Problems 23 & 24

  1. Dec 23, 2008 #1
    Problem 23:
    If a certain set of orthonormal kets, [tex] |1> |2> |3> [/tex], are used as the base kets, the operators A and B are represented by

    A = \left( \begin{array}{ccc} a & 0 & 0 \\
    0 & -a & 0 \\
    0 & 0 & -a \end{array} \right)

    B = \left( \begin{array}{ccc} b & 0 & 0 \\
    0 & 0 & -ib \\
    0 & ib & 0 \end{array} \right).


    A and B commute. Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket?

    Problem 24:
    Prove that [tex] (1 / \sqrt{2})(1 + i\sigma_x) [/tex] acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle [tex] -\pi / 2[/tex]. (The minus sign signifies that the rotation is clockwise.)
  2. jcsd
  3. Dec 23, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hi quantumkiko! :wink:

    Show us what you've tried, and where you're stuck, and then we'll know how to help. :smile:
  4. Dec 23, 2008 #3
    Hi Tim!

    In problem 23, I don't know how to represent the simultaneous eigenkets [tex] |a, b> [/tex]. I just know how to solve the eigenvalues for each operator using the characteristic equation (some are degenerate). I also know that for two commuting observables, their simultaneous eigenkets form a complete set. Therefore, their simultaneous eigenkets are automatically orthogonal. That's all.

    For problem 24, I think we have to show that the result of letting the operator [tex] (1 / \\sqrt{2})(1 + i\\sigma_x) [/tex] act on a spinor is equivalent to a rotation operator acting on the same spinor. For a spinor of unit length, I used the matrix representation [tex] \left( \begin{array}{c} \cos \theta & \sin\theta \end{array} \right) [/tex] (I think this is where I was wrong.) Since the angle of rotation is [tex] -\pi / 2 [/tex], the rotation matrix will be given by,

    [tex] \left( \begin{array}{cc} cos(-\pi / 2) & sin(-\pi / 2) \\ -sin(-\pi / 2) & cos(-\pi / 2) \end{array} \right) = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) [/tex].

    If I let this operator act on the spinor, the resulting s
    Last edited: Dec 23, 2008
  5. Dec 23, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    Hint: in problem 23, just look at the bottom right-hand 2x2 square of A …

    it's a multiple of the unit matrix!

    so its eigenkets are … ? :smile:
  6. Dec 23, 2008 #5
    It's eigenkets are [tex] \left( \begin{array}{c} 1 \\ 1 \end{array}\right) and \left( \begin{array}{c} -1 \\ -1 \end{array}\right) [/tex]?
  7. Dec 23, 2008 #6


    User Avatar
    Science Advisor
    Homework Helper

    waah! :cry:

    think … if C is the 2x2 unit matrix,

    for what vectors or kets V is CV = V? :biggrin:
  8. Dec 23, 2008 #7
    Oh, for all kets V! So how does that fit into finding the simultaneous eigenstates of A and B?
  9. Dec 23, 2008 #8


    User Avatar
    Science Advisor
    Homework Helper

    Well, there's one obvious simultaneous eigenstate …

    and once you've found the other two eigenstates of B, they're bound to be eigenstates of A also. :smile:

    (i'm logging out now for a few hours :wink:)
  10. Dec 23, 2008 #9
    I got it! The obvious one is [tex] \left( \begin{array}{c} 1 & 0 & 0 \end{array} \right) [/tex] while the others are [tex] \left( \begin{array}{c} 0 & 1/\sqrt{2} & i/\sqrt{2} \end{array} \right) [/tex] and [tex] \left( \begin{array}{c} 0 & -1/\sqrt{2} & -i/\sqrt{2} \end{array} \right) [/tex]. Thank you very much!

    Now how about Problem # 24?
  11. Dec 23, 2008 #10


    User Avatar
    Science Advisor
    Homework Helper

    erm … they're the same!! :redface:
    Le'ssee …
    Well … to prove it's a π/2 rotation …

    the obvious thing to do is to square it! :biggrin:
  12. Dec 23, 2008 #11
    Oh yeah, I should really get different eigenkets, not just multiples of one of the other. So the other two should be [tex]
    \left( \begin{array}{c} 0 & 1/\sqrt{2} & i/\sqrt{2} \end{array} \right)
    [/tex] and [tex]
    \left( \begin{array}{c} 0 & i/\sqrt{2} & 1/\sqrt{2} \end{array} \right)
    I was thinking that they won't be orthonormal, but I forgot that one of the [tex] i [/tex]'s changes sign when doing the inner product.

    I got Problem # 24 also. Thank you!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Sakurai, Chapter 1 Problems 23 & 24