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Quantum Mechanics: Eigenstate

  1. Feb 16, 2015 #1
    1. The problem statement, all variables and given/known data
    Calculate ##\triangle S_x## and ##\triangle S_y## for an eigenstate of ##\hat{S}_z## for a spin##-\frac12## particle. Check to see if the uncertainty relation ##\triangle S_x\triangle S_y\ge \hbar|\langle S_z\rangle|/2## is satisfied.

    2. Relevant equations
    ##S_x =\frac12(S_+ +S_-)##
    ##S_y = \frac{1}{2i}(S_+-S_-)##

    3. The attempt at a solution

    I am not sure what I have to do in this problem. For the matrix representation I have that:

    ##S_x = \frac{\hbar}{2}\left[\begin{array}{ c c }0 & 1 \\1 & 0\end{array} \right]##
    ##S_y = \frac{\hbar}{2i}\left[\begin{array}{ c c }0 & -1 \\1 & 0\end{array} \right].##
     
  2. jcsd
  3. Feb 16, 2015 #2

    Orodruin

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    What is the definition of uncertainty? I suggest you start from that and see what you can find based on it.
     
  4. Feb 16, 2015 #3
    I was more concerned on the first part of the question.
     
  5. Feb 16, 2015 #4

    Orodruin

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    Yes? The uncertainty is ##\Delta S_x##, how is it defined? Later you can go on to the uncertainty relation, which as the name suggests is a relation of uncertainties.
     
  6. Feb 16, 2015 #5
    Ops, I thought you were talking about the uncertainty relation. But regarding ##\triangle S_x## the uncertainty is defined as ##\sqrt{\langle S_x^2\rangle -\langle S_x\rangle^2}##, where ##S_x = \frac{\hbar}{2}P_1 -\frac{\hbar}{2}P_2##.

    So all I do is do matrix multiplication with ##S_z##, i.e.

    ##(S_x) (S_z)=\frac{\hbar}{2}\left[\begin{array}{ c c }0 & 1 \\1 & 0\end{array} \right]
    \left[\begin{array}{ c c }1 & 0 \\0 & -1\end{array} \right].##
     
    Last edited: Feb 16, 2015
  7. Feb 16, 2015 #6

    Orodruin

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    So what do you get when you evaluate the expectation values for an eigenstate of ##S_z##? How do you compute the expectation value of any operator in a given state?
     
  8. Feb 16, 2015 #7
    To compute the expectation value of any operator ##\hat{\mathbb{O}}## for a particle in the state ##|\phi\rangle## is defined as ##\langle \hat{\mathbb{O}}\rangle = \langle \phi|\hat{\mathbb{O}}|\phi\rangle.## But what will the eigenstate for ##S_z## be?
     
  9. Feb 16, 2015 #8

    Orodruin

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    I suggest you take one of the ones required by the problem statement, i.e., any of the eigenstates of ##S_z##.
     
  10. Feb 16, 2015 #9
    That doesn't help me understand.
     
  11. Feb 17, 2015 #10

    Orodruin

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    So let us try it this way: What are the eigenstates of ##S_z##?

    Edit: Also, you should perhaps look for a different way of computing your expectation values. I suspect the one you quoted will not be very helpful. How would you compute ##P_1## and ##P_2## and how would you compute ##\left< S_x^2 \right>##?
     
  12. Feb 17, 2015 #11
    Well, in the book it states that ##\hat{\mathbb{J}}_+|\lambda,m\rangle## is an eigenstate of ##\mathbb{J}_z## (where I was told that ##\mathbb{J}_z## and ##\mathbb{S}_z## are interchangeable)##.

    So I can't use ##S_x = \frac{\hbar}{2}P_1 -\frac{\hbar}{2}P_2##? I do not know any other way of computing the expectation values other than that definition.
     
  13. Feb 17, 2015 #12

    Orodruin

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    Yes, but it requires that you already are familiar with another eigenstate. Are you familiar with how to find the eigenvectors of a matrix?

    Not unless you figure out how to comput ##P_1## and ##P_2##. What about the way you quoted in post #7?
     
  14. Feb 17, 2015 #13
    Yes, I am familiar with how to find eigenvectors. In quantum mechanics, I have difficulty in setting up the problem correctly. Computing it, I can do but setting it up I need a lot more practice with.



    That I know how to compute, its just that I am having trouble finding ##|\phi\rangle##. Since we have ##S_x##, i.e., ##
    S_x = \frac{\hbar}{2}\left[\begin{array}{ c c }0 & 1 \\1 & 0\end{array} \right].## Thus, I need to compute ##\langle\phi|S_x|\phi\rangle##.
     
  15. Feb 17, 2015 #14

    Orodruin

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    So, how do you represent ##S_z## in matrix form? What are the eigenvectors of that matrix? (The eigenvectors of the matrix represent the eigenstates.)
     
  16. Feb 17, 2015 #15
    In matrix form ##
    (S_z)=\frac{\hbar}{2}
    \left[\begin{array}{ c c }1 & 0 \\0 & -1\end{array} \right].##
     
  17. Feb 17, 2015 #16

    Orodruin

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    ... and so the eigenvectors are ...
     
  18. Feb 17, 2015 #17
    Will the eigenvectors just be ##|+z\rangle = {1\choose 0}## and ##|-z\rangle ={0 \choose 1}?##
     
  19. Feb 17, 2015 #18

    Orodruin

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    You tell me. Is ##S_z |+z\rangle = \lambda_+ |+z\rangle## for some constant ##\lambda_+##?
     
  20. Feb 17, 2015 #19
    Yup, it does satisfy that. I made this more difficult than it is. -__-. Thank you very much!
     
  21. Feb 17, 2015 #20

    Orodruin

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    Just to resolve the problem itself: What do you get for ##\langle S_x\rangle## and ##\langle S_x^2 \rangle##, respectively? What is the resulting uncertainty relation?
     
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