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Represent Action of Stern Gerlach Operator as a Matrix

  1. Jan 31, 2015 #1
    1. The problem statement, all variables and given/known data

    Given the series of three Stern-Gerlach devices:

    2hnm34n.jpg

    Represent the action of the last two SG devices as matrices ##\hat{A}## and ##\hat{B}## in the ##|+z\rangle, |-z\rangle## basis.

    2. Relevant equations

    ##|+n\rangle = cos(\frac{\theta}{2})|+z\rangle + e^{i\phi}sin(\frac{\theta}{2})|-z\rangle##

    3. The attempt at a solution

    A Stern-Gerlach device with one beam blocked off can be represented as a projection of the incoming state onto the outgoing state. A projection operator can be written as the outer product of the selected basis state with itself, e.g. ##|+z\rangle\langle\,+\,z\,|## would be the projection operator for an SGz device with the minus size blocked.

    The projection matrix of the first SGn is then:

    ##
    \hat{A}\rightarrow{|+n\rangle\langle\,+\,n\,|} =
    \begin{pmatrix}
    cos(\frac{\theta}{2}) \\
    e^{i\phi}sin(\frac{\theta}{2})
    \end{pmatrix}
    \begin{pmatrix}
    cos(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})
    \end{pmatrix} =
    \begin{pmatrix}
    cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\
    e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})
    \end{pmatrix}
    ##

    Therefore, the beam coming out of the positive side of the SGn should be:

    ##
    |+n\rangle = \hat{A}|+z\rangle \rightarrow
    \begin{pmatrix}
    cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\
    e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})
    \end{pmatrix}
    \begin{pmatrix}
    1 \\
    0
    \end{pmatrix} =
    \begin{pmatrix}
    cos^{2}(\frac{\theta}{2}) \\
    e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
    \end{pmatrix}
    ##

    Should this be equal to the given ##|+n\rangle##?

    I feel as though I'm missing something completely obvious or misunderstanding the question.

    Thank you for any help you can provide.
     
  2. jcsd
  3. Jan 31, 2015 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Hello and welcome to PF!
    Yes, the output of SGn should be |+n>, up to a normalization factor.
     
  4. Jan 31, 2015 #3
    So would ##\frac{1}{cos(\frac{\theta}{2})}## make for a suitable normalization factor? Is there a more formal way of determining this normalization factor?
     
  5. Jan 31, 2015 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes
    Not that I can think of at the moment. The usual way to find the normalization factor for a state |ψ>is to just look at <ψ|ψ>. But, in your example, you can spot the normalization factor by inspection.
     
  6. Jan 31, 2015 #5
    Thanks for you help. I still don't see how ##\langle\,+\,n\,|+n\rangle## would give me the normalization factor though. I'm getting ##cos^{2}(\frac{\theta}{2}) + sin^{2}(\frac{\theta}{2}) = 1## which is what is expected.
     
  7. Jan 31, 2015 #6

    TSny

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    Homework Helper
    Gold Member

    Let |ψ> be the output of GSn.
     
  8. Jan 31, 2015 #7
    Ahh I see and I set this equal to 1. Thank you for your help!
     
  9. Jan 31, 2015 #8
    This didn't work. I got a factor of ##\frac{1}{cos^{2}(\frac{\theta}{2})}## when evaluating ##\langle\psi|\psi\rangle## with ##|\psi\rangle## being the output of SGn.

    ##
    \begin{pmatrix}
    cos^{2}(\frac{\theta}{2}) &
    e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
    \end{pmatrix}
    \begin{pmatrix}
    cos^{2}(\frac{\theta}{2}) \\
    e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})
    \end{pmatrix} =
    cos^{2}(\frac{\theta}{2})[cos^{2}(\frac{\theta}{2}) + sin^{2}(\frac{\theta}{2})] = cos^{2}(\frac{\theta}{2})
    ##
     
    Last edited: Jan 31, 2015
  10. Jan 31, 2015 #9

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The normalization factor gets applied to both occurences of ψ, so you need the square root of ##\langle\psi|\psi\rangle## as normalization factor.
     
  11. Feb 1, 2015 #10
    Thank you. How does this then relate back to the matrix representation? Would it just be ##\frac{1}{cos(\frac{\theta}{2})}\begin{pmatrix}cos^{2}(\frac{\theta}{2}) & e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) \\ e^{i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2}) & sin^{2}(\frac{\theta}{2})\end{pmatrix}##
     
  12. Feb 1, 2015 #11

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    I don't think it makes sense to normalize the matrix. The norm smaller than 1 indicates that some particles get lost, which has a physical meaning.
     
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