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Overlap of two spin one-half states.

  1. Sep 11, 2016 #1
    1. The problem statement, all variables and given/known data

    Consider a spin state |n; +> where n is the unit vector defined by the polar and azimuthal angles θ and φ and the spin state |n'; +> where n' is the unit vector defined by the polar and azimuthal angles θ' and φ'. Let γ denote the angle between the vectors n and n':

    n⋅n'
    = cos(γ)

    Show by direct computation that the overlap of the associated spin states is controlled by half the angle between the unit vectors:

    |<n';+|n;+>|^2 = cos^2(γ/2)

    2. Relevant equations

    The spin one-half state along an arbitrary direction can be written as n = (cos(θ/2) , exp(iφ)sin(θ/2))
    I am assuming that this is the only definition needed in order to solve the problem. I don't think knowing the spin operator would help in this case.

    3. The attempt at a solution

    I can understand the relation given in the problem conceptually, however the mathematics just wouldn't give me the result. My attempt is writing down the two vectors in terms of the two angles (two for each), and then doing the multiplication. At this stage, it is not much more than a math problem to be honest however I would like to see the correct approach/proof nevertheless. Thanks in advance for anyone taking the time to help out.
    There is no need to post the whole solution, you are welcome to do so, however the statement of the correct approach would be enough.

    Source: http://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps3.pdfhttp://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps3.pdf [Broken]
    Second question in the problem set.
     
    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Sep 11, 2016 #2

    TSny

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    The link you posted doesn't work for me. I think this will work
    http://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps3.pdf

    The problem is tedious but fairly straightforward. You have the right idea:
    You will also need to carry out the scalar product on the left side of n⋅n' = cos(γ) in order to relate cos(γ) to the polar and azimuthal angles of n and n'.
     
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