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Need help understanding this: Deriving expressions for spin 1/2 systems

  1. Feb 7, 2012 #1
    So this is my first quantum mechanics class and so far we have spent about a month (well 3 weeks) on just the mathematics.

    So in my book, the author explores the ideas presented by the stern gerlach experiment. It related the beam of silver atoms traveling through the separators to light traveling through two crossed polarizes by making a parallel that even though you can eliminate the x component, it can be returned if the light goes though a medium that "rotates" the light. This made a lot of sense to me and I was very happy with the analogy. From this, they constructed the expressions for spin-1/2 eigenkets and operators.

    So now the book is moving to the idea of deriving these expressions through the mathematics in quantum mechanics.

    So first, since there are two even distribution after passing through a separator, thus from the probabilities you get.

    [itex]|\left\langle +|S_x;+ \right\rangle |=|\left\langle -|S_x;+ \right\rangle| = \frac{1}{\sqrt{2}}[/itex]
    Where [itex]|S_{x};+> [/itex] and [itex]|S_{x};- > [/itex] are the eigenkets for spin up and spin down.

    This next step is where I am confused.
    [itex]|S_x;+> =\frac{1}{\sqrt{2}}|+ > +\frac{1}{\sqrt{2}}e^{i \delta_1}|- >[/itex]

    And then they say that the expression for the minus ket must be orthogonal since it is mutually exclusive and they get...
    [itex]|S_x;->=\frac{1}{\sqrt{2}}|+>-\frac{1}{\sqrt{2}}e^{i \delta_1}|->[/itex]
    This I think I DO get but why are they representing one term as a wave? This logical leap doesn't make sens to me. Is there some physical interpretation that would help me?

    Thanks so much in advance for your help!

    EDIT: The angled brackets were screwing up the Latex so I just used ">"
     
    Last edited: Feb 7, 2012
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  3. Feb 7, 2012 #2

    Physics Monkey

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    What do you mean by "representing one term as a wave?"
     
  4. Feb 7, 2012 #3
    Well from all the physics I have done so far, [itex]e^{i\delta_1}[/itex] represents a wave correct? From Euler's formula.
     
  5. Feb 7, 2012 #4

    Physics Monkey

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    I would say it represents a complex number of unit magnitude. It certainly plays an important in wave physics, but I wouldn't say that it directly represents a wave.

    Terminology aside, the rules of quantum mechanics state that the coefficient of each state can be a complex number so long as the sum of the square magnitudes is one. Also, the overall phase doesn't matter. If you demand that + and - have equal probability of 1/2 and you use the overall phase of the wavefunction to make the coefficient of + real, then you should be able to convince yourself that the only freedom you have left is a complex phase given by [itex] e^{i \delta_1} [/itex].

    To determine the physical meaning of this phase, you could calculate the expecation values of Sx and Sy in the states you wrote down.
     
  6. Feb 7, 2012 #5
    I believe that I understand. What you say makes sense. Thank you!
     
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