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Sequential Stern-Gerlach experiment to differentiate between two samples

  1. Sep 26, 2008 #1
    1. The problem statement, all variables and given/known data
    Compare the following situations (a) and (b):
    (a) A beam of atoms, half of which are preselected as “+ in z”, the
    other half as “− in z”, is sent through a Stern-Gerlach apparatus
    that
    ...(A) sorts in the z direction.
    ...(B) sorts in the x direction.

    (b) A beam of atoms, all of which are preselected as “+ in x”, is sent
    through a Stern-Gerlach apparatus that
    ...(A) sorts in the z direction.
    ...(B) sorts in the x direction.

    2. Relevant equations
    | x+ > = [tex]\frac{1}{\sqrt{2}}[/tex] | z+ > + [tex]\frac{1}{\sqrt{2}}[/tex] | z- >

    | z+ > = [tex]\frac{1}{\sqrt{2}}[/tex] | x+ > + [tex]\frac{1}{\sqrt{2}}[/tex] | x- >

    | z- > = [tex]\frac{1}{\sqrt{2}}[/tex] | x+ > - [tex]\frac{1}{\sqrt{2}}[/tex] | x- >

    \\Edit :The Question is whether we can differentiate between situation (a) and (b) by knowing the results of the experiment alone. \\

    3. The attempt at a solution
    I would think that the measurement results do not enable us to tell the situation apart.
    In (a), (A) will results in 50:50 selection and x-selector (B) will sort +/- x atoms 50:50.
    In (b), (A) will results in 50:50 selection and x-selector (B) will sort now +/- z atoms 50:50.

    However, I found the answer to be 'yes'. I am thinking that we can only tell the situation apart when there was no measurement done in (A) stage.
    Can someone explain?

    With Thanks,
    Felicity
     
    Last edited: Sep 27, 2008
  2. jcsd
  3. Sep 26, 2008 #2
    Sorry... having described your two experimental set-ups, what's the actual question? What will the outcomes of the two arrangements be?
     
  4. Sep 27, 2008 #3
    Sorry. The Question is whether we can differentiate between situation (a) and (b) by knowing the results of the experiment alone.
     
  5. Sep 28, 2008 #4
    Hmm. Well I've slept on the question... and have come to the conclusion that I don't know the answer :tongue: There's one suggestion I can make, however, that sounds sensible to me but shouldn't be accepted uncritically. (Hence, I'll be less oblique than I usually am in homework help, or else I might send you looking in completely the wrong direction!)
    The problem might really be about the time evolution of the system. You don't have to have a 50:50 sum of eigenstates in order to be in a superposition of states; and all that an eigenstate of z tells you about the x- and y- components is that they aren't eigenstates. Thus, it might be the case that the probability of getting spin -up or -down in the x-direction given that they were previously in an up-eigenstate of x is different to the case in which nothing about the history of the x-component is known.
    Sorry I can't be more conclusive, but I hope that helps!
     
  6. Sep 28, 2008 #5
    Muppet,
    I somehow see the point about probabilites being different. I have not learned about the superposition of state yet so I am still lost.
    Thank you all the same!
     
  7. Sep 28, 2008 #6
    From your post above I can see that you have- you just don't know it!
    Terminology: An eigenfunction of an operator is a function that, when operated upon by that operator, gives you itself mulitplied by some constant number called an eigenvalue. If a wave function is an eigenfunction of an operator corresponding to a physical quantity (an observable), then it is said to be in an eigenstate of that quantity.
    If a system is described by a wavefunction that is a linear combination of eigenstates (like your spin states above) then it is said to be in a superposition of states.
     
  8. Sep 28, 2008 #7
    Okay. Ah, well..
     
  9. Sep 28, 2008 #8
    Anything else I can help with as I'm online??
    The other suggestion I could make is that if you need help with a conceptual point you don't properly understand, you can always raise it in the QM forum. A lot more people check that.
     
  10. Oct 6, 2008 #9
    Hihi..
    Thanks for your offer! It is much better now :D
    Btw, I'd like to close my question. I found out that the question was rather misleading.

    I initially thought
    "(a) A beam of atoms, half of which are preselected as “+ in z”, the
    other half as “− in z”, is sent through a Stern-Gerlach apparatus
    that
    ...(A) sorts in the z direction.
    ...(B) sorts in the x direction."
    means
    Source ---> [ (A) + (B) ] ---- measurement result since it is a Stern-Gerlach apparatus.

    Apparently, the correct interpretation is
    Source ---> [A]
    Source ---> ---- two measurement results.

    Then, we can tell the situation (a) and (b) apart.
     
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