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Spin 1/2-Raising and Lowering operators question

  1. Jul 16, 2012 #1
    Spin 1/2--Raising and Lowering operators question

    Hi,

    Quick question regarding raising and lowering operators.

    Sakurai (on pg 23 of Modern QM), gives the spin 1/2 raising and lowering operators [itex]S_{+}=\hbar \left|+\right\rangle \left\langle-\right|[/itex] and [itex]S_{-}=\hbar \left|-\right\rangle \left\langle+\right|[/itex].

    Acting with the raising operator on, say, the spin down state, you get
    [itex]S_{+} \left|-\right\rangle = \hbar \left|+\right\rangle [/itex]. The physical interpretation of this is that the raising operator increases the spin component by one unit of [itex]\hbar[/itex].

    This makes sense to me but when I try to explicitly verify this I run into a misunderstanding.

    Let's say I apply [itex]S_{+}[/itex] to [itex]\left|-\right\rangle[/itex] and get [itex]\hbar \left|+\right\rangle [/itex].
    To then "measure" the eigenvalue of this spin-up state, would you not apply the [itex]S_{z}[/itex] operator, which would give another factor of [itex]\hbar[/itex]:
    [itex]S_{z} S_{+} \left|-\right\rangle = S_{z} \hbar \left|+\right\rangle = \frac{\hbar^{2}}{2} \left|+\right\rangle[/itex]

    Or is it a mistake to apply [itex]S_{z}[/itex] after applying the raising operator? If not, how does the extra factor of [itex]\hbar[/itex] disappear?

    Thanks.
     
    Last edited: Jul 16, 2012
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  3. Jul 17, 2012 #2

    Bill_K

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    Re: Spin 1/2--Raising and Lowering operators question

    tjny699, I believe most people would say that S+ is ħ times the raising operator. The raising operator just turns |-> into |+>.
     
  4. Jul 17, 2012 #3
    Re: Spin 1/2--Raising and Lowering operators question

    Hi Bill_K,

    Thanks for your response.

    Do you mean that [itex]S_{+}[/itex] should be defined without the [itex]\hbar[/itex]?

    I always see it defined as in the original post. Also, later on the raising operator is defined in terms of the [itex]S_{x}[/itex] and [itex]S_{y}[/itex], which definitely have to have the [itex]\hbar[/itex]. Or perhaps I misunderstood you?
     
  5. Jul 17, 2012 #4

    vanhees71

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    Re: Spin 1/2--Raising and Lowering operators question

    That's right, you just have to normalize your states to 1, and everything is fine.
     
  6. Jul 17, 2012 #5
    Re: Spin 1/2--Raising and Lowering operators question

    So the correct definition is without [itex]\hbar[/itex]? Is there a reason that Sakurai includes it then?
     
  7. Jul 17, 2012 #6

    Bill_K

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    Re: Spin 1/2--Raising and Lowering operators question

    S± and Sz have dimensions of angular momentum and their definition must include ħ. The raising and lowering operators are dimensionless and do not include ħ. If Sakurai calls S+ the raising operator, I would disagree, they are only proportional to each other.
     
  8. Jul 17, 2012 #7
    Re: Spin 1/2--Raising and Lowering operators question

    Thanks, Bill. [itex]S_{+}[/itex] is called the raising operator in a number of sources that I've taken a look at. Naming aside, is it correct to apply [itex]S_{z}[/itex] after applying [itex]S_{+}[/itex] as I did in the first post? Sorry to linger on this but I'm trying to understand how this works in detail.

    The spin down state when acted upon by [itex]S_{+}[/itex] becomes the spin up state, and to measure it's spin we apply [itex]S_{z}[/itex]? Does this not give an extra factor of [itex]\hbar[/itex]?
     
    Last edited: Jul 17, 2012
  9. Jul 17, 2012 #8

    Bill_K

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    Re: Spin 1/2--Raising and Lowering operators question

    Then I would have to disagree with every one of them. Unless they are working in units where ħ = 1.
     
  10. Jul 17, 2012 #9
    Re: Spin 1/2--Raising and Lowering operators question

    OK, I guess my other question still stands--after applying [itex]S_{+}[/itex] to a spin down state do we need to then apply [itex]S_{z}[/itex] to measure the spin of the resulting state? If so, does this not lead to an extra factor of [itex]\hbar[/itex]
     
  11. Jul 18, 2012 #10
    Re: Spin 1/2--Raising and Lowering operators question

    I think I understand your question, but if I've misinterpreted it, let me know.

    The normalization of the state doesn't affect the eigenvalue of an operator. To see this, remember the definition of an eigenvalue [itex]\lambda[/itex]:

    [tex]A\Psi = \lambda\Psi[/tex]

    Now try applying that to a scaled version of [itex]\Psi[/itex]:

    [tex]A(c\Psi) = c(A\Psi) = c(\lambda\Psi) = \lambda(c\Psi)[/tex]

    So you can see, the eigenvalue of [itex]c\Psi[/itex] is also [itex]\lambda[/itex], for any value of [itex]c[/itex].

    Applied to your question, if you have an operator [itex]S_z[/itex] and states [itex]|+\rangle[/itex] and [itex]|-\rangle[/itex], such that [itex]S_z|+\rangle = \hbar |+\rangle[/itex] and [itex]S_z|-\rangle = -\hbar|-\rangle[/itex], then we have:

    [tex]S_z(\hbar|+\rangle) = \hbar(\hbar|+\rangle)[/tex]

    So the spin is still just [itex]\hbar[/itex].
     
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