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Relation between a measurement and the operators

  1. Mar 7, 2008 #1
    Now, here is the problem. (Capital letters indicate operators, lower letters are states, * indicates Hermitian conjugate)
    Say we know that state | p > = cos(a) |0> + sin(a) |1> (0<a<PI, a is in R)
    Two operators : M1= |0><0| , M2=|1><1|, apperatantly they satisfy the completeness equation. M1*M1+M2*M2 = I

    According to some textbooks, if we perform ONE opeartor on |p>, we should have TWO outcomes |0> or |1>, with certain probabilities. Here is the details:

    If I use one operator on the state, say M1,
    then: the state AFTER the operation is: M1 |p> / sqrt(<p | M*M |>p) = cos(a) |0> / cos(a) = |0> ........(1)
    the probability of this happening is PM1= <p | M*M |>p = cos(a)^2 .........(2)

    NOW, according to (1), it is impossible to get state (outcome) |1> after the operation.
    SO, where do the after-state (outcome) |1> go?
    I can get the outcome of |1> only with the other operator M2.
    M2 |p> / sqrt(<p | M*M |>p) = sin(a) |1> / sin(a) = |1> .......(3)
    with the probability PM2=sin(a)^2..............(4)

    To me, it seems that :
    The language of "use operator on state |p>" is not correct.
    A measurement can NOT contain ONLY a subset of the operators (in this case, contains only M1, not M2),
    A measurement have to be composed of a SET of operators which satisfies completeness equation.


    Is this the case in real-world experiments? We can not say "perform M1 on p", but have to say "perform the measurement which has M1 and M2 on p" or "measure the spin in the |0> to |1> axis"?

    BTW, put it in another way. If we keep M1 unchanged, but change M2 to | 1/sqrt(2)( |0> - |1>) > < 1/sqrt(2)( <0| - <1|)|, (they do not satisfy the compleness equation). SO, a measurement containing ONLY M1 and M2 can not be made in real life. Is this right?
     
  2. jcsd
  3. Mar 7, 2008 #2

    Doc Al

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    Staff: Mentor

    I'd say that if the operator representing the quantity that you are measuring is M, and it's eigenvectors are |0> (eigenvalue = M1) and |1> (eigenvalue = M2) (and that's the complete set of eigenvectors for that operator), then for any initial state | p > = cos(a) |0> + sin(a) |1>, the projection operators acting on that state can give you the probability amplitude of a measurement of M giving any particular value (M1 or M2).
     
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