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Superposition of eigen

  1. Mar 9, 2005 #1
    I know this question isn't supposed to be hard but I can't figure it out for the life of me.

    If a certain wavefunction is made by superposition of three eigenfunctions of the momentum operator (F1, F2, and F3): wavefunction=0.465F1+0.357F2+0.810F3. The eigenvalues of those eigenfunctions are f1=+0.10, f2=-0.47, and f3=+0.35. What is the probability of a single measurement giving a momentum of +0.10? What is the probability of a single measurement giving a momentum of -0.20? and What is the expectation value of the momentum of the particle?
     
  2. jcsd
  3. Mar 9, 2005 #2

    dextercioby

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    Homework Helper

    Apply the 3-rd principle and the definition of expectation value.

    Daniel.
     
  4. Mar 9, 2005 #3

    Doc Al

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    Staff: Mentor

    The probability of a getting a particular eigenvalue when making a measurement is proportional to the (complex) square of the coefficient for that eigenfunction in the wavefunction:
    [tex]\Psi = C_1 F_1 + C_2 F_2 + C_3 F_3[/tex]
    Assuming the wavefunction is normalized (as is the one in this example), then the probability of obtaining a value of f1 is [itex]{C_1}^*C_1[/itex].
    The only possible values for a measurement are the eigenvalues associated with eigenfunctions that appear in the wavefunction (with non-zero coefficients).
    The expectation value is the weighted average of all possible measurements:
    [tex]<p> = {C_1}^*C_1 f_1 + {C_2}^*C_2 f_2 + {C_3}^*C_3 f_3[/tex]
     
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