Probabilities for orthonormal wavefunctions

  • Thread starter kac9
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  • #1
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Ok I have two orthonormal wavefunctions of a system, [tex]\psi[/tex] 1 and [tex]\psi[/tex] 2 and [tex]\widehat{A}[/tex] is an observable such that

[tex]\widehat{A}[/tex] |[tex]\phi[/tex] [tex]_{n}[/tex] > = a[tex]_{n}[/tex] |[tex]\phi[/tex] [tex]_{n}[/tex] >

for eigenvalues a sub n

what are the probabilities p1(a1) and p2(a2) of obtaining the value a sub n in the state |psi1> and |psi2> respectively in terms of only phi sub i and psi sub 1(or 2)
 

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  • #2
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If I understand your question-

You can't work out the probabilities from knowledge of the eigenvalues of an operator.
 
  • #3
George Jones
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If I understand your question-

You can't work out the probabilities from knowledge of the eigenvalues of an operator.

But the probability that a measurement yields a particular eigenvalue can be expressed in terms of the associated eigenstate and the state of the system, which, if I have interpreted the original post correctly, kac9 has given.

kac9: I'm having trouble guiding you to the answer without just writing down the answer. This is a basic postulate of (shut up and calculate) quantum mechanics. It must be in your notes and text. If you're using Griffiths, it's equation [3.43].
 
  • #4
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George: Ah yes, the state is either in psi1 or psi2- my mistake.
 
  • #5
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Well- a hint would be to write down the identity operator in terms of |phi>

I=sum_i |phi_i><phi_i|

|psi_1>=I|psi_1>
 

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