Probabilities for orthonormal wavefunctions

[kac9]

Ok I have two orthonormal wavefunctions of a system, $$\psi$$ 1 and $$\psi$$ 2 and $$\widehat{A}$$ is an observable such that

$$\widehat{A}$$ |$$\phi$$ $$_{n}$$ > = a$$_{n}$$ |$$\phi$$ $$_{n}$$ >

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)

 

[christianjb]

If I understand your question-

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

 

[George Jones]

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].

 

[christianjb]

George: Ah yes, the state is either in psi1 or psi2- my mistake.

 

[christianjb]

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>