# Probabilities for orthonormal wavefunctions

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)

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

George Jones
Staff Emeritus
Gold Member

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].

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

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>