Expectation formula in Dirac notation.

[david.makcenz]

Expectation value of operator A is given by following formula in Dirac notation.

<A> = <x|A|x>

where
A : Operator
<A> : Expectation value of A
|x> : State

Somehow I am unable to convince myself that this formula is true.
Would someone please explain it to me?

Thanks

 

[blkqi]

$$\langle \hat{A} \rangle = \langle x|\hat{A}|x \rangle = \int \psi_{x}^{"*"} \hat{A} \psi_{x} \,d \tau$$

This is the expectation value postulate of quantum mechanics.

If the wavefunction $$\psi_{x}$$ is not an eigenfunction of the operator $$\hat{A}$$ then the measured value of the observable $$A$$ is variable. When $$\psi_{x}$$ is normalized such that
$$\int \psi_{x}^{"*"}\psi_{x} \,d \tau=1$$
the expectation value above yeilds the average measurement of all possible measurements of $$A$$. (Note that $$d \tau$$ indicates an integration over all space.) This works because, since $$\hat{A}$$ is an Hermitian operator, its eigenvectors form an orthonormal basis. It follows that any function can be written as an infinite linear combination of eigenfunctions of $$\hat{A}$$.