The ** expectation value** of a quantum-mechanical operator is the weighted average value of its observable, and is defined as:

The above equation has roots in probability theory, where the expectation value or expected value of an observable is , with being the probability of observing the outcome .

From eq30, , and so

We further postulate that eq34 is valid for an infinite dimensional Hilbert space.

###### Question

Why is ?

###### Answer

Consider an operator with a complete set of orthonormal basis eigenfunctions . So, any eigenfunction can be written as a linear combination of these basis eigenfunctions, i.e. . According to the Born rule, the probability that a measurement will yield a given result is , where . So,

We have used the orthonormal property of in the 2^{nd }and 3^{rd} equalities. is interpreted as the probability that a measurement of a system will yield an eigenvalue associated with the eigenfunction . Therefore,

###### Question

Using the Schrodinger equation, show that the expectation value of the Hamiltonian is .

###### Answer

Multiplying both sides of eq40 on the left by and integrating over all space, we have

If the wavefunction is normalized, the above equation becomes .