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## Homework Statement

I really do not understand why the expectation value of an observable such as position is

<x> = [itex]\int\Psi*(x)\Psi[/itex]

## Homework Equations

If Q is an operator then

<Q> = = [itex]\int\Psi*(Q)\Psi[/itex]

cn = <f,[itex]\Psi>[/itex]

## The Attempt at a Solution

What I understand this is saying is that since x is a linear transformation and [itex]\Psi[/itex] is an eigenvector, by x[itex]\Psi[/itex] would be a vector in position space. Then taking the expectation value is like taking the inner product of [itex]\Psi[/itex], and a position eigenfunction. But why would that give an average value? What I vaguely understand is that [itex]\int\Psi^{2}[/itex], is the probability density and multiplying operator by probability would give the average value. But I'm still confused as connecting linear algebra with probability and quantum, I'm having a hard time with that.