Why is the expectation value of an observable what it is (the formula)

[gothloli]

Homework Statement

I really do not understand why the expectation value of an observable such as position is
<x> = $\int\Psi*(x)\Psi$

Homework Equations

If Q is an operator then
<Q> = = $\int\Psi*(Q)\Psi$
cn = <f,$\Psi>$

The Attempt at a Solution

What I understand this is saying is that since x is a linear transformation and $\Psi$ is an eigenvector, by x$\Psi$ would be a vector in position space. Then taking the expectation value is like taking the inner product of $\Psi$, and a position eigenfunction. But why would that give an average value? What I vaguely understand is that $\int\Psi^{2}$, 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.

 

[mfb]

It is like a weighted average, where the weight appears via the wave function in the integral.
It just works this way. Why? Ask nature.

 

[vanhees71]

It's important to understand this. It's the key of the probabilistic meaning of the quantum mechanical state. An observable is represented by a self-adjoint operator in Hilbert space, and there's a complete set of (generalized) eigenfunctions $u_j(x)$ of this operator. The (generatlized) eigen values (or better "spectral values") of the operator are the possible values the observable can take, and the probability to measure the $j$th eigenvalue, given the system is prepared in a state described by the normalized wave function $\psi$, is given by
$$P_j=\left |\int \mathrm{d}^3 x u_j^*(x) \psi(x) \right|^2.$$
The expectation value is then given by
$$\langle O \rangle = \sum_j o_j P_j.$$
Using the completeness of the (generalized) wave functions you can then easily show that you can write this expectation value in the way given in your question.

 

[gothloli]

Ok, I understand, that makes sense.

 

[paranormal]

As a scientist, it is important to understand the mathematical foundations and concepts behind quantum mechanics. The expectation value of an observable is a fundamental concept in quantum mechanics and is derived from the mathematical framework of linear algebra and probability theory.

To understand why the expectation value of an observable is given by the formula <Q> = \int\Psi*(Q)\Psi, we need to first understand the concept of an operator. In quantum mechanics, operators are used to describe physical observables such as position, momentum, and energy. These operators act on wave functions, which represent the state of a quantum system.

In the formula, Q represents the operator for the observable we are interested in, such as position. The wave function, \Psi, represents the state of the system. The complex conjugate of the wave function, \Psi*, represents the probability amplitude of the system.

When we take the inner product of \Psi and Q\Psi, we are essentially multiplying the probability amplitude by the operator. This gives us the average value of the observable Q for the system described by the wave function \Psi. This is because the probability amplitude represents the likelihood of finding the system in a particular state, and multiplying it by the operator gives us the average value of that observable.

The integral in the formula represents the sum of all possible states of the system. This is because in quantum mechanics, the state of a system is described by a wave function, which can take on an infinite number of values. Thus, to calculate the average value of an observable, we need to consider all possible states of the system, hence the integral.

In summary, the expectation value of an observable is calculated using the formula <Q> = \int\Psi*(Q)\Psi, which is derived from the mathematical concepts of linear algebra and probability theory. It represents the average value of the observable for a given system, taking into account all possible states of the system.