The expression $\int | \Psi\left(x\right)|^2dx$ gives the probability of finding a particle at a given position.
If wave function gives the probabilities of positions, why do we calculate "expectation value of position"?
I don't understand the conceptual difference, we already have a wave function of a position. Expectation value is related to probabilities.
So what is the differences between them? And why do we calculate expectation value for position, although we have a function for probability of finding a particle at a given position?
Best Answer
In position-space (that is, when your functions are functions of x), the function $\int|\Psi|^2$ gives the probability of finding the particle in a given range. The expectation value of x is where you'd expect to find the particle. It is often essentially the weighted average of all the positions where the probability density, $|\Psi|^2$, is the weighting function (that's not exactly what it is, but it's a useful analogy). Similarly, you can find the expectation value for any measurable quantity. In this space, the difference between the two is that the expectation value is a number that represents the expected average position of the particle over many measurements whereas the probability is a number that gives you the probability for finding the particle within the limits of integration.
However, you can use any different basis. For example, you could choose momentum-space, $\left|\Psi\right>$ is $\Psi(p)$ (quantum physicists please don't kill me for that affront to notation). In momentum space, the integral $\int|\Psi|^2$ is now the probability of the particle having a given range of momenta. However, the expectation value of x is still the average measurement of x. What, you ask, is the point? The expectation value is a number that can be found in any basis that represents the "on-average" value of a measurement. The probability found by $\int|\Psi|^2$ is the probability that a particle will be found existing within a specified range of values for the basis you are using.
$\int_{x_1}^{x_2}|\Psi|^2dx$ is "there is #% chance that the particle will be found between $x_1$ and $x_2$"
$\left<\Psi\right|x\left|\Psi\right>$ is "the expected average position of the particle over a large number of sample measurements is at $x$=#"
$|\Psi|^2(x)$ is a function "the probability per unit length of finding the particle at this position is #%"