# Expectation Value vs Probability Density

1. Oct 16, 2014

### jaydnul

I know the difference between the expectation value and probability density, but how do you calculate the probability density of an observable other than position? For position, the probability of the particle being in a particular spot is given by $|\Psi|^2$, which is the probability density, and the average of all those probabilities is given by $\int \Psi^*x\Psi$ correct?

Now for all the other observables (momentum, energy, etc...) the expectation values are straight foward, but how do I calculate the probability densities?

Thanks

Last edited: Oct 16, 2014
2. Oct 16, 2014

### Staff: Mentor

3. Oct 16, 2014

### jaydnul

So if i were given $\Psi(x)$, to find the momentum probability density I would convert to momentum space using a Fourier transform? What about the energy probability density?

4. Oct 16, 2014

### atyy

Let's restrict to projective measurements for simplicity. The potential outcome is represented by a certain state. The probability of the outcome is the square of the projection of the state being measured onto the state representing the potential outcome. This rule is called the Born rule.

For energy, you can expand the state in terms of energy eigenfunctions. Then the probability of a given energy will be the square of the coefficient for the corresponding eigenfunction.

You will often hear the only expectation values can be predicted in quantum mechanics. Then you will ask why only averages, and not the distributions themselves? For most distributions, one can reconstruct them from the expectation values of their moments or cumulants. This ability to reconstruct distributions from expectation values is why it is often said that only expectation values can be predicted in quantum mechanics.

Last edited: Oct 16, 2014
5. Oct 17, 2014

### naima

And a probability is an expectation value of 0 and 1 (this is the way you measure them)