Interpretation of $e|\psi|^2$ as electron density

Question

In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to some bizarre consequences", and that Born's statistical interpretation of $|\psi|^2$ as a probability density is more satisfactory.

I am aware that a position measurement of an electron leads to (in the Copenhagen interpretation) a collapse of the wave function into a position eigenstate $x$ with the probability given by $|\psi(x)|^2$, and that it is known from some scattering experiments that the electron behaves as a point-like particle.

However, the electron density is experimentally observable, e.g. by X-ray scattering. One could argue that X-ray scattering is done with a large ensemble of atoms, so that what we actually observe is the average over many position measurements.

I am wondering if this the only "bizarre consequence", since this distinction seems to be a minor difference to me. My questions aims at clarifying the difference between the probabilistic interpretation of $|\psi(x)|^2$ and the interpretation as an electron density.

Answer 1

The statement that the right interpretation of $|\psi(x)|^2$ is probabilistic means that the value of this expression may only be "measured" in the same sense as other probabilistic distributions – by a repeated experiment starting with the same initial conditions. Every probability distribution may be approximately reconstructed by "throwing the dice" many times and recording the frequencies.

However, $|\psi(x)|^2$ cannot be measured in one repetition of the situation. There can't exist a gadget that would show $|\psi(x)|^2$ on its display. That's why we say that the wave function (or its squared absolute value) isn't observable: it's not observable in one copy of the situation. This word "observable" may be interpreted colloquially as well as technically. Technically, an "observable" is a Hermitian operator. The squared absolute value of the wave function isn't an operator, so it's not an observable, so it can't be measured.

The charge density is a classical quantity in classical physics and may be an observable in quantum field theory – like for other observables, only the probabilities of different values may be predicted. However, in one-particle quantum mechanics, a particle is either located in the volume $dV$ or it's not. Nothing in between is possible and a measurement may answer whether the question is Yes or No. The probability of Yes is given by $|\psi(x)|^2 dV$.

If you understand the things above and your wording indicates that you do, then you understand why the right interpretation of the wave function is probabilistic. This assertion means nothing else than the fact than the fact that to "measure" $|\psi(x)|^2$ at a given point, one needs to repeat the same situation many times and use the laws of probability and statistics. The function $|\psi(x)|^2$ doesn't correspond to a property of the region of space that may be measured instantly, by one measurement in one repetition of the experiment.

Answer 2

Obviously, the interpretation of $e |\psi|^2$ as an electron density will hold as soon as we consider an ensemble of electrons, since the probability density then results in a statistical distribution. In the case of just one electron we can still interpret $e |\psi|^2$ as an electron density as long as we think about longer time scales. Please note, that this only works as all observables (that I can think of now) depend linearly on the electron charge.

In any other case the electron still has to be considered point-like and is not smeared out in space, only that its position in the phase space is not determined. This means e.g. that the detection of an electron will always (after the aforementioned collapse of the wave function) result in one single point of detection.