Properties of the Wave Function
Probability Amplitude
The idea of the wave function is that it represents the probability amplitude of a system. The two names are in that sense interchangeable, but "wave function" is the more commonly used of the two. This interpretation is due to Max Born and the heuristic model that goes along with it is often referred to as the Copenhagen interpretation of quantum mechanics. You can read more about it here.
If we know the probability amplitude for an electron in hydrogen, then just as with other studies of waves, the physical observable is proportional to the square of that amplitude. Recall that the square of the amplitude of the electric field of a light wave relates to brightness, for instance.
In the case of the wave function (probability amplitude), the square of it is usually understood to represent the probability density. It is called a probability density because it give the probability of measuring a particle per unit space. If we search for an electron in a 1D box, for instance, we must search for it over a range of values of x. The probability of finding the particle within a range x=a to x=b is:
Normalization of the Wave Function
If you digest what the last equation implies, then if the range over which we search includes the whole box, we must get P=1. This just means if we search throughout the whole box in which a single electron exists, we must find it exactly once. This property is required of the wave function. A properly scaled wave function that has this property is said to be 'normalized'. Mathematically, this means:
Operations on the Wave Function
I decided to present the operator notation in the previous section to illustrate a point about the wave function. The wave function should be seen as containing all the information in a system relating to any observable quantities such as position, energy, momentum, etc, yet information must be extracted from it by use of an operator.
The idea is that operations are performed on the wave function to obtain the values of observable quantities. For instance, the way you can see Schrödinger's equation is that when the Hamiltonian operator operates on the wave function it returns the energy states of a system.
You will notice when we do our first examples that the nature of such an equation is that the wave function part disappears from the expression after the Hamiltonian has operated and algebraic cancellation has been performed. If this doesn't happen, you do not have a valid wave function!
Probability of What?!
While the above mathematics can get us answers about what we expect to measure, many people - like the originators of quantum theory and anybody with a curious mind - would like a better understanding of what form the electron actually takes. We have discussed field excitations before. How exactly a wave function relates to a field excitation is very mathematically complex. They are not the same thing, but are certainly related.
Historically, Schrödinger suggested that the probability density of an electron, for instance, related to where the actual charge was. He supposed that the square of the wave function times the electron charge gave the charge density in space. The same was suggested for mass. He refined his ideas over time, and in a few years (1928) wrote instead:
The classical system of material points does not really exist, instead there exists something that continuously fills the entire space and of which one would obtain a ‘snapshot’ if one dragged the classical system, with the camera shutter open, through all its configurations, the representative point in q-space spending in each volume element dτ a time that is proportional to the instantaneous value of ψψ∗.Physicists don't like the idea that the electron's charge is spread out in space along with the wave function since they suggest there should be a "self-interaction" term that would cause one part of that spread-out charge to repel other parts of it. Schrödinger and others, after years of reflection, were more comfortable with the idea of a localized electron that moves ergodically (randomly, but governed by probability) in a discontinuous fashion through space in any infinitesimal time interval. While I am not a world expert on quantum theory, there are certainly shortcomings to that interpretation. They are the following: 1) We have to allow for both the localized existence of the electron and discontinuous motion - meaning that it can go from here to there without traversing the space between. 2) The need to avoid the self-interaction requires us to nonetheless accept the self-energy of a localized electron, which seems to be just as big a problem. While clever mathematics can seemingly do away with the problem of the charge of an electron crammed into a very small space (since an electron's diameter may in fact be zero when measured in scattering type experiments), they are comfortable ignoring it. I would suggest that even a delocalized charge distribution may in fact not be able to interact with itself for the same reason that field collapse must be allowed. The entity, whatever its form, can not be seen as a composition of anything smaller. In other words, the argument for field collapse is that it must be instantaneous since we can't leave a part of a quantum out there while the rest of it has become localized and has deposited energy and momentum somewhere. The same argument, as strange as it may sound, aught to hold for a delocalized electron. A parallel argument would be that we can't think about a left half of a quantum interacting with a right half since halves of quanta can't in principle exist as individual units by definition. Needless to say, the exact interpretation of the probability density is an ongoing discussion and papers are still being published today on the topic.