As the Mythbusters might say: WARNING, SCIENCE CONTENT. Proceed at your own risk.
What is a quantum mechanical wave function? This discussion recently came up at work in the form of the following question: is the particle (e.g. an electron) the same thing as the wave function describing it? The question may have some unresolvable philosophical and metaphysical elements which I’m not entirely prepared to discuss here. But there is some real physics there too. For example, many working physicists commonly interpret the electron probability density of electron orbitals in a hydrogen atom (given by the absolute square of the wavefunction) as a literal “electron cloud” — a smear of charge and mass, best described by a density function. This thinking is understandable because it is a good heuristic way to think of a rather abstract thing. This semi-classical approach is also reinforced in most undergraduate quantum mechanics classes for the same reason. But is this what the formalism of quantum mechanics tells us? Not really. And I’m afraid that the analogy, while providing a good initial intellectual foothold, can lead to misconceptions down the road.
My argument against the “electron is the wave function” thinking is threefold:
1) Think in terms of states, processes, and probability amplitudes, not wave functions:
What is really calculated in quantum mechanics are probability amplitudes, not wave functions and probability amplitudes describe processes for particles, not the particles themselves. The absolute square of the probability amplitude then gives the probability for that process. Some might argue that, in some metaphysical sense, the particles are defined by their processes. Perhaps. But I assert that the particles are defined independent of their processes. Something closer to the particle itself might be the state of the particle as represented by the state vector. But even that just represents one possible physical configuration of a pre-existing entity. A particle, with pre-existing properties, is defined by its allowed states and how it propagates between different states (i.e. its overlap with other states).
2) The wave function and “electron orbitals” are special cases of a probability amplitudes:
The wave function, calculated using Schroedinger’s equation for non-relativistic problems like the hydrogen atom and a 1D particle-in-a-box, is really just shorthand way to define an ensemble of probability amplitudes for many possible processes. For example, the ground state wavefunction for a particle in a box is a probability amplitude for a process whereby a particle is prepared with the ground state energy and measured at location x. That point x could be anywhere on the x-axis, so you have an infinite ensemble of amplitudes, described by a function e.g. Psi(x), a.k.a. “the wavefunction.” So a wavefunction (as it is usually used in an undergraduate physics class) typically is a probability amplitude for a process where some prepared particle state measured at some specific location in space. That is, it is some state projected onto the position basis (it is also common to call state vectors projected onto the momentum basis wave functions as well). The common fuzziness of the wave function and resultant probability distributions represent the result of repeated observations of the particle in question over an ensemble of equally prepared states. But every individual measurement would give, by construction, the particle at exactly one point. The reason the “electron cloud” thinking works so well in chemistry and various kinds of nanoscale physics is that one is usually really dealing with ensembles of essentially equally prepared particles. In cases where it appears you are viewing individual atoms that appear fuzzy like with scanning tunneling microscope (STM), there is a tacit time averaging as the STM probes the material. So, again, you are really sampling the same electrons many times over and averaging their positions in relatively coarse time bins (based on the electron’s frequency).
3) By treating the wave function as the particle itself, you violate the assumptions you used to obtain the wave function in the first place:
Perhaps a more concrete way to see that quantum mechanics in no way formally endorses thinking of the particle as the wave function itself, is by examining the Hamiltonian, the very tool used in Schroedinger’s equation to calculate the wave function. In typical undergraduate examples of wave function calculations, the Hamiltonian (usually just the kinetic plus potential energy) is written for a point particle, not a smeared out distribution of charge and mass. To start a calculation with a point particle (i.e. a point of mass and charge) then to assume the “cloud” produced by the calculation is a cloud of mass and charge violates the very assumption you used to get the cloud in the first place. It is worth noting that performing calculations starting with mass and charge clouds is certainly possible and is often done in the context of manybody wave functions in solids. But it is usually just a calculational approximation to avoid coping with very high dimensional state vectors. But for the electron in the presense of a proton, the Hamiltonian is written as the kinetic plus potential energy of a single point charge and mass, not a cloud, so the electron’s wave function in hydrogen cannot be a distribution of mass and charge. As I said before, electron wavefunction is an ensemble of probability amplitudes that describe the processes for a point particle.
There is another bag of worms regarding what is meant by a “particle.” This is a slippery term that can mean different things in different contexts. Formally, in physics, specifically quantum field theory (so called second quantization), a particle is a “quantum of the field.” That is, a particle is the smallest normal mode excitation for a quantized field, where that field has the right properties like mass, charge, spin, etc. for the particle in question. But in spite of this fancier definition, if you still just calculate amplitudes for process. Like with ordinary quantum mechanics, if you ask about the amplitude of a point particle (like an electron) to be measured at a point, you (surprise, surprise) get all those properties converging to a point. For a more formal description of how to visualize the wave functions (spatial projections) of field quanta, check out this paper written by my colleague Scott Johnson and myself: ‘Visualizing the phonon wave function,’ S.C. Johnson and T.D. Gutierrez, Am. J. Phys. v.70, p. 227 (2002).