The meaning of the wave function
When Schrödinger first
discovered his equation he discovered the conservation law of
Eq. (21.8) as a consequence of his equation. But he
imagined incorrectly that $P$ was the electric charge
density of the electron and that $\FLPJ$
was the electric current density, so he thought that the electrons
interacted with the electromagnetic field through these charges and
currents. When he solved his equations for the hydrogen atom and
calculated $\psi$, he wasn’t calculating the probability of
anything—there were no amplitudes at that time—the interpretation
was completely different. The atomic nucleus was stationary but there
were currents moving around; the charges $P$ and currents $\FLPJ$ would
generate electromagnetic fields and the thing would radiate light. He
soon found on doing a number of problems that it didn’t work out quite
right. It was at this point that Born
made an essential contribution to our ideas regarding quantum mechanics.
It was Born who correctly (as far as
we know) interpreted the $\psi$ of the Schrödinger equation in terms of
a probability amplitude—that very difficult idea that the square of
the amplitude is not the charge density but is only the probability per
unit volume of finding an electron there, and that when you do find the
electron some place the entire charge is there. That whole idea is due
to Born. The wave function $\psi(\FLPr)$ for an electron in an atom does not,
then, describe a smeared-out electron with a smooth charge
density. The electron is either here, or there, or somewhere else, but
wherever it is, it is a point charge. On the other hand, think of a
situation in which there are an enormous number of particles in
exactly the same state, a very large number of them with exactly the
same wave function. Then what? One of them is here and one of them is
there, and the probability of finding any one of them at a given place
is proportional to $\psi\psi\cconj$. But since there are so many
particles, if I look in any volume $dx\,dy\,dz$ I will generally find
a number close to $\psi\psi\cconj\,dx\,dy\,dz$. So in a situation in
which $\psi$ is the wave function for each of an enormous number of
particles which are all in the same state, $\psi\psi\cconj$ can
be interpreted as the density of particles. If, under these
circumstances, each particle carries the same charge $q$, we can, in
fact, go further and interpret $\psi\cconj\psi$ as the density of
electricity. Normally, $\psi\psi\cconj$ is given the dimensions
of a probability density, then $\psi$ should be multiplied by $q$ to
give the dimensions of a charge density. For our present purposes we
can put this constant factor into $\psi$, and take $\psi\psi\cconj$
itself as the electric charge density. With this understanding,
$\FLPJ$ (the current of probability I have calculated) becomes
directly the electric current density. So in the situation in which we can have very many particles in
exactly the same state, there is possible a new physical
interpretation of the wave functions. The charge density and the
electric current can be calculated directly from the wave functions
and the wave functions take on a physical meaning which extends into
classical, macroscopic situations. Something similar can happen with neutral particles. When we have the
wave function of a single photon, it is the amplitude to find a photon
somewhere. Although we haven’t ever written it down there is an
equation for the photon wave function analogous to the Schrödinger
equation for the electron. The photon equation is just the same as
Maxwell’s equations for the electromagnetic field, and the wave
function is the same as the vector potential $\FLPA$. The wave
function turns out to be just the vector potential. The quantum
physics is the same thing as the classical physics because photons are
noninteracting Bose particles and many of them can be in the same
state—as you know, they like to be in the same state. The
moment that you have billions in the same state (that is, in the same
electromagnetic wave), you can measure the wave function, which is the
vector potential, directly. Of course, it worked historically the
other way. The first observations were on situations with many photons
in the same state, and so we were able to discover the correct
equation for a single photon by observing directly with our hands on a
macroscopic level the nature of the wave function. Now the trouble with the electron is that you cannot put more than one
in the same state. Therefore, it was long believed that the wave
function of the Schrödinger equation would never have a macroscopic
representation analogous to the macroscopic representation of the
amplitude for photons. On the other hand, it is now realized that the
phenomenon of superconductivity presents us with just this situation.