IID.  The Wave Function.

So that you have some way of thinking about the wave function, we will give a means of visualizing it.  We will not attempt to give a deeper understanding of the significance of the wave function because that requires some mathematical sophistication.  But one may be able to gain this, at least to some extent, from section IV.  (It has to do with the fact that there is a whole nonspacetime reality that lies beneath our apparently solid world.)

1. Classical Mechanics.

In classical mechanics, matter consisted of point particles.  The mathematics determined the paths through space, the trajectories, of each of these point particles.  This is easy enough to visualize.

2. Visualizing the Wave Function for an Isolated “Electron.”

In a sense, there is an equally easy way to visualize the wave function.  Instead of visualizing an electron (or any other particle) as a point, we can visualize it as spread out in space, as if it were a cloud of negative charge, or a mist.  The mist is more opaque in some places and less opaque in others, tailing off to complete transparency—no mist—over most of space.  The equations of quantum mechanics determine the trajectory of the cloud as a whole.  In your television set, for example, electrons are shot from an electrode to the screen.  This is to be visualized as a cloud of charge, of approximately atomic dimensions (nanometers across), moving from the electrode to the screen.

3. Shape of the Wave Function. 

The equations of motion also determine the shape of the cloud, where it is more transparent, where it is less transparent.  This is not of much importance in the television example.  But if we go inside a hydrogen atom, the shape of the cloud becomes very important.  If the electron cloud assumes one shape—centered around the proton in the middle—it has the lowest possible energy.  If it assumes another shape, the cloud has the next lowest energy, and so on.  The equations of quantum mechanics determine how the shape changes once the initial shape is given.  (In an atom, interestingly enough, the shape doesn’t change; the cloud is stable.)

4. Scattering of an Electronlike Wave Function. 

Another interesting example is that of an electron (electronlike wave function) shot at a proton (protonlike wave function).  The electronlike wave function comes in as a fairly well-defined, compact cloud.  When it hits the proton, most of the electronlike wave function simply continues past it as a still-compact cloud.  But because of the electric force, some of the electronlike wave function is scattered outward in concentric waves centered on the proton, as in figure IID.1. 

Figure IID.1.  An electronlike wave function partially scattered in a concentric circle from a proton.  The wave function initially came in from the left along the dashed line and hit the proton at the center of the circle.

Thus the electronlike wave function becomes spread out over a wide region of space.

5.  Parts or Branches of the Wave Function. 

As a final example, we consider the wave function associated with light.  Suppose we coat a flat piece of glass with a thin layer of silver and put it in the path of a light beam, at an angle of 45 degrees to the beam.  If the layer has just the right thickness, half the beam will go through the glass plate and half will be reflected upwards.  The light beam actually consists of many individual compact lightlike wave functions all traveling in the same direction.  If we look at just one wave function, we find that the half-silvered mirror splits it into two parts.  One part continues through the mirror and the other part is reflected upwards.  

Figure IID.2. A half-silvered mirror splitting a photon wave function into two parts or branches.  One branch is reflected upward along a vertical path and the other continues through the mirror along the horizontal path.  Note that this figure does not properly convey the implications of the two-branch wave function.  See figures IIE.1 and IIE.2 in section IIE.

Because the wave function has branched into two separate parts, we call the two parts branches.  In many situations there can be more than two branches.

It is most important to note that this simple visualization is misleading when the wave function splits.  In fact, it conceals the primary mystery of quantum mechanics.  We will explain further in section IIE. 

6. Conservation Law. 

There is a conservation law for the wave function.  In the light case, it follows from the mathematics of quantum mechanics that the total amount (suitably defined) of the unreflected wave function plus the total amount of the reflected wave function equals the total amount of the wave function before it hit the mirror.  In the electron scattering case, the total amount of the electron wave function that passes by the proton plus the total amount that is concentrically scattered by the proton equals the total amount of the wave function before it hit the proton.  This conservation law is related to conservation of probability (section IIG).