Monday, July 2, 2007

The wavefunction

We consider a system of a single particle acted upon by some
forces. In classical mechanics the state of the system at any time is
specified by the position and velocity of the particle at that time.
The subsequent motion is then uniquely determined for all future
times by solution of Newton’s second law of motion, which tells us
that the acceleration is the force divided by the mass.
In quantum theory the state of the system is specified by a
wavefunction. Instead of Newton’s law we have Schrodinger’s
equation. This plays an analogous role because it allows the
wavefunction to be uniquely determined at all times if it is known at some initial time. Thus quantum mechanics is a deterministic
theory of wavefunctions, just as classical mechanics is of positions.
The wavefunction of a particle exists at all points of space. It
consists of two numbers, whose values, in general, vary with the
point considered. We shall find it convenient later to picture these
two numbers by regarding the wavefunction as a line on a plane,
like that shown in figure 7. The two numbers are then the length
of the line and the angle it makes with some fixed line. We shall
refer to these numbers as the magnitude and the angle of the
wavefunction.

As mentioned in the previous section, the wavefunction at a
given point determines the probability for the particle to be at that
point. In fact, the relation between the wavefunction and the probability
is very simple: the probability is proportional to the square
of the magnitude of the wavefunction. It does not depend in any
way on the angle of the wavefunction.
The classical notion of a particle’s position is therefore related to
the magnitude of the wavefunction. What about the classical
velocity? Not surprisingly, this is related to the angle. In fact, the
velocity is proportional to the rate at which the angle of the
wavefunction varies with the point of space, i.e. with x. The reason for this is discussed in Appendix 4 (but only for readers with the
necessary mathematical knowledge). Note that here we are speaking
of the actual velocity, not the uncertainty in the velocity which,
as discussed earlier, is proportional to the width of the peak in the
probability.
For easier visualisation of what is happening it is useful to
simplify the idea of a wavefunction by thinking about its so-called
real part, which is the projection of the wavefunction along some
fixed line, as shown in figure 7. For example, the real part of the
wavefunction corresponding to the probability distribution of
figure 5 might look like figure 8. The dashed line in this figure is
the magnitude of the wavefunction. The rate of oscillation of the
real part is proportional to the velocity of the particle.
We shall see later that it is necessary to have a method of
‘adding’ wavefunctions. The method we use can be understood by
reference to figure 9. We wish to add the wavefunctions represented
by the lines in figures 9(a) and (b). To do this we join the beginning
of the first line to the end of the second; then the line joining the
beginning of the second to the end of the first is the line that
represents the sum of the two wavefunctions. This is illustrated in
figure 9(c). It is not hard to show that, with this definition, it is
irrelevant which line is called the first and which the second. We
now notice the important fact that this definition is not the same
as using ordinary addition to add the numbers associated with each
wavefunction. In particular, the magnitude of the sum of two
wavefunctions is not the same as the sum of the magnitudes of the
wavefunctions. As an example of this, whereas, since magnitudes
are always positive, the sum of two magnitudes is always greater
than either, this is not necessarily the case for the magnitude of the
sum, as is seen in figure 10. Note, however, that the real parts of
wavefunctions do add just like ordinary numbers.

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