05-11-2012, 12:34 PM
The Schrödinger Wave Equation
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So far, we have made a lot of progress concerning the properties of, and interpretation of
the wave function, but as yet we have had very little to say about how the wave function
may be derived in a general situation, that is to say, we do not have on hand a ‘wave
equation’ for the wave function. There is no true derivation of this equation, but its form
can be motivated by physical and mathematical arguments at a wide variety of levels of
sophistication. Here, we will offer a simple derivation based on what we have learned so
far about the wave function.
The Schr¨odinger equation has two ‘forms’, one in which time explicitly appears, and so
describes how the wave function of a particle will evolve in time. In general, the wave
function behaves like a wave, and so the equation is often referred to as the time dependent
Schr¨odinger wave equation. The other is the equation in which the time dependence
has been ‘removed’ and hence is known as the time independent Schr¨odinger equation
and is found to describe, amongst other things, what the allowed energies are of the
particle. These are not two separate, independent equations – the time independent
equation can be derived readily from the time dependent equation (except if the potential
is time dependent, a development we will not be discussing here). In the following we
will describe how the first, time dependent equation can be ‘derived’, and in then how the
second follows from the first.
The Finite Potential Well The infinite potential well is a valuable model since, with the minimum amount of fuss, it shows immediately the way that energy quantization as potentials do not occur in nature. However, for electrons trapped in a block of metal, or gas molecules contained in a bottle, this model serves to describe very accurately the quantum character of such systems. In such cases the potential experienced by an electron as it approaches the edges of a block of metal
Scattering from a Potential Barrier
The above examples are of bound states, i.e. wherein the particles are confined to a limited
region of space by some kind of attractive or confining potential. However, not all
potentials are attractive (e.g. two like charges repel), and in any case, even when there
is an attractive potential acting (two opposite charges attracting), it is possible that the
particle can be ‘free’ in the sense that it is not confined to a limited region of space. A
simple example of this, classically, is that of a comet orbiting around the sun. It is possible
for the comet to follow an orbit in which it initially moves towards the sun, then
around the sun, and then heads off into deep space, never to return. This is an example
of an unbound orbit, in contrast to the orbits of comets that return repeatedly, though
sometimes very infrequently, such as Halley’s comet. Of course, the orbiting planets are
also in bound states.
A comet behaving in the way just described – coming in from infinity and then ultimately
heading off to infinity after bending around the sun – is an example of what is known as
a scattering process. In this case, the potential is attractive, so we have the possibility of
both scattering occurring, as well as the comet being confined to a closed orbit – a bound
state. If the potential was repulsive, then only scattering would occur.