If you read the first section on the failings of classical physics, then you will recall that we discussed the problem of explaining the spectrum of light that radiates from an iron bar that's heated, and that to do so, it was necessary to introduce the idea that atoms vibrating in the bar could only take on certain discrete energies.  Here we will pursue this notion of "energy quantisation" a little further.  We will attempt to show how this comes about through a different example, and this example is an electron trapped in a box with infinitely hard walls.  This simple system is shown in the diagram below.

In the previous sections we discussed wavefunctions, and noted that the magnitude of the wavefunction squared gives the probability of finding a quantum object at a particular location.  Now, since we have assumed that the walls of our system above are infinitely hard, the electron cannot penetrate the wall and so the probability of finding the electron at the wall is zero.  This means that the wavefunction of the electron must also be zero at the walls.  Note that we have come to this conclusion without writing down or solving any equations!

Now for the tricky bit, and you will have to take this on trust from me.  Why?  because I have to take it on trust myself.  It just happens to work!  The point is, there is an analogous situation to our electron in the box with infinitely hard walls, and it just happens to be the modes of vibration of waves on a string, with both ends of the string pinned down (e.g. a guitar string).  Pinning down the ends is the "string equivalent" of saying that the wavefunction of the electron is zero at the boundaries of the wall.

Let's take a closer look at our string.  If we "pluck" our string, then it turns out that there are certain basic modes of vibration of that string, and that any generalised vibration of the string (for example generated by plucking the string at some random point) can be split up into these fundamental components.  These are shown in the following diagram

The first mode shown in this diagram is called the "fundamental mode", the second is called the "first harmonic", the third is called the "second harmonic", and so on.  Any mode of vibration of the string can built up by adding together these harmonics with various amplitudes of vibration.  Note that the wavelength of the vibration decreases as the number of the harmonic increases. The various waves in the figure above are called "standing waves", since they - in effect - stand on the string without moving along the string, and they are the only ones that satisfy the requirement that the wave amplitude is zero at the point where the string is tied down.

Now, do you recall, from the section on wave-particle duality, that we can define a de-Broglie wavelength for electrons?  To recap, the de Broglie wavelength is given by

λ = h / p

where h is Planck's constant and p is the momentum of the electron.  The de-Broglie wavelength expresses the extent to which the electron behaves like a wave.

Now for the totally stunning bit!  It turns out that the de-Broglie wave for our electron in the box behaves exactly the same way that the standing waves do on the guitar string.  Because of the requirement that the wavefunction be zero at the walls of the box, only certain de-Broglie wavelengths are possible within our box!  This is illustrated in the figure below.

The energy of a wave varies as the wavelength varies.  So, for our standing waves on the guitar string, only certain vibrational energies are possible.  The same applies for our electron in the box - since it can only possess de Broglie waves of certain wavelengths, so the energy of the electron can only assume certain discrete values.

We have therefore obtained the result that our electron in the box can only take on certain energies.  That is, we have arrived at the result of energy quantisation, and it turns out that this is a consequence of confining our electron to a certain region of space.  We've also arrived back roughly where we started, which was seeking an explanation for the ultra-violet catastrophe and the failings of classical physics.

If you didn't follow the above discussion on energy quantisation, don't worry.  The key message is as follows:

In quantum systems where a particle is confined, then that particle can only take on certain energies.  This is a general result and the confinement need not be by infinitely hard walls - it can be a more "general" type of confinement where there are no fixed boundaries.  The concept of energy quantisation is one of the fundamental differences between the quantum world and the classical world in which we live.