We now have a complete picture of the basics of quantum mechanics.

In classical physics we had particles and waves. Light was an example of a wave, and the electron was an example of a particle.

In quantum mechanics, our first lesson was that everything was a wave. Thus the electron was also a wave, and the evolution of this wave would be given by the laws of wave mechanics rather than Newton's laws of motion.

Our second lesson was that the waves of quantum theory had a strange property: their strength came in discrete units. Thus we can have a wave of strength zero or strength one, but not strength 'half'.

Putting these two lessons together will bring out most of the surprising features of quantum theory.

It may not be too puzzling that the electron was found to be a wave. We have already noted that the electron's wavelength is typically very small in most laboratory situations. So it is not surprising that we missed the up and down motion of the wave until our instruments became precise enough to measure very small distances.

The discreteness of the wave is more puzzling. One would have thought that a wave could have *any* amplitude. If the amplitude was indeed arbitrary, then electrons would not have come in discrete units: by slightly increasing or decreasing the amplitude of the electron wave, we could get \( 1.1\) electrons or \( 0.9 \) electrons.

But the discreteness of the number of electrons is a well observed fact. So we are forced to accept that the strength of the electron wave cannot be arbitrary. It is then a pleasing fact that the strength of the electromagnetic wave also comes in discrete units, since we get the same set of rules rules for all objects in nature.

This discreteness is sometimes called the 'particle' nature of the electron, since even in classical theory particles were observed to come in discrete units. It is true that if we cannot resolve small distances then the electron wave may look approximately like a particle moving according to Newton's laws. But we emphasize again that this approximate behavior is *not* what is called the particle-like behavior of the electron; this term refers to the discreteness of the strength of the electron wave.

In chapter 1 we had noted that the wavelength of the electron wave depends on the momentum \(p\), through the de Broglie relation \( \lambda = h/p\). If all entities follow the same rules in quantum theory, then electromagnetic waves should also satisfy such a relation. But what is the momentum of an electromagnetic wave?

In classical physics, electromagnetic waves do carry momentum. Consider an electromagnetic wave with wavelength \( \lambda \). If we increase the *strength* of this wave, then its momentum \( p \) increases. So in classical physics there is no equation relating \( \lambda \) to \( p \); we need to know the \( \lambda \) *and*
the strength of the wave to determine its momentum.

But with quantum theory, we have seen that electromagnetic waves comes in discrete units: photons. A photon has the minimum strength allowed for the wave with a given wavelength. This photon carries a momentum \( p \). The wavelength \( \lambda \) of the the photon is related to the momentum \( p \) carried by the photon via the de Broglie relation \( \lambda = h/p \).

In fact de Broglie arrived at his relation by starting with *electromagnetic* waves, and then conjecturing that other waves should have the same behavior. His derivation is a very simple one, and we include it later in the technical appendices to this course.

Most of the puzzling aspects of quantum theory come from putting together the two basic lessons that we noted above: wave behavior and the discreteness of this wave. Our next task will be to understand one such aspect of the theory - the nature of measurement.