The photoelectric effect suggested that light consisted of photons. On the other hand, the classical theory of electrodynamics, which was based on Maxwell's equations stated that light consisted of waves. Waves and particles are completely exclusive concepts in classical physics. However, one can design experiments in which both properties can be demonstrated.

The slit is large compared to the wavelength of the light we will not be able to differentiate between the particle and the wave picture, since in both cases we predict a broad area to be illuminated on the screen opposing the slit (perhaps with some fuzzy border where the shadow of the slit edges appears).

If we make the slit small enough to be comparable to the wavelength of the light (of the order of 1/2 to 1 micron), we will be able to distinguish between the wave and particle picture. The experiment clearly shows an interference pattern on the screen, which is predicted from the wave picture. The particle picture would predict just a narrow illuminated region on the screen which is not observed.

The Double Slit Experiment:

Another way to demonstrate the wave nature of light is the double slit experiment. There, two narrow slits are placed next to each other with a distance of the order of the wavelength of the light. The particle picture would predict just two narrow beams of light passing through the slits and illuminating two narrow strips on the screen, while the wave picture predicts an interference pattern with a main maximum in the center (between the slits) and several higher order maxima to either side. The experiment proofs the wave picture to be correct.

These interference experiments proof that light consists of electromagnetic waves! What else is there to proof? Particles and waves are mutually exclusive, or are they?

The Photoelectric Effect:

As we have seen in the previous chapter, the photoelectric effect proofs that light consists of photons. Can we now set up a single experiment that proofs at the same time that light is both particles and waves?

The way to do this is to use the photoelectric effect to detect the light particles as they arrive at the screen. Instead of a screen we will set up an array of little photomultiplier cells (photodiodes can also be used) which we can read out electronically. They will count the number of detected photons, and tell us where and at which time a photon is registered. The number of photons registered at a given spot can then be used to calculate the intensity of the light at that particular spot on the screen. We can then recreate the single slit experiment with single photon detection. In order to measure one single photon at a time we have to turn down the intensity of our light source to such a level that only one photon is on its way through the slit at any given time. The numbers of photons registered in each channel of the array reflects exactly the distribution of the intensity we expect from the interference experiment using waves. Yet we can detect and count each individual light particle separately.

We have thus set up an experiment that proofs at the same time that light consists of particles (because we can count them individually) and of waves (since the intensity distribution at the detector array reflects the interference pattern). When we pass just one photon through the slit at a time we find that in some mysterious way the photon interferes with itself and changes its direction of flight. While it is impossible to predict where exactly it will hit the detector array, one can say that its probability is high to hit at the maxima of the interference pattern.

The Davisson-Germer Experiment:

Since light comes in the form of waves and particles at the same time, could it be possible that material particles (like electrons, protons, etc.) may also have a wavelike nature at the same time? De Broglie, a French physicist predicted this in his famous hypothesis, and related the wavelength of the material particle to its linear momentum, p, by:



Proof of this relation was obtained in 1926 by the Davisson-Germer experiment. It is a type of interference experiment. If one wants to perform an interference experiment with electrons this would not be possible with a mechanical set-up. The slit-width would have to be too narrow to be able to manufacture since according to de Broglie's relation the electronic wavelength is of the order of a few Angstroms (even for slow electrons). The solution was to use a single crystal (Davisson and Germer used a Ni crystal) and use the periodic Coulomb potential the crystal lattice presents for the electron to do the interference experiment. In the actual set-up an electron beam was collimated onto the surface of the Ni crystal and the reflected electrons measured by a detector at a variable angle with respect to the incoming electron beam. While most of the electrons were scattered back at theta=0 (main maximum) a higher order maximum was obtained at a certain angle theta which proofed the existence of an interference pattern predicted from the wavelike nature of the electrons.

The Wavefunction:

Due to the particle-wave duality we are now faced with the problem how to correctly set up a theory that describes the motion of a light or material particle. Classical mechanics would not help since its equations of motions are exclusively valid for particles. Classical electrodynamics is exclusively valid for waves. A new concept had to be born, and was formulated by Schrödinger in his wave equation.

The microscopic particle was described by a so-called wavefunction. The significance of the wavefunction is that its absolute square described the probability function to find the particle at any given location in space. Schrödinger's equation is an equation of motion for the wavefunction.

One important feature of the wavefunction is that it can be superimposed with itself and with other wavefunctions the same way that electromagnetic waves can interfere with each other. This allows for the single photon to interfere with itself, but also allows for a single electron to split itself up in a linear combination of wavefunctions.

The interference of waves can be illustrated in a number of simple graphs: When two waves of different wavelength interfere they will produce a wave with a new wavelength which is just the average of the two old wavelengths, and an envelope with the difference wavelength. An acoustic example would be the beating that one can hear when two instruments produce sounds of slightly different frequencies. In trying to tune in to a given standard frequency the musician will listen to the beating of the two sounds and tune his instrument such that the beating periods get longer and longer until they vanish in the case of perfect consonance.

In the case of multiple frequencies superimposed on each other the resulting pattern yields a (periodic) main maximum with lower amplitudes before and after it.

This can be extended to very many different frequencies, combining together to form a so-called wave packet. The wave packet can be viewed as a model for creating a particle out of waves (although there are some serious theoretical shortcomings, and this picture may not be carried through completely). A particle with high kinetic energy would then be represented by a linear combination of waves with high frequency and thus show a low width of its associated wave packet, while a low energy particle would have a relatively broad width due to the dominant contribution of low frequency waves.