I: Prerequisites: Quantum Mechanics and the Electronic States in Solids

Quantum Mechanics


The two-slit experiment plays a central role in quantum mechanics. As Feynman has said if someone does not understand the two-slit experiment he does not understand quantum mechanics. It is worth pointing out that it is a “gedanken” experiment, i.e. it exists only in our minds (for pedagogical purposes), but there are real diffraction experiments that are completely equivalent to it, so we always treat it as a real experiment.

The electron two-slit experiment is a replica of the electromagnetic (photonic) two- slit experiment originally performed by the English physicist Thomas Young in 1801. We describe it below. The apparatus involves a screen on which two pinholes (S^ S2) have been punctured. The screen is illuminated by a lamp emitting spherical waves of wavelength X which fall onto the pinholes, see figure 1.1. Before we proceed further, it is important to clarify what is meant by “pinholes”. If a is the width of the pinholes we must have a « X = wavelength of light, otherwise if a S> X the features we are going to present will be absent. As the initial spherical wave falls onto the pinholes the latter become secondary sources and emit secondary spherical waves with their origin being the pinholes S,, S2.

Due to the design of the apparatus, the secondary sources are coherent which means that their frequency is the same and there is no time lag between them. Put simply, the photons from Sp S2 propagate in phase. As a consequence, any developed phase difference between the two secondary spherical waves can only come from a different path travelled by each. If the distances from their sources to the same point P on a second screen are rb r2, then the path difference at P is |r2 — r21, see figure 1.1 again. We remind the reader that a spherical wave has the form r_lexp[i(/cr-o:>f)].

Therefore, when the phase difference is an integer number of wavelengths there is an enhancement of waves and a bright spot appears on the screen at P. On the other hand, when the phase difference is an odd integer of half-wavelengths, there is a cancellation of waves and a dark spot appears on the screen. On the screen therefore an alteration of bright and dark fringes occurs which we will call interference fringes, see figure 1.2. An analysis found in all introductory optics books shows that the maxima are given by ym = mDA/d where d is the slit spacing, D is the distance between the two screens and m is an integer.

The two-slit experiment

FIGURE 1.1 The two-slit experiment: two coherent waves from the slits S| and S2 meet at a point P on a screen where they interfere constructively or destructively.

The exact algebraic relations which give the maxima and minima are not of real interest to us. The important point is that enhancement and cancellation can only occur with waves. Suppose we did not have a spherical wave in front of the two-slit screen but instead a series of pistols firing bullets (i.e. particles) to the screen at all angles and that the screen was impenetrable by the bullets. What type of distribution would we get on the second screen? We would get two gaussian-type distributions—shown in figure 1.2—each gaussian being positioned opposite each pinhole. When the experiment was performed by Davisson and Germer using electrons they did not get the bullet-type distribution but rather interference fringes, as shown in figure 1.2. Actually, as pointed out previously, they did not perform a two-slit experiment because they could not construct pinholes smaller than the wavelength of electrons but instead they performed an equivalent diffraction experiment of an electron beam off a metal surface. Therefore, electrons behave as waves. But what is it that

The expected pattern on the screen would be a set of interference fringes if the electrons are waves (dotted curve) or a set of two Gaussians if the electrons behave as particles

FIGURE 1.2 The expected pattern on the screen would be a set of interference fringes if the electrons are waves (dotted curve) or a set of two Gaussians if the electrons behave as particles.

is vibrating in electrons? We postpone the answer to this question for a moment to continue with the two-slit experiment.

Suppose we decrease the intensity of our electron gun (see later the field assisted electron emission phenomenon) so that only one electron passes at a time through the screen, i.e. perform the experiment as if electrons are fired by a gun. What do we see on the second screen? We still see the interference fringes but made out of dense spots. But we never see spots, i.e. electrons, on the minima of the fringes where the intensity was originally zero. Hence it seems that even a single electron can produce interference, that is a single electron can have wave properties while being a particle. All this is really mind blowing and the interpretation given by what is called the Copenhagen school of quantum mechanics— now accepted universally—is the following.

When electrons are measured they behave as particles but when they propagate in vacuum or in materials before measurement they behave as waves and so their position, upon measurement, is not predetermined but only probabilistically known. We must emphasize that when we talk about an electron wave it is wrong to think of the electron as a diffusive charge cloud. The correct picture or interpretation is that, due to the extremely small lengths involved, we can never tell exactly where it is, we can only talk about the probability of finding it in an infinitesimal space around a given position r. It is this probability that behaves as a wave. Then it is the act of measurement that makes this probability certainty. In the two- slit experiment each electron gives two spherical probability waves (rj) and *F 2 (r2)—one from each pinhole—and the probability P of finding the electron in a small volume dV centered on any point given by its position vector r is

where *F = 4^ + 'F,

It must be clear from the discussion so far that the result of one position measurement can only be zero or one (0 or 1). Equation 1.1 becomes meaningful if many electrons are thrown onto the two-slit screen, i.e. if repeated measurements are made and then any positive value between 0 and 1 can occur. A more abstract interpretation of the experiment exists for which the reader can refer to advanced textbooks on Quantum Mechanics.

We can now come back to our original question of what it is that is vibrating in the electron wave. The answer is Ч'(г) but only in the sense of equation 1.1. There is often an initial misunderstanding by some students that the electron is a “charge cloud”. This is essentially wrong. In fact Schroedinger—whose equation we will discuss immediately below—originally thought that |VF|2 was the charge density. Later developments by Born corrected that and completed the architecture of the Copenhagen school. On the other hand, *F(r) should not be looked at as expressing only lack of information on behalf of the observer but rather as expressing a non-local nature of the electron.

The interference fringes are not produced by one electron going through one pinhole and another through the other pinhole. Remember the fringes are produced even when the electrons are thrown one by one onto the two - slit screen. If the two spherical waves emanating from the pinholes were not produced by the same electron they would not be coherent and interference fringes would not be produced. Therefore, the wavefunction *F refers to a single electron and not to a collection of many as a statistical average. The usual question asked in the lecture rooms is: how does the electron know that there is a second hole? The answer is that the electron is not a classical particle and it has a non-local character. In fact, non-locality is an inherent property of all particles. We will not delve any more into the particle-wave duality. We will accept on the one hand that electrons are countable and on the other we will deduce the properties of solids using the electronic wavefunctions ЧТ

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