Two-slit experiment

The first experiment is the two-slit experiment of electron interference illustrated in Figure 3.1. In this experiment, a thermal electron gun emits high-speed electrons shooting at an electron-sensitive plate (shown at the right side of the figure). Between the plate and the electron gun there is a thin wall which has two slits. Electrons are absorbed if they hit somewhere other than these two slits on the wall, only those electrons that go through the slits can arrive at the plate and generate sparks.

Figure 3.1: Two-slit experiment of electron interference.

The experiment goes like this: if slit 1 is opened and slit 2 is closed, the distribution of electrons which have arrived at the plate equals distribution 1, shown in the figure. On the other hand, if slit 2 is opened and slit 1 is closed, the distribution of electrons which have arrived at the plate is the curve labeled distribution 2. Now if both slits are opened, classical mechanics predicts that the joint distribution shall be the sum of distribution 1 and distribution 2, but quantum mechanics predicts differently. The classical account goes as follows. Assuming that the initial momenta of electrons at the electron gun have random distribution, a particular electron will traverse either through slit 1 or through slit 2 (but not both) on account of its initial momentum at the electron gun. Moreover, where this particular electron will hit is independent of where the other electrons will hit (assuming the electron stream is not very dense so that the collisions between electrons can be neglected). Consequently, the joint distribution should be the sum of distribution 1 and distribution 2. According to classical mechanics, the fate of an electron is determined right at the start of the electron gun, although we may not be able to know its fate technically.

The experiment outcome is not that which is predicted by classical mechanics! Instead, the distribution is a pattern of interference quite similar to that of light or water waves going through two slits (the undulating gray curve shown in the figure). For one thing, there are positions (e.g. the point marked with $x$ in the figure) that are very likely to be hit with either slit 1 or slit 2 is closed but are never hit if both slits are opened. This phenomenon cannot be explained in classical mechanics: the fact that an electron that should have hit $x$ when slit 2 is closed (that is, an electron that possesses the initial momenta to go through slit 1) is somehow pushed away from $x$ simply because slit 2 is opened.

At first sight, one might argue that this particular electron could be indeed pushed away by other electrons that go through slit 2. But this is not the case. In fact, the undulating distribution remains the same even if the electron gun is throttled down so that it will emit only one electron at a time, and also when the interval between two emission is prolonged in such a way that there can never be two electrons flying at the same time. The ``lonely'' electron nevertheless seems to interfere with itself. Indeed, according to quantum mechanics, a particle interferes only with itself.

Now we encounter the first strangeness of quantum mechanics: if this electron has a particular initial momentum such that it will arrive at position $x$ if slit 2 is closed, how come it is expelled from $x$ if slit 2 is opened? To avoid hitting $x$, the electron seems to ``know'' that slit 2 is opened, so that it may ``decide'' where it should hit. Or maybe it goes through slit 1 and slit 2 at the same time?

But, according classical physics, isn't it the case that an electron can go through either slit 1 or slit 2 but not both? To corroborate or falsify this hypothesis of exclusiveness, one can introduce a position detector near slit 1 so that whenever an electron comes through slit 1, a spark is generated. In order for the electron to be able to continue its journey to the plate, the position detector has to employ some sort of nondestructive measurement technique, such as shining a light on the electron. In this way one knows whether the electron goes through slit 1 or slit 2. It turns out that it is indeed possible to check whether an electron goes through slit 1 or slit 2. But in this case, the undulating distribution disappears and the curve predicted by classical mechanics is observed. Classical mechanics becomes suddenly correct again.

A common ``explanation'' of this is: since one has to use photons to detect the position of passer-by electrons and to determine the position of electrons highly accurately (so that one knows with certainty that a particular electron goes through slit 1 but not slit 2), one has to use light with a shorter wave-length (and therefore higher frequency $\nu$). According to quantum mechanics, we know that the energy of a photon is

$$E=h\nu,$$

so photons with higher frequency must have higher energy. As a consequence, collisions between photons and the electron will push the electron back to position $x$. Sadly, this cannot explain everything. For one thing, why should an electron go back to $x$ and not somewhere else when the position detector is turned on? Moreover, it also does not explain what happens to the electrons when one is not ``watching'' (with position detector turned off)? Does the electron go through either slit 1 or slit 2? Indeed, a haunting question in quantum mechanics can be formulated simply: what happens to a physical system when nobody is watching? It shows that the presumably ``objective'' physical reality depends on the observer's ``way of looking.''

In fact, electrons have properties of both wave (going through slit 1 and slit 2 simultaneously) and particles (going through either slit 1 or slit 2). This is usually called the wave-particle duality in quantum mechanics. For practical purposes, it is enough to assume that an electron does somehow ``know'' whether both slits are opened. To determine the properties of a wave (e.g. wave length or frequency), we have to assume that the wave extends into infinity. So these properties are holistic. Since waves's properties are holistic, this ``knowledge'' must be holistic as well. In more concrete terms, this ``knowledge'' is described with a wave function. One should bear in mind, however, that the only properties a system can manifest are those of particles (in this experiment, sparks).