Figure5 double slit interference for light

 But a closer look reveals that something strange happens in interference experiments.

Thus although there is something wavelike, a single electron doesn't seem to be smeared over a large region on the screen as a wave should do. The individual electron hits apparently randomly a definite point . Thus the interference pattern seems at first look to reflect only a collective statistical behavior.

and/or

What could we do to test this assumption?

1. We could lower the intensity of the electron beam so much that electrons pass only one after the other through the slits so that we eliminate the electron-electron interaction.

One can do the that. If one waits long enough so that a large number of hits occur, the resulting interference pattern is the same as the one one gets with a high intensity beam. So, whatever creates the interference pattern, it cannot be an electron-electron interaction. Now let’s consider the second alternative; namely let's assume that the reason is the interaction between the electron and the slit edge.

How could we notice a possible effect of electron-slit edge interaction on the interference pattern?

We can do the following:

Figure 6, patterns for single slit A and single slit B

Now we reverse the roles of both slits and do the same experiment by letting whole electrons pass through slit B. We observe a similar pattern with a main bright stripe with its center located behind the slit B ( line 6b in Figure6).

Now what would you expect if we open both slits?

For an individual electron that goes through slit A it should be irrelevant if the slit B is open or not and vice versa. So we would expect a superposition of both patterns on the screen (the dashed line in figure 6 and the dashed line in figure 7)

Figure 7 comparison of double slit interference pattern (solid line) and the superposition of 6a and 6b (dashed line)

The observed resulting interference pattern is however not the superposition but the pattern in figure 5 (solid line in figure 7). There are now dark stripes where there were bright regions in the separate experiments and vice versa . If we still want to keep the picture of electrons with definite trajectories passing through one or the other slit, we have to come to the following strange conclusion:The electron that passes for example through the slit B changes its trajectory depending on whether slit A is open or not. Thus it is as if the electron doesn’t interact only with the edges of the slit it passes through, but it also interacts with the slit IT DOES NOT pass through.

The strange fact is that the comparison of the pattern of a single slit with the interference pattern of the double slit suggests that the effect of the interaction with the edge of the FAR SLIT on the electron is not less then the effect of the edge of the slit it passes through.

If we don’t want to bother ourselves with such strange and complicated electron-slit edge interactions when there is a simple way to describe the pattern in terms of ordinary wave interference, we have to come to the following conclusion:

Even one single electron is an extended wave and goes through both slits simultaneously.

However if we try to determine through which slit the electron goes through by bringing the scintillating screen immediately behind the slits we can actually observe that the electron goes only through one of the slits and not through both slits simultaneously. One can leave the screen where it is and can obtain the which-way information without stopping the electron using a cloud chamber¹ If we do that we observe that the interference pattern on the screen, namely the wave behavior, is destroyed.

Actually the interference experiment with electrons was made later in 1927 after the discovery of matrix mechanics by Heisenberg and the discovery of Schrodinger equation by Schrodinger and the conceptual problems that arise when one tries to explain the double slit experiment was discussed in this form much later by Feynman. However I took the freedom to discuss it in advance in the book because it demonstrates most dramatically the conceptual problems of the quantum mechanics including the wave particle duality.

Tus there appeared to be 4 possible logical combinations as candidates for approaching the wave-particle problem.

1.There is only a particle .

2.There is only a wave.

3. There is a wave and a particle as two coexisting entities (pilotwaves). In this approach it is assumed that particles are guided by waves.

4. Wave and particle are two complementary incompatible faces of the same entity. One or the other face comes in appearance depending on the experiment one conducts.

Experimental evidence rules the alternative 1 out. There is certainly something wavelike that interferes. Viewpoint 2 was advocated by Schrodinger . Viewpoint 3 was advocated by De Broglie and viewpoint 4 was defended by Bohr. There was no successful pilotwave theory at these times. ²

Viewpoint 2 was given up because of problems we will discuss in chapter 6. So that in the end viewpoint 4 became the official viewpoint. Let’s first consider alternative 2 more closely to understand its advantages and shortcomings.

Assume that the free electron behaves as an extended wave but it changes its form and turns into a sharply localized peak when it interacts with other particles/waves, namely with atoms in the scintillating screen in our case. You can imagine the wave function of the electron as a cloud condensing to a droplet during a position measurement. This process is called the collapse or reduction of the wave function.

This picture seems at first look to resolve the wave/particle dilemma but raises a new question.

Why does the condensation of the cloud/wave representing a single electron occurs for each electron at a different position on the scintillating screen? What determines the localization point ?

To gain insight into the process of the interaction of the wave with a screen one had to find the answer to the question of how the electromagnetic fields that are present in the scintillating screen effect the wave function. The answer to this question was not only important to understand how the electron wave can change its form so drastically and turn into a sharply localized peak when it hits the screen, but it was also important to understand the properties of atoms since atoms are entities made up of electrons captured in the attractive field of the nucleus.

The answer proposed by Erwin Schrodinger was a wave equation . Its discovery is one of the important mile-stones in the history of physics. The next chapter is dedicated to the Schrodinger equation. The equation led to a deep understanding of the properties of the atoms but unfortunately it seemed that it didn’t help to understand the collapse of the of the wave function during position measurement.

1. A cloud chamber contains over-saturated vapor of water . If a charged particle flies in it, condensation of water occurs in the form of micro droplets around the particle. The path of the particle becomes visible.

2. A successful pilotwave theory was developed by David Bohm later in 1950’s. long after the Copenhagen interpretation of quantum mechanics was established and became an inseparable part of undergraduate text books. Despite its internal consistency Bohm’s approach is unfortunately not without problems. We will come back to Bohm’s interpretation in chapter 12 where we discuss the alternatives to Copenhagen interpretation.