We know that if we want to perform a \"double slit\" experiment with electrons,

Question

We know that if we want to perform a "double slit" experiment with electrons, the "slits" need to be spaced very close together, which is why in practice we scatter electrons off of atoms in a crystal lattice. Let's imagine that I can somehow create two actual slits that are of a similar size to the crystal atomic spacing for this problem, so that our setup is completely analogous to Young's double slit experiment for light. We know that the proper mathematical description of this problem shows the electron wave function- which we approximate as an incoming plane wave-diffracting through both slits. The two diffracted wave patterns then interfere with each other while heading toward the detection screen, creating a non-uniform. "lumpv" cos2-like probabilitv densitv function at the screen. The physical interaction with the screen "collapses" the wave function, and the electron is observed to appear (i.e. interact with) a single position on the screen. That position is random within the constraints that statistical repetition of the process will produce results that agree with the cos2-like probability densitv function. Someone decides that he wants to see if the electron actually passes through slit 1 or slit 2 for a given trial, so he sets up a thin screen just behind slit 1. The screen is thin enough that the electron will lose almost no kinetic energy when passing through it, but it will register a hit, or no hit on each trial When our person performs the experiment, he indeed sees that every electron seems to either pass through slit 1 (he sees a hit on his thin screen), or slit 2 (no hit detected). Most interestingly, he sees that the statistical pattern of observations on the main screen far from the two slits no longer shows the characteristic "lumpy" cos2-like interference pattern. It just looks like a bright spot directly in front of slit 1, and another bright spot directly in front of slit 2. When he takes away his thin screen behind slit 1, however, the 'lumpy" cos2-like interference pattern returns. Explain why the introduction of the thin screen behind slit 1 kills the interference pattern using the concept of wave function collapse.

Explanation / Answer

After you put a detector behind one of the slits, the basis of states becomes a tensor product of the states of the particle and the ones of the detector (went through this slit or not).

So the cross terms that appear when you evaluate the probability of having reached a given position at the final screen coming from one or another slit, are now orthogonal (there is no overlapping of the detector states) and don't contribute to the probability, so you recover the classical particle like result. The classical particle like result is that the particle either came out of first slit or second slit.

In other words, the detector basically causes an abrupt change in the state of the particle which is called the wave function collapse because you are already measuring the state with the detector before it hits the screen.

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