Monday's Lesson - Season Finale

Today's lesson is on the extent of strangeness in quantum mechanics.

You have our good friend Tom Foss to blame for bringing this idea to my mind. In his blog, he asked his readers to identify their favorite unsung heroes of science. I immediately thought of Yakir Aharonov, and if I had posted a comment there, it would have spared you this post here. But as it turns out Aharonov is still alive (as far as I can tell), and Tom restricted discussion to dead scientists. So here you go:

Those of you who have taken Modern Physics are familiar with the double-slit experiment (those of you who haven't - what are you waiting for?!) If you perform the double-slit experiment with an electron beam, the beam passes through both slits and the parts of the electron-beam-wave passing through the two slits interfere with each other, forming an interference pattern: lots of electrons where there is constructive interference, few electrons where there is destructive interference. If we picture the electrons as classical particles, with definite positions, passing through one slit or the other, we cannot explain the interference between the two paths. Conclusion: the electron beam is described by a wavefunction, and the wave passes through both slits at once.

OK, we can picture waves being in more than one place at once, like a water wave striking more than one point on the shore, but it it gets stranger. Suppose we put in detectors to determine which slit the electron went through. Will it go through both slits? No. The detector will show us which slit the electron went through. Then how can the two paths interfere? They don't. Without the detectors, you get interference, with them, you don't. The detectors are said to "collapse the wavefunction" so it no longer passes through both slits.

Wavefunction collapse is a little weird, but just wait. What if we only put one detector in one slit (call it slit A), and no detector in the other slit (slit B). You might guess that only the electrons passing through slit A would have their wavefunctions collapse - they are the only ones meeting a detector, after all. However, the actual result is exactly the same as the previous case, when there were detectors in both slits. Odd. The electrons passing through slit B seem to be affected by the detector at slit A, where they don't go.

The electrons being affected by something in a region of space they didn't go through seems a little strange, but at least the wavefunction of these electrons was there before it collapsed. In the Aharonov-Bohm Effect, the electrons are affected by a magnetic (or electric) field in a region they never go AND where the wave doesn't go - the wavefunction is zero in the region of the field!

Imagine the electron beam passing on either side of a solenoid, rather than through two slits. The beam still has two paths - left and right of the solenoid - so we get an interference pattern. Suppose we turn on a magnetic field inside the solenoid (where the electrons don't go) but the field outside the solenoid (where the electrons do go) is zero. This magnetic field changes the interference pattern that is observed. The behavior of the electrons is affected by a field they never pass through!

Quantum Mechanics is strange indeed. Strange, but true.

Appendix. For those of you with a bit of E&M background, read on:
In classical E&M, the electric and magnetic fields are the quantities that exert forces and affect a particle's path. The scalar potential, V(r), and vector potential, A(r) are introduced as mathematical toys to be used to calculate the fields. In our quantum example, even though the magnetic field outside the solenoid is zero, the vector potential is not zero. The phase of the electron wavefunction is affected directly by the vector potential - it is not just a mathematical toy.