Electron spin

Introduction

Figure 2.1 - Spin vector

One of the most basic experiments we can do to highlight the difference between classical and quantum systems is to measure the spin of an electron.

Electron spin can be visualised as a unit three-dimensional vector associated with the particle, representing an axis of rotation. We label the arrowhead as N(orth) to highlight the fact that spin can also be seen as a magnetic property.

Nb. These are unit vectors that can point in any direction. In the following, we are suppressing the third dimension.

Preparation

Figure 2.2 - Prepared directions

We fix an electron's position in space (this actually can be done in the laboratory). In order to prepare an electron in a particular direction, we could surround the electron in a powerful magnetic field - see figure 2.2.

The magnetic field forces the electron's spin to end up in the desired direction after a certain amount of time. In fact the spin vector precesses around the desired position, radiating energy and spiralling in. The stronger the field, the quicker this all happens.

Classical detection

Suppose that an electron has been prepared, as above, in some unknown direction and we want to be able to measure, or detect the electron's spin.

Figure 2.3 - Classical detection

We could again surround the electron with a known magnetic field, as in figure 2.3. We expect, as in the preparation stage, that the electron's spin will eventually line up as expected, which we denote $|North-Up \rangle$.

We can measure the amount of energy that is radiated away in this process - the larger the energy radiated, the larger the angle away from the final position, $|North-Up \rangle$.

We would expect to be able to produce the graph on the right-hand side of figure 2.3, where the amount of energy radiated ranges continuously from zero to a maximum at an angle, $\theta$, away from $|North-Up \rangle$.

THIS DOES NOT HAPPEN!

Actual detection

Figure 2.4 - One photon

What actually happens is that, whatever angle the electron is initially prepared at, when we come to apply the magnetic field only one of two things happen:

• No photon is emitted by the electron
• Exactly one photon is emitted

If a photon is emitted, then its associated frequency corresponds to the amount of energy that would be radiated if the electron had been prepared in the $|North-Down \rangle$ position.

Note that the actual result - that is, one photon emitted or no photon emitted - doesn't depend on either the prepared angle or the detection angle. In fact, the outcomes of any experiment are probabilistic. (In our case, given some prepared angle of an electron, there is some probability that a photon is emitted (and a probability that a photon isn't emitted).

It is this probability that depends on the angle. Qualitively, the smaller the angle (between prepared and detection states) the less likely that a photon is emitted.

The maximum probability, $\frac{1}{2}$, occurs at $\theta = \pi$.

So, information about the prepared angle can be statistically recovered from repeated experiments, but to re-iterate, only one of two outcomes can occur per detection.