In the Dark

As the Christmas holiday draws to a close and I begin thinking about the possibility that sooner or later, in due course, at some point in the future, in the fullness of time, all things considered, when all is said and done, in the end, I will have to start teaching again. Thinking about this is preferable to thinking about the stack of exam marking that I will have to contend with shortly.

One of the modules I am down to teach in the Spring Semester is particle physics, a subject I haven’t taught for well over a decade, so I have been looking through a box of old notes on the subject. Doing so I remembered that I had to explain neutrino oscillations, a process in which neutrinos (which have three distinct flavour states, associated with the electron, mu and tau leptons) can change flavour as they propagate. It’s quite a weird thing to spring on students who previously thought that lepton number was always conserved so I decided to start with an analogy based on more familiar physics.

A charged fermion such as an electron (or in fact anything that has a magnetic moment, which would include, e.g. the neutron)  has spin and, according to standard quantum mechanics, the component of this in any direction can  can be described in terms of two basis states, say “up” for the +z-direction and “down” for the opposite (-z) represented schematically like this:

In this example, as long as the particle is travelling through empty space, the probability of finding it with spin “up” is  50%, as is the probability of finding it in the spin “down” state, the probabilities defined by the square of the amplitudes. Once a measurement is made, however, the state collapses into a definite “up” or “down” wherein it remains until something else is done to it. In such a situation one of the coefficients goes to zero and the other is unity.

If, on the other hand, the particle  is travelling through a region where there is a magnetic field the “spin-up” and “spin-down” states can acquire different energies owing to the interaction between the magnetic moment of the particle and the magnetic field. This is important because it means the bits of the wave function describing the up and down states evolve at different rates, and this  has measurable consequences: measurements made at different positions yield different probabilities of finding the spin pointing in different directions. In effect, the spin vector of the  particle performs  a sort of oscillation, similar to the classical phenomenon called  precession.

The mathematical description of neutrino oscillations is very similar to this, except it’s not the spin part of the wavefunction being affected by an external field that breaks the symmetry between “up” and “down”. Instead the flavour part of the wavefunction is “precessing” because the flavour states don’t coincide with the eigenstates of the Hamiltonian that describes the neutrinoes. For this to happen, however, different neutrino types must have intrinsically different energies  (which, in turn, means that the neutrinos must have different masses), in quite  a similar way similar to the spin-precession example.

Although this isn’t a perfect analogy I thought it was a good way of getting across the basic idea. Unfortunately, however, when I subsequently asked an examination question about neutrino oscillations I got a significant number of answers that said “neutrino oscillations happen when a neutrino travels through a magnetic field….”.

Sigh.

Neutrinos have no magnetic moment so don’t interact with  magnetic fields, you see…

Anyhow, I’m sure there’s more than one reader out there who has had a similar experience with an analogy that wasn’t perhaps as instructive as hoped. Feel free to share through the comments box…

This entry was posted on January 7, 2025 at 6:33 pm and is filed under Biographical, Education, The Universe and Stuff with tags neutrino oscillations, precession, Quantum Mechanics, spin. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.