This and the next posts will further build on the concepts introduced in my previous post on particle spin. This post in particular will focus on some of the math we’ll need to understand what quantum mechanics is all about. The first topic is about the quantum-mechanical equivalent of the phenomenon of precession. The other topics are… Well… You’ll see… š

**The Larmor frequency**

The motion of a spinning object in a force field is quite complicated. In our post on gyroscopes, we introduced the concepts of *precession* and *nutation*. The concept of precession is illustrated below for the Earth as well as for a spinning top. In both cases, the external force is just gravity.

Nutation is an additional movement: on top of the precessional movement, a spinning object may wobble, as illustrated below.

There seems to be no analog for nutation in quantum mechanics. In fact, the terms nutation and precession seem to be used interchangeably in quantum physics, although they are very different in classical physics. But let’s not complicate things and, hence, talk about the phenomenon of precession only.

We will not re-explain the phenomenon of precession here but just remind you that the phenomenon can be described in terms of (a) the angle between the symmetry axis and the momentum vector, which we’ll denote by Īø, and (b) the angular velocity of the precession, which we’ll denote by Ļ_{pĀ }= dĻ/dt, as shown below. The **J** in the illustration below is theĀ angular momentum of the object. Hence, if we’d imagine it to be an electron, thenĀ **J**Ā would be the *spinĀ *angular momentum only, *not* its orbital angular momentumāalthough the analysis would obviously be valid for the *orbital *and/orĀ *totalĀ *angular momentum as well.

OK. Let’s look at what’s going on. The *angular displacement* ā which is also, rather confusingly, referred to as the angle of precessionĀ ā in the time intervalĀ Īt is, obviously, equal to ĪĻ = Ļ_{p}Ā·Īt.Ā Now, looking at the geometry of the situation, and using the small-angle approximation for the sine, one can also see that ĪJ ā (JĀ·sinĪø)Ā·(Ļ_{p}Ā·Īt). In fact, going to the limit (i.e. for infinitesimally smallĀ ĪĻ and ĪJ), we can write:

dJ/dt = Ļ_{p}Ā·JĀ·sinĪø

But the angular momentum cannot change if there’s no *torque*. In fact, the time rate of change of the angular momentum is *equalĀ *to theĀ torque. [You should look this up but, if you don’t want to do that, note that this is just the equivalent, for *rotational* motion, of the F = dp/dt law for *linear* motion.] Now, in my post on magnetic dipoles, I showed that the torqueĀ **Ļ**Ā on a loop of current with magnetic momentĀ **Ī¼Ā **in an external magnetic field **B**Ā is equal toĀ **Ļ** = **Ī¼**Ć**B**. So the magnitude of the torque is equal to |**Ļ**| = |**Ī¼**|Ā·|**B**|Ā·sinĪø = Ī¼Ā·BĀ·sinĪø. Therefore,Ā Ļ_{p}Ā·JĀ·sinĪø = Ī¼Ā·BĀ·sinĪø and, hence,

Ļ_{pĀ }= Ī¼Ā·B/J

However, from the generalĀ **Ī¼**/**J** = āgĀ·(q_{e}/2m)**Ā **equation we derived in our previous post, we know that Ī¼/JĀ ā for an atomic magnet, that is āĀ must be equal to Ī¼/J = gĀ·q_{e}/2m. So we get the formula we wanted to get here:

Ļ_{pĀ }= gĀ·(q_{e}/2m)Ā·B

This equation says that the angular velocity of the precession is proportional to the magnitude of the external magnetic field, and that the constant of proportionality is equal to gĀ·(q_{e}/2m). It’s good to do the math and actually *calculateĀ *the precession frequency *f*_{p}Ā = Ļ_{p}/2Ļ. It’s easy.Ā We had calculated q_{e}/2m already: it was equal to 1.6Ć10^{ā19}Ā C divided by 2Ā·9.1Ć10^{ā31Ā }kg, so thatās 0.0879Ć10^{12Ā } C/kg or 0.0879Ć10^{12Ā }(CĀ·m)/(NĀ·s^{2}), more or less. š Now, g is dimensionless, and B is expressed in *tesla*: 1 T = (NĀ·s)/(CĀ·m), so we get the s^{ā1}Ā dimension we want for a frequency. For g = 2 (so we look at the *spin *of the electron itself only), we get:

*f*_{p}Ā =Ā Ļ_{p}/2Ļ = 2Ā·0.0879Ć10^{12}/2Ļ ā 28Ć10^{9 }= 28 gigacycles *per tesla = *28Ā GHz/T

This is a number expressed* per unit *of the magnetic field strength B. Note that you’ll often see this number expressed as 1.4 megacycles per *gauss*, using the olderĀ *gaussĀ *unit for magnetic field strength: 1 tesla = 10,000 gauss. For a nucleus, we get a somewhat less impressive number because the proton (or neutron) mass is so much bigger: it’s a number expressed in *mega*cycles per tesla, indeed, and for a proton (i.e. a hydrogen nucleus), it’s about 42.58 MHz/T.

Now, you may wonder about the numbers here. Are they astronomical? Maybe. Maybe not. It’s probably good to note that theĀ strength of the magnetic field in medical MRI systemsĀ (magnetic resonance imaging systems) is only 1.5 to 3 tesla, so it’s a rather large unit. You should also note that the clock speed of the CPU in your laptop ā so that’s the speed at which it executes instructions ā is measured in GHz too, so perhaps it’s not so astronomic. I’ll let you judge. š

So… Well… That’s all nice. The key question, of course, is whether or not this classical view of the electron spinning around a proton is accurate, quantum-mechanically, that is. I’ll let Feynman answer that question provisionally:

“According to the *classical* theory, then, the electron orbitsāand spinsāin an atom should precess in a magnetic field. Is it also true quantum-mechanically? It is essentially true, but the meaning of the āprecessionā is different. In quantum mechanics one cannot talk about the *direction* of the angular momentum in the same sense as one does classically; nevertheless, there is a very close analogyāso close that we continue to call it precession.”

To distinguish classical and quantum-mechanical precession, quantum-mechanical precession is usually referred to as *Larmor precession*, and the frequencies above are often referred to as ** Larmor frequencies**. However, I should note that, technically speaking, the term

*Larmor frequencyĀ*is actually reserved for the frequency I’ll describe in the next section.Ā I should also note that theĀ Ļ

_{pĀ }= gĀ·(q

_{e}/2m)Ā·B is usually written, quite simply, asĀ Ļ

_{pĀ }= Ī³Ā·B. Of course, the gamma is

*Ā notĀ*the Lorentz factor here, but the so-called gyromagnetic ratioĀ (aka as the magnetogyric ratio):Ā Ī³ =Ā gĀ·(q

_{e}/2m). Ohājust so you know: Sir Joseph Larmor was a British physicists and, yes, he developed all of the stuff we’re talking about here. š

At this point, you may wonder if and why all of the above is relevant. Well… There’s more than one answer to this question, but I’d recommend you start with reading the Wikipedia article on NMR spectroscopy. š And then you should also read Feynman’s exposĆ© on the Rabi atomic or molecular beam method for determining the precession frequency. It’s really fascinating stuff, but you are sufficiently armed now to read those things for yourself, and so I’ll just move on. Indeed, there’s something else I need to talk about here, and that’sĀ *Larmor’s Theorem*.

**Larmor’s Theorem**

We’ve been talking single electrons only so far. Now, you may fear that things become quite complicated when many electrons are involved and… Well… That’s true, of course. And then you may also think that things become even more complicated when *external fields *are involved, like that external magnetic field we introduced above, and that led our electrons to precess at extraordinary frequencies. Well… That’s *notĀ *true. Here we get some help: Larmor proved a theorem that basically says that, if we can work out the motions of the electronsĀ *withoutĀ *the external field, the solution for the motionsĀ *withĀ *the external field is the no-field solutionĀ *with an added rotation about the axis of the field*. More specifically, for an external magnetic field, the added rotation will have an angular frequency equal to:

Ļ_{LĀ }= (q_{e}/2m)Ā·B

So that’s the same formula as we found for the angular velocity of the precession if g = 1, so that’s very easy to remember. TheĀ Ļ_{L}Ā frequency, which is the precession frequency for g = 1, is referred to as theĀ *Larmor frequency*. The proof of the above is remarkably easy, but… Well… I don’t want to copy Feynman here, so I’ll just refer you to the relevant *Lecture*Ā on it. š

**Diamagnetism**

I guess it’s about time we relate all of what we learned so far to properties of matter we can relate to, and so that’s what I’ll do here. We’re not going to talk aboutĀ *ferromagnetismĀ *here, i.e. the mechanism through which iron, nickel and cobalt and most of their alloys become *permanentĀ *magnets. That’s quite peculiar and so we willĀ *notĀ *discuss it here. Here we’ll talk about the very weak quantum-mechanical magnetic effectĀ ā a thousand to a million times less than the effects in ferromagnetic materialsĀ ā that occurs inĀ *allĀ *materials when placed in an external magnetic field.

While the effect is there inĀ *allĀ *materials, it’s stronger for some than for others. In fact, it’s usually so weak it is hard to detect, and so it’s usually demonstrated using elements for which the diamagnetic effect is somewhat stronger, like bismuth or antimony. The effect is demonstrated by suspending a piece of material in a non-uniform field, as illustrated below. **The diamagnetic effect will cause a small displacement of the material, away from the high-field region, i.e. away from the pointed pole.**

I should immediately add that some materials, like aluminium, will actually be attracted to the pointed pole, but that’s because of yet another effect that *notĀ *all materials share: paramagnetism. I’ll talk about that in another post, together with ferromagnetism. So… Diamagnetism: what is it?

The illustration below shows our spinning electron (q) once again. It also shows a magnetic field **B** but, unlike our analysis above, or the analysis in our previous post, we assume the external magnetic field is not just there. We assume it *changes*, because it’s been turned on or offāhopefully *slowly*: if not, we’d have eddy-current forces causing potentially strong impulses.

But so we’ve got some *changeĀ *in the magnetic flux , and so we know, because of Faraday or MaxwellĀ āĀ you choose šĀ ā that we’ll have some circulation of **E**, i.e. theĀ *electricĀ *field. The magnetic flux is B times the surface area, and the circulation is the average tangential component E times the length of the path. Because our model of the orbiting electron is so nice and symmetric, we can write Faraday’s Law here as:

EĀ·2ĻĀ·r =Ā ād(BĀ·ĻĀ·r^{2})/dt āĀ E =Ā ā(r/2)Ā·dB/dt

A field implies a force and, therefore, a *torque* on the electron. The torque is equal to the force times the lever arm, so it’s equal to (āq_{e}Ā·E)Ā·r = āq_{e}Ā·EĀ·r. Of course, the torque is also equal to the rate of the change of the angular momentum, so dJ/dt must equal:

dJ/dt = āq_{e}Ā·EĀ·r = Ā q_{e}Ā·(r/2)Ā·(dB/dt)Ā·r = (q_{e}Ā·r^{2}/2)Ā·(dB/dt)

Now, the assumption is that the field goes from zero to B, so ĪB = B. Therefore, ĪJ must be equal to:

ĪJ =Ā (q_{e}Ā·r^{2}/2)Ā·B

You should, in fact, derive this more formally, by integratingābut let’s keep things as simple as we can. š What does this formula say, really? **It’s the extra angular momentum from the ‘twist’ that’s given to the electrons as the field is turned on**. Now, this

*added*angular momentum makes an

*extra*magnetic moment which, because it is an

*orbital*motion, is justĀ q

_{e}/2mĀ times the angular momentum that’s already there. But more angular momentum means the magnetic moment has changed, according to theĀ

**Ī¼**= (q

_{e}/2m)Ā·

**J**formula we derived in our previous post, so we have:

ĪĪ¼ = ā(q_{e}/2m)Ā·ĪJ

TheĀ minus sign is there because of Lenz’ law: theĀ *added*Ā momentum is *opposite *to the magnetic fieldā*a*nd, yes, I know: it’s hard to keep track of all of the conventions involved here. In any case, we get the following grand equation:

So we found that theĀ *induced*Ā magnetic moment is directly *proportional to* the magnetic fieldĀ B,Ā and *opposing it*. Now that is what explains why our piece of bismuth does what it does in that non-uniform magnetic field. Of course, you’ll say: why is stronger for bismuth than for other materials? And what about aluminium, or paramagnetism in general? Well… Good questions, but we’ll tackle them in the next posts. š

Let me conclude this post by copying Feynman’s little *exposĆ©* on why the phenomenon of diamagnetism is so particular. In fact, he notes that, because we’re talking a piece of material here that can’t spin ā so it’s held in place, so to say āĀ we should have “no magnetic effects whatsoever”. The reasoning is as follows:

This is very interesting indeed. This classical theorem basically says that the energy of a system should not be affected by the presence of a magnetic field. However, we know magnetic effects, such as the diamagnetic effect, are there, so these effects are referred to as ‘quantum-mechanical’ effects indeed: they can*notĀ *be explained using classical theory only, even if all of what we wrote above used classical theory only.

I should also note another point: why do we need a *non*-homogeneous field? Well… The situation is comparable to what we wrote on the Stern-Gerlach experiment. If we would have a *homogeneousĀ *magnetic field, then we would only have a torque on all of the atomic magnets, but no *net *force in one or the other direction. Thereās something else here too: you may think that the forces pointing towards and away from the pointed tip should *cancel each other out*, so there should actually be *no *net movement of the material at all! Feynmanās analysis works for *one *atom, indeed, but does it still make sense if we look at the whole piece of material? It does, because weāre talking an *induced *magnetic moment thatās *opposing *the field, regardless of the orientation of the magnetic moment of the individual atoms in the piece of material. So, even if the individual atoms have opposite momenta, the extra *induced *magnetic moment will point in the same direction for all. So that solves that issue. However, it does *not *address Feynmanās own critical remark in regard to the supposed ‘impossibility’ of diamagnetism in classical mechanics.

But I’ll let you think about this, and sign off for today. š I hope you enjoyed this post.