That first post I wrote, it sounds like I’m going to try to teach the world. Probably that won’t work. As I’m still looking for a place to dump my thoughts on this, I decided to moderate my ambition. This is all just a record of where I am.
One of my recent focuses has been working through problems in Chapter 3 of Sakurai. That’s the angular momentum chapter. I’ve done 18 problems there since December. They are less impressive than I remember; much easier in most cases.
A related system that I’ve come back to thinking about in the last couple days is Nuclear Magnetic Resonance. Way back before I started my physics degree, I struggled desperately to learn NMR as a biochemist. I thought it an incredibly cool technique and I still think so. The one big thing that has changed since then is that I’ve learned a lot about quantum mechanics. My first great brush with NMR was with classical expressions of bulk magnetization and I’ve been trying in the last few days to reconcile my current physicist’s intuition with those ghetto understandings from my previous life.
NMR has everything to do with angular momentum and rotation. It depends entirely on nuclear spin. Spin is a pure quantum mechanical concept that never quite fits into classical frameworks which can be approximately understood as ‘spinning’ like what a child’s top does. But, it isn’t quite like that because trying to describe spin as ‘spinning’ tends break reality a little. In a way, you can regard the ‘spinning’ of a half integer spin atomic nucleus as a circulation of charge in a loop, giving you a tiny magnetic dipole that wants to turn much like a compass needle when it is immersed in a magnetic field.
So, that’s what NMR does: you supply a powerful external magnetic field and the tiny atomic compass needles tend to orient along that field.
That would be the simplest way to regard it. If you learn more about quantum, you realize that it isn’t quite like that. A half integer spin like a hydrogen nucleus exists in a superposition of two states of spin angular momentum. That’s like saying that it can only be described as having one of two orientations and that any orientation that it takes is actually some mixture of those two where if anything happens to the nucleus, you can only ever see one orientation or the other. It’s a compass needle that can only ever be found to point either with or against the magnetic field.
This would be all well and good except that you can’t just sit down and look at the needle the way you can with a magnetic compass. When these things are less than half and angstrom in size, you have basically no way of ever directly knowing which way they point at any given time under any circumstance.
How does an NMR machine find which way these atomic needles are pointing? That is basically all you do in NMR: read out atomic compass needles.
Under normal conditions, the state of a sample sitting in an NMR machine is basically a thermal bath. All the atoms that can respond to the field do and they point either along the field or against it. Under these circumstances, the orientation pointing with the field is a low energy orientation, but every so often an atom absorbs energy and flips its spin to point against the field. It can then lose that energy again and flip to point back along the field. This happens continuously in the sample and the difference between the number of spins pointing with the field and against it describes the magnetization of the sample. If you were to take this sample out of the NMR machine instantaneously, you would find that it briefly has a magnetic field of its own much like a bar magnet. This magnetization would not last very long because, without the huge magnetic field of the NMR machine, the ‘with’ and ‘against’ populations would rapidly equalize in number and the magnetization would disappear. Conversely, if the NMR magnet is more powerful, fewer atoms can get the energy to flip against that field, making the difference between the ‘with’ and ‘against’ populations numerically larger, increasing the strength of the magnetization.
Now, how does a spin flip? It turns out that this is really important because this is how an NMR machine can communicate with an atomic compass needle to find out which way it’s pointing.
The only way that information goes into or leaves this system is by interactions with light. When a photon, a single corpuscle of light, impinges upon an atom, that photon can be absorbed to give the atom its energy. For NMR, this is important because absorbing a photon as large as the difference in energy between the two spin orientations can cause the atom to flip from one orientation to the other. This occurs from the lower energy state to the higher (from point along the field to pointing against it). The atom pointing against the field can then choose to radiate another photon of the same energy in order to return to pointing along the field. The NMR machine is designed to detect these photons that radiate out when the spin flips back along the field.
If that weren’t enough, the fact is that one atom flipping its spin is one photon and detecting one photon (particularly of the very low energy associated with spin flipping) can sometimes be pretty difficult. NMR machines are therefore built to massively enrich the signal. They hit all the present atoms with a large radio frequency pulse that causes everything to flip states all at once, which literally scrambles up the magnetization. Once the machine turns off the radio frequency pulse, the population is no longer in a thermal equilibrium and it has to give up the absorbed energy in order to return to that thermal state pointing along the magnetic field. So, the NMR machine looks for the massive pulse of energy that the sample gives up as it relaxes back to a more normal state.
The resulting signal that comes out of the sample is a radio wave broadcast called a free induction decay and is sometimes likened to the ringing of struck bell. The envelope of the signal looks like a sine wave decaying away.
This signal is where NMR got its name. It describes the driving of a magnetic system by an energy input at the resonant frequency of the system… hence, nuclear magnetic resonance. You strike the bell at its resonant frequency and then it sits there and rings.
I was sitting and thinking about this post and decided that I could add a small extension which helps detail some of the quantum mechanics that are going on. I said that a radio frequency burst ‘scrambles up the magnetization’ when I described the excitation that leads to the free induction decay. It is actually somewhat more sophisticated than simply ‘scrambling up.’ It turns out that the RF pulse is applied at a particular polarization with respect to the static magnetic field of the system. And, in classical terms, since the light in the RF pulse contains a magnetic field component, the direction of that field is additive with the static field to give a ‘torque’ on the magnetic dipoles of the half-integer spin atoms present in the NMR bomb. This is quantum mechanically an experiment where you have ‘prepared a state.’ If you consider the static field to be the z-axis of the system, the RF pulse essentially creates a new mixture of spins pointing along the x-axis of the system, perpendicular to the direction of the static field.
This x-axis mixture of states is prepared at a particular point in time by the RF pulse to a quantum mechanical eigenstate that is one of the two states in a temporary system not pointed along the z-axis static field. Those states are effectively a population of closed boxes, out of interaction temporarily with the rest of the universe after the RF field is turned off. When the RF pulse is turned off, the entire population of atomic spins is sitting in a prepared state with respect to the static field, in which they can only ever be found to point either with or against the z-axis. This prepared state is some coherent mixture of the z-axis ‘with’ and ‘against’ eigenstates which can evolve over time per the time dependent Schrodinger equation until they interact with the rest of the universe again, either by emitting or absorbing a photon and dropping into a z-axis eigenstate.
The free induction decay shows an exponential envelop because of the constant probability for a spin flip occurring among atoms present in the prepared state. The spin flip has a particular probability of occurring that does not vary over time during the observation, but the chances of seeing that flip occur are proportional to the number of atoms that are in the state: as the atoms flip into eigenstates of the static field, the number of atoms remaining in the prepared state decreases. As an example, if you have a certainty of seeing one spin flip every six seconds for the prepared state, a population of three atoms in that state means that you have a certainty of seeing a spin flip once every two seconds until one of the atoms has flipped, reducing the probability to once every three seconds until the next has flipped, reducing the probability for the final flip to occur once every six seconds. For a big population, you reach essentially a gaussian behavior for the event, which gives a smooth exponential decay curve for the decrease of events happening over time. This is what the exponential envelop shows: that the chances of the event occurring is decreasing simply because the numerical density of the prepared state atoms is decreasing exponentially.
The free induction decay also shows a sinusoidal envelop on top of the exponent because of the quantum mechanical evolution of the prepared state population. The wave function is a coherent mixture of the two eigenstates of the z-axis system where the antenna of the experiment has greatest probability of detecting flips from only the highest energy of those two states, since the lower energy of the two is under no pressure to flip (the higher energy being the ‘against’ state to within the phase constant). So, you see the flipping probability modulated by the period of the evolution of the closed box state, which is bobbing between the high energy ‘against’ state and the lower energy ‘with’ state… which happens to be the Larmor frequency of the static field. You can think of it this way: only the ‘against’ state can emit a photon in a spin flip… the ‘with’ state can only absorb a photon to reach the ‘against’ state, which can’t happen if no photons are being injected, even though the flip is presumably equal probability. As the wave function of the coherent state evolves in time to reach the ‘against’ eigenstate, you have a greater chance of seeing photons emitted, which is what the NMR experiment is sensitive to detect. The result is that the exponentially decreasing possibility of seeing atoms flipping to eigenstates of the static field are modulated by the sinusoidal occupancy of atoms in the constituent eigenstates of the prepared state wave function.
Kinda wicked, huh? The shape of the free induction decay literally tells you all the quantum mechanics of the system!