Quantum Mechanics | Physics | Simplified
Spin: Explained
Spin was a latecomer to the quantum mechanical party. Even after Schrödinger wrote his infamous equation and everything seemed to be working perfectly, nobody realized that spin existed. For the unimitated: the spin of an electron, simply put, is a form of angular momentum. Objects that are spinning, rotating. or orbiting another body are ones with angular motion. Now with angular motion, comes angular momentum. This is slightly different from the linear momentum that you might be familiar with. Linear momentum is conventionally called just “momentum”. However, with quantum mechanics, it becomes extremely important to demonstrate the difference between these two. If you haven’t already, I’d strongly recommend reading the following as a prelude to quantum mechanics.
The thing about spin though is that certain particles have an inherent angular momentum. Or at least they seem to behave like they are spinning. They don’t actually spin. If such particles had inherent and infinite angular momentum, they would violate the law of conservation of energy.
The particle itself is not spinning, it just behaves like it does.
So what gives? Where does this behavior of angular momentum come from? Special relativistic effects in quantum mechanics! This is an extremely interesting topic that I’d like to do justice to. So, I suppose I’ll hoard it off till another date. Though what I will establish, is this: it’s misleadingly known that relativity and quantum mechanics don’t play well with each other. Interestingly, quantum mechanics and special relativity work great together. It’s general relativity that’s arrogant.
At this point, I’d also like to point out that lots of different particles have spin. But, we’ll limit ourselves to electrons since they’re probably the easiest to explain. Now with electrons, some of you might be familiar with the existence of two conceivable possibilities: a spin-up state and a spin-down one. When an electron is measured it could be in either of the following states:
|↑⟩ = spin up → clockwise
|↓⟩ = spin down → anticlockwise
What this means is that our electron can “have” angular momentum that makes it seem either spinning clockwise or anticlockwise. It’s always one of the two. Never both (or at least so we thought). Here’s the caveat. If our electron — our quantum system — is just left alone then it is said to be in a superposition of both these states, In other words, the electron isn’t |↑⟩ or |↓⟩, it’s |↑⟩ and |↓⟩. Now in the situation where our particular quantum system isn’t being measured, we can write the wave function of our electron as a superposition of both these states. The way that we can do that is to add the two states together.
Essentially what we’re doing is giving a representation of the overall state of our quantum system. But here’s the thing, let’s imagine that we go and actually measure the spin of this electron. We’ll only find it either in the spin-up state or the spin-down one. What happens is that our wave function, which was initially a superposition of both these states, collapses into just one. This phenomenon is known as the collapse of the wave function. Here’s the important thing though: we’ll never find our quantum system in a superposition of two states when we measure it.
This is where the number in front of the kets comes in. These numbers are directly linked to the probability of us finding the electron to be in a particular spin state. In other words, squaring the number that is in front of the spin-up state will give us the probability of finding our electron in that particular state. And equivalently with the spin-down state. Notice that adding their squares gives a total of 1 — which is what probability is measured relative to.
So essentially the probability of measuring our electron in the spin-up state is 0.75. While the probability of measuring it in the spin-down state remains 0.25. It’s important to understand that this is as far as we can go. Before measuring a quantum system, we cannot, with absolute certainty, predict its outcome. We can only predict the probabilities of the outcome.
All we need to know is that our quantum system can be in a superposition of states and supposedly in more than one state at one time. However, that only holds for as long as it’s left alone and nothing is interacting with it. If you somehow manage to measure its spin, you’ll find that it’s in either one of the two states. As soon as you make a measurement of the electron’s spin, its wavefunction collapses.
But how?
How can we tell that these electrons behave as if they’re spinning? What if it’s just God pulling the strings? Enter: the Stern-Gerlach experiment. Before we head into this, we’re going to need one piece from classical electromagnetism theory: the motion of a charged particle across space creates a magnetic field. I won’t delve too deep into the working of this experiment but as far as we’re concerned today, the actual experiment was carried out with a beam of silver atoms. One would expect there to be no interaction with an external magnetic field but when Stern and Gerlach directed the beam of atoms into a region of a nonuniform magnetic field, a magnetic force was experienced by the electrons. They found that the field separated the beam into two distinct parts, indicating just two possible orientations of the magnetic moment of the electron. Spin up and spin down. Of course, there are a few intricacies that I’ve left out here but if you’d like to know more about it then check this out.
But how does the electron obtain a magnetic moment if it has zero angular momentum? Well here’s where the postulate of intrinsic angular momentum comes.
Spin is quantized. In other words, when a measurement is made, the magnitude of the spin must be a particular value. A subtlety here: what it means for the magnitude to be particular is that it’s a discrete quantifiable amount. The magnitude is not definite. What it means for the magnitude of the spin to be “definite” is that it would be the same across all particles. This is evidently not the case.
This is equivalent to our electron only being allowed to spin clockwise or anticlockwise at a very particular rate. It cannot spin any faster or any slower. But here’s the important bit. We’ve already established that other particles also have spin. But not all particles are quantized the way electrons are. In other words, not all particles show two possible spin measurements. Photons, for example, have three possible spin states.
Mathematical description
Some of you might be familiar with electrons being called spin-½ particles, and photons being called spin-1 particles. What does this mean? Well, whenever you hear that a particle is a spin-n particle, n must take values of integers or half integers. In other words,
The “n” here represents the magnitude of angular momentum that a particle can have. Specifically, n allows us to find the maximum angular momentum:
n x ħ is the maximum angular momentum
where ħ is the modified Planck’s constant. So a spin-1 particle can be found with a maximum angular momentum of 1 times ħ which is nothing but ħ itself. However, the value only corresponds to the magnitude of the spin state. The number of possible spin states is given by this:
n-1 = other possible spin states
It is important to note that you stop subtracting 1 once you reach the -n value. For example, a spin-1 particle (like the photon) will have three possible spin states: 1, 0, -1. We stop at -1 since it’s the negative of n, which is 1 here. This example is a bit dodgy, however. For photons, the 0 state is not allowed. Though that’s for a completely different reason. But the point still stands. This is how you calculate all possible spin states. Similarly, for a spin-2 particle, there would be the following states: 2, 1, 0, -1, -2.
A ½ -spin particle will have two possible spin states: ½, -½. You get the point.
Once these states are calculated, multiply with the modified Planck’s constant to find the maximum possible angular momentum. Child’s play. To summarize:
nħ = maximum angular momentum
n-1= other possible states (take to -n value)
The one thing I’d like to point out, however, is that all particles with a half-integer spin behave very differently than those with an integer spin. In fact, particles that have a half-integer spin are defined as fermions and those with an integer spin (n = 1, 2, 3, and so on) are defined as bosons. An electron, therefore, is an example of a fermion. A photon, on the other hand, is a boson.
That’s all for spin. I hope this article has offered insight into the obscure workings of quantum mechanics. I’d appreciate your support. Thank you for reading.
