# An informal introduction to spin and kets

Subatomic particles have an intrinsic property that is quite strange to us. Called *spin*, this property has the dimensions of angular momentum. It is a rather strange name because the word spin does *not* imply that the particle is spinning on its own axis. All we *can* know is that it’s a type of angular momentum and it is different from the orbital and spin angular momenta we associate with bigger objects like the earth or the wheel of a moving bicycle. Spin is as fundamental as mass to subatomic particles and there is no macroscopic analogue to it.

Spin is a vector. Like any other vector, we can express it with components along $${x}$$, $${y}$$, and $${z}$$ directions. If we label the spin vector with **S**, then, **S** = S_{$${x}$$}$$\bf{i}$$ + S_{$${y}$$}$$\bf{j}$$ + S_{$${z}$$}$$\bf{k}$$, where S_{$${x}$$}, S_{$${y}$$}, and S_{$${z}$$} are the $${x-}$$, $${y-}$$, and $${z-}$$ components. Curiously, it’s useful to describe the spin by a vector that resides in what is called the *Hilbert space*. Though abstract, such vectors are found to be quite handy in understanding the world of atoms.

Despite being so exotic, spin is something that can be measured in a lab. For example, you can measure the maginitude of the spin of an electron by sending a bunch of neutral silver atoms through a strong magnetic field with a sharp gradient. You can use silver atoms as a proxy for electrons because the spin angular momentum of a silver atom is effectively due to that of a single electron.

Silver atoms, though electrically neutral, possess a magnetic moment and hence experience deflection in a magnetic field. You expect the silver atoms to be deflected over a wide range of angles because that’s what happens to bigger magnets that we play with. You can compute the magnitude of the spin angular momentum of silver atoms, and hence that of electrons, by measuring their deflection.

Pretend that you have a system that sets up a magnetic field, send silver atoms through it, measure the deflection, and calculate and display value of spin angular momentum. Your job is just to switch on the system, and set the direction of the magnetic field. Just keep in mind that the direction of magnetic field determines the component of spin you measure: if you set up the magnetic field in $${z-}$$ direction, you measure S_{$${z}$$} and so on.

Let’s say that you start the machine and set up the magnetic field in the $${z}$$ direction. You send one elctron at a time and note down the value of S_{$${z}$$ } for each electron. You will soon find out that the silver atoms are deflected in only two directions, but not by a multitude of angles like the magnets. The values of S_{$${z}$$} corresponding to these angles of deflection are $$+\hbar/2$$ and $$-\hbar/2$$.

After 100 silver atoms, you find that these two values are more or less evenly distributed; that is, *about* 50 silver atoms have $$+\hbar/2$$, and the rest have $$-\hbar/2$$. You repeat the experiment for about 10,000 silver atoms, and the story doesn’t change: their S_{$${z}$$} values are either $$+\hbar/2$$ or $$-\hbar/2$$, and the values are *more or less* evenly distributed.

Let’s pretend that you somehow are able to collect the silver atoms after they exit the magnetic field and put them into different boxes based on the magnitude of their spin angular momentum. So, you have a box with silver atoms with S_{$${z}$$} = $$+\hbar/2$$ and another with S_{$${z}$$}= $$-\hbar/2$$. On the box containing silver atoms with spin $$+\hbar/2$$ (call it Box 1), you write ‘$${+z}$$’. You write ‘$${+}$$’ because all the silver atoms inside have a positive value of spin, and ‘$${z}$$’ to indicate the direction of the magnetic field you had set up. Similarly, on the box having silver atoms with spin $$-\hbar/2$$ (call it Box 2), you write ‘$${-z}$$’. You don’t write the actual value of spin because the numeral part is same for all the silver atoms; what changes is the sign.

All the silver atoms in Box 1 have the same spin value, and as far as spin is concerned, they are in the same state. Let’s label this spin state $$|$$$${+z}$$$$\rangle$$, where the ‘$${+}$$’ and ‘$${z}$$’ have the meanings explained earlier. The symbols ‘$$|$$’ and ‘$$\rangle$$’ indicate that $$|$$$${+z}$$$$\rangle$$ is a *ket* — a type of vector that exists in what is called the Hilbert space. Kets, along with a few other constructs, form the foundation of *matrix mechanics* — a refreshingly different and elegant approach to quantum mechanics.

Kets are abstract vectors, but the information they provide is concrete. If someone tells you that electron is in state $$|$$$${+z}$$$$\rangle$$, what they mean is this: if you measure the spin angular momentum of the electron, say, by setting up a strong magnetic field in $${z-}$$ direction, you get S_{$${z}$$} = $$+\hbar/2$$. Similalry, an electron whose spin state is described by the ket $$|$$$${-x}$$$$\rangle$$ has S_{$${x}$$} = $$-\hbar/2$$. Of course, to measure S_{$${x}$$}, you would have to set up the magnetic field in $${x-}$$ direction. In general, labels like $$|$$$${+z}$$$$\rangle$$ and $$|$$$${-x}$$$$\rangle$$ convey two things: the direction of magnetic field, and the value one would obtain upon measurement.

Kets we have encountered so far provide clear information only about the state of the intrinsic spin angular momentum of the electron. They do not say anything about other important physical properties like the position, linear momentum, and orbital angular momentum of the electron. However, a more comprehensive description of the particle should involve the spatial part of the wave function describing position, linear momentum and so on.

Suppose you have a box full of silver atoms with their spin in state $$|$$$${-x}$$$$\rangle$$. Out of curiosity, you decide to measure the value of S_{$${z}$$} for each of them. The spins of all silver atoms are in the same state, all of them pass through the same magnetic field, and you naturally expect them to be have the same value of S_{$${z}$$}. You have a surprise again: you notice that *about* half of them have S_{$${z}$$} = $$+\hbar/2$$ and the rest S_{$${z}$$} = $$-\hbar/2$$. In other words, when we measure S_{$${z}$$} of silver atoms in state $$|$$$${-x}$$$$\rangle$$, about half of them end up in state $$|$$$${+z}$$ $$\rangle$$, and the rest in $$|$$$${-z}$$$$\rangle$$. We can capture the results of this experiment in a single equation:

The above equation is loaded with information. The LHS is a label for an ensemble of silver atoms with S_{$${x}$$} = $$-\hbar/2$$. That means, if you take an ensemble of silver atoms in state |$${-x}$$ $$\rangle$$, and measure their spin angular momentum by setting up a magnetic field in {x-} direction, then you would get $$-\hbar/2$$. The kets |$${+z}$$ $$\rangle$$ and |$${-z}$$ $$\rangle$$ — called the *basis states* — are the states in which the silver atoms will be found after passing through the magnetic field. The |probability that a silver atom will be found in state |$${+z}$$ $$\rangle$$ (or |$${-z}$$$$\rangle$$) is:

In general, the probability that a silver atom will be found in state $$|$$$${+z}$$ $$\rangle$$ is given by the square of the absolute value of the coefficient of $$|$$$${+z}$$ $$\rangle$$. Similarly, the square of the absolute value of the coefficient of $$|$$$${-z}$$ $$\rangle$$ gives the probability that a silver atom will be found in state $$|$$$${-z}$$ $$\rangle$$. For the state described in first of the above two equations, the chances of finding the silver atom in states $$|$$$${+z}$$ $$\rangle$$ and $$|$$$${-z}$$ $$\rangle$$ are same because those states have the same magnitude of the coefficient. However, we can never say with certainty that a given silver atom will end up in state $$|$$$${+z}$$ $$\rangle$$ (or $$|$$$${-z}$$ $$\rangle$$).

This ambiguity is indeed startling. If you have 100 silver atoms, and measure their masses under identical conditions, you *will* obtain the same value for each of them. You do not find that 52 (say) silver atoms have one mass and the rest have another mass. However, if you have 100 silver atoms in the same spin state, and measure their spin angular momentum under exactly same conditions, you do not get the same value. Why is it like this? Frankly, there is no answer to this question. All we can say is: “This is how nature works at the subatomic level.” More importantly, such observations show us that we can’t try to understand the subatomic world based on our experience gained in the macroscopic world.

We can interpret states like those expressed in eqn. (1) to be made of two other states $$|$$ $${+z}$$$$\rangle$$ and $$|$$ $${-z}$$$$\rangle$$. Such states, called *superposition states*, are real and can even be prepared in a lab. In fact, they are the building blocks of quantum computers and people have figured more than one way of preparing them. Surprisingly, quantum systems exist in superposition states only until they are observed, and collapse to one of the basis states upon observation.

Expressions like that in eqn. (1) beautifully capture the randomness that defines the subatomic world and accurately predict the statistical outcome of a large number of identical experiments. However, they fail to predict whether a given silver atom will end up in state $$|$$ $${+z}$$$$\rangle$$ or $$|$$ $${-z}$$$$\rangle$$! This is true not only for a simple system that is captured by eqn. (1), but for all quantum mechanics. This is not a bug, but a feature because it accounts for the fact that the subatomic world is inherently random and quantum mechanics captures it remarkably well.

**Acknowledgement:**

Jayanth Vyasanakere reviewed the initial draft of the article and provided valuable feedback.

### Author:

Madhukara S Putty