Questions regarding particle spin

Question

I have two queries regarding the spin of particles.

1. Can we talk on the conservation of spin? Citing this, the answerer writes that spin is intrinsic angular momentum, but what exactly do you mean by intrinsic angular momentum?

2. At many places, I have seen spin being described as 'a spatial orientation' of the particle (most profoundly in Stephen Hawking's Brief History of Time), that spin of ½ means that only on rotating a particle twice, will it completely be symmetric to its original configuration. But how does this definition relate to the angular momentum aspect of spin or magnetic aspect of spin?

Answer 1

Spin is an intrinsic property of particles. For example: an electron has spin $1/2$, always, regardless of its condition or its surroundings; that's why we say that spin is an intrinsic property. This is not strange or uncommon if you think about it: an electron has also charge equal to $e$, always; in fact charge is another intrinsic property of a particle.

But we don't just say that spin is an intrinsic property, no no no, we also say that spin is the intrinsic angular momentum of a particle, why is that? Well we cannot answer this question completely without getting into the mathematical details, but I will try my best:

You have to understand that in Quantum Mechanics different observables¹ respect different rules of behaviour, for example: you cannot know simultaneously the position of a particle and its momentum, or you cannot know simultaneously the angular momentum of a particle along different axis; these rules are described by certain mathematical structures; we tend to say that different sets of observables obey different algebras. From the experiments about spin, like the Stern-Gerlach experiment, we see that spin, as an intrinsic property, respects the exact same rules of behaviour that the orbital angular momentum respects; so since this two different observables respect the same rules, the same algebra, we say that they are two different form of the same thing; this is the logic behind it.

And at last, regarding your second point, that description of spin that you are talking about is a portion of the rules that I was talking about; specifically we can show, with an ungodly amount of math for which you are not prepared, that the spin of a particle with spin $1/2$ can be described by a two dimensional vector, sometimes called spinor. This then implies the symmetry that you were talking about. This is of course related with angular momentum for the reasons I mentioned above. And regarding the "magnetic aspects" of the spin that is another, different, phenomena. I suggest you to take a look at the g-factor, but basically: in classical mechanics charged things that are rotating produce a magnetic field, or more precisely they have a magnetic dipole, this is also true in the context of quantum mechanics; experiments show that a charged particles with spin has a magnetic dipole, that can be described using the g-factor.

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[1]: By observables we mean quantities that can be measured: like position, momentum, orbital angular momentum, spin, etc.

Answer 2

Spin cannot be visualised as a ball spinning because it would need to spin faster than light ( which special relativity forbids) to generate the magnetic field associated with for example the spin of an electron. Angular momentum in classical mechanics means mass x velocity x radius but what quantum mechanical spin shows is that angular momentum can exist without velocity or radius.

Answer 3

Maybe why particles are assigned intrinsic spin will help: Conservation of angular momentum is a law in classical physics. When it became necessary to invent quantum mechanics in order to explain experiment, i.e. the photoelectric effect, the black body radiation and particularly the spectra of atoms, the macroscopic conservation laws, as momentum, energy and angular momentum , were expected to also be laws in the new theory, for continuity from the microscopic scale to the classical mechanics scale.

Quantization presented no problems for energy and momentum in explaining data. Angular momentum of nuclear interactions and decays of particles presented a problem, as it seemed that angular momentum was not conserved. BUT, they found out that if they assigned an intrinsic angular momentum to protons, neutrons, electrons, photons .. then the angular momentum conservation law was upheld always. That is how intrinsic spin came to exist. It is an axiomatic proposal that works in keeping the same conservation laws in the classical and quantum states. The appropriate group models were found and all data up to now are consistent with the axiomatic hypothesis of intrinsic spin.

So for 1.

It is conservation of total angular momentum that was at stake. There is no conservation of spin, it is intrinsic to a particle description as intrinsic as its mass and charge.

For 2.

All of quantum physics started with the Bohr model of the atom, electron around a positive nucleus. It has an angular momentum,by definition, and Bohr quantized it in order to be able to explain the spectra of the atoms. The mathematical behavior of spin 1/2 particles is as you describe, in continuation of a hypothesis that angular momentum is a rotation about an axis.

Answer 4

I will talk about the quantum field theoretic aspect of spin since its intrinsic nature is best described in this theory.

In quantum field theory, there are two ways to find that the spin is conserved, the first way is the reason (not historically) we say that the spin is analogous to angular momentum, and the second is the reason we say that the spin is an intrinsic property:

1. You have to study the conserved quantities of a field (what we call "Noether currents") under certain transformations. To translation in time is associated the Energy. To translation in space is associated the Momentum. And to rotations in space is associated the total angular momentum. Note that it is the total angular momentum because when we do the calculations, we have the classical angular momentum plus another term. This other term is what we call the "spin" of the field, and particles (that are excitations of this field) inherit this spin. Here we see that what we call spin is really analogous to an angular momentum since it is an additionary term to the classical angular momentum.

2. You have to study the representation theory of the $\text{Spin}(3)$ group, more precisely its algebra $\mathfrak{spin}(3)$. I will not enter into the details because it is complicated but what said nwolijin in the comments is also true since $\mathfrak{spin}(3)= \mathfrak{su}(2)\simeq \mathfrak{so}(3)$. Concretely, and this will be easy to understand, a field is described by a mathematical object: to scalar fields, scalars at any point in space-time; to vector fields, vectors at any points in space-time; and to spinor fields, spinors at any point in space-time. The component dimension of the field tells us its spin: scalar fields have one component so the spin is $0$; vector fields have four components but only three of them are relevant to describe because of the equations of motion, so the spin is $1$. For spinor fields have only two components and by calculations, we find that its associated spin is $1/2$. Note that here the spin is really intrinsic in the sense that it is encoded in the very structure of the field.

What is the relation between these two approaches? Well in the computation of the square of the value of the first, we find a direct application of the second. In this sense, we had correctly guessed that the additional term of the angular momentum is the spin.