Spin and Angular Momentum

Question

What is the difference between spin and angular momentum, are they linked i.e. spin is a form of angular momentum. And is spin what I think it is (the way the electron/particle actually spins)

Answer 1

Spin and orbital angular momentum are two different things, as already pointed out in Aniket's answer, but there is a good reason why we still call spin a "spin".

This is because the Einstein-de Haas-Richardson experiment shows that electron spin is indeed of the nature of an angular momentum, although not exactly due to a "spinning electron".

In fact, when Goudsmit and Uhlenbeck proposed their theory of electron spin, they did imagine a "spinning electron", just to find out from Lorentz that a charged spinning sphere would radiate and have the wrong self-energy for an electron. They were so embarrassed about their mistake that they wanted to cancel the paper they had written, but their advisor, Ehrenfest, had already sent it out. His comment was: "You don't yet have a reputation, so you have nothing to lose". Regardless, the important part that got them the Nobel was that they recognized the existence of a new quantum degree of freedom for the electron, and the name "spin" stuck.

The Einstein-de Haas experiment relates spin and angular momentum in a very simple way:

Suspend a ferromagnetic rod by a thin string inside a coil and connect the coil to a power source. As the coil's magnetic field magnetizes the ferromagnetic rod, the rod rotates. Change the current direction, the rod rotates again. By the conservation of angular momentum this rotation must be compensated by an equal and opposite change in angular momentum within the magnetized material. Since magnetization is produced by alignment of electrons' spin, it follows that spin must be of the nature of angular momentum. (Actually at the time of the experiment Einstein and de Haas were trying to confirm Ampère's hypothesis that magnetization is due to microscopic currents and Lorentz's theory that Ampère's currents were due to electrons).

In addition, the Stern-Gerlach experiment shows that spin, like angular momentum, carries a magnetic moment. The conclusion is that the electron's spin is a quantum degree of freedom of the nature of angular momentum that carries a magnetic moment. It characterizes the electron's state independent of its position(or momentum)-dependent wave function, or as you observed, it is intrinsic. The orbital angular momentum, on the other hand, concerns the spatial wave function and is the analog of the classical angular momentum.

Answer 2

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.

Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).

In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.

As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct so far as spin obeys the same mathematical laws as quantized angular momenta do. On the other hand, spin has some peculiar properties that distinguish it from orbital angular momenta:

1. Spin quantum numbers may take half-integer values.
2. Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower.
3. The spin of a charged particle is associated with a magnetic dipole moment with a g-factor differing from 1. This could only occur classically if the internal charge of the particle were distributed differently from its mass.

You can also check these links from Wikipedia: LINK 1 LINK 2

Answer 3

There is a very interesting Am J Phys paper by Ohanian, titled "What is spin?". You can find free PDF copies on google, in case you don't have academic access.

He points out that ALL forms of angular momentum, even spin, arise from linear momentum via the relationship $\vec r \times \vec p$. In other words, even spin is orbital angular momentum.

This is contrary to what you might have heard, that spin is a magical property hiding inside the point particle that makes the electron. What Ohanian points out is that, yes, the electron is a point particle, but it has a weird quantum wave equation. The "spinniness" is an unavoidable, circulating linear momentum encoded in the spatial properties of the wave function of the electron and cannot be removed.

But no, you can't think of the electron spin as if it is a ball, spinning around. Rather the best analogy is to the circular polarization of light (i.e. photons). Spin is a fundamental wave property that has no representation in terms of rigid bodies.