## PHY456H1F: Quantum Mechanics II. Lecture 13 (Taught by Prof J.E. Sipe). Spin and spinors (cont.)

Posted by peeterjoot on October 24, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Multiple wavefunction spaces.

Reading: See section 26.5 in the text [1].

We identified

with improper basis kets

Now introduce many function spaces

with improper (unnormalizable) basis kets

for an abstract ket

We will try taking this Hilbert space

Where is the Hilbert space of “scalar” QM, “o” orbital and translational motion, associated with kets and is the Hilbert space associated with the components . This latter space we will label the “spin” or “internal physics” (class suggestion: or perhaps intrinsic). This is “unconnected” with translational motion.

We build up the basis kets for by direct products

Now, for a rotated ket we seek a general angular momentum operator such that

where

where acts over kets in , “orbital angular momentum”, and is the “spin angular momentum”, acting on kets in .

Strictly speaking this would be written as direct products involving the respective identities

We require

Since and “act over separate Hilbert spaces”. Since these come from legacy operators

We also know that

so

as expected. We could, in principle, have more complicated operators, where this would not be true. This is a proposal of sorts. Given such a definition of operators, let’s see where we can go with it.

For matrix elements of we have

What are the matrix elements of ? From the commutation relationships we know

We see that our matrix element is tightly constrained by our choice of commutator relationships. We have such matrix elements, and it turns out that it is possible to choose (or find) matrix elements that satisfy these constraints?

The matrix elements that satisfy these constraints are found by imposing the commutation relations

and with

(this is just a definition). We find

and seeking eigenkets

Find solutions for , where . ie. possible vectors for a given .

We start with the algebra (mathematically the Lie algebra), and one can compute the Hilbert spaces that are consistent with these algebraic constraints.

We assume that for any type of given particle is fixed, where this has to do with the nature of the particle.

latex 1/2$ particle} \\ s = 1 &\qquad \text{A spin particle} \\ s = \frac{3}{2} &\qquad \text{A spin particle}\end{aligned} $

is fixed once we decide that we are talking about a specific type of particle.

A non-relativistic particle in this framework has two nondynamical quantities. One is the mass and we now introduce a new invariant, the spin of the particle.

This has been introduced as a kind of strategy. It is something that we are going to try, and it turns out that it does. This agrees well with experiment.

In 1939 Wigner asked, “what constraints do I get if I constrain the constraints of quantum mechanics with special relativity.” It turns out that in the non-relativistic limit, we get just this.

There’s a subtlety here, because we get into some logical trouble with the photon with a rest mass of zero ( is certainly allowed as a value of our invariant above). We can’t stop or slow down a photon, so orbital angular momentum is only a conceptual idea. Really, the orbital angular momentum and the spin angular momentum cannot be separated out for a photon, so talking of a spin particle really means spin as in , and not spin as in .

## Spin particles

Reading: See section 26.6 in the text [1].

Let’s start talking about the simplest case. This includes electrons, all leptons (integer spin particles like photons and the weakly interacting W and Z bosons), and quarks.

states

Note there is a convention

For shorthand

One can easily work out from the commutation relationships that

We’ll start with adding into the mix on Wednesday.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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