12.10 Pauli spin matrices

This subsection returns to the simple two-rung spin ladder (doublet) of an electron, or any other spin particle for that matter, and tries to tease out some more information about the spin. While the analysis so far has made statements about the angular momentum in the arbitrarily chosen -direction, you often also need information about the spin in the corresponding and directions. This subsection will find it.

But before getting at it, a matter of notations. It is customary to indicate angular momentum that is due to spin by a capital . Similarly, the azimuthal quantum number of spin is indicated by . This subsection will follow this convention.

Now, suppose you know that the particle is in the
spin-up

state with angular
momentum in a chosen direction; in other words that it is in the
, or , state. You want the effect of the
and operators on this state. In the absence of a
physical model for the motion that gives rise to the spin, this may
seem like a hard question indeed. But again the faithful ladder
operators and clamber up and down to your rescue!

Assuming that the normalization factor of the state is
chosen in terms of the one of the state consistent with the
ladder relations (12.9) and (12.10), you have:

By adding or subtracting the two equations, you find the effects of and on the spin-up state:

It works the same way for the spin-down state :

You now know the effect of the and angular momentum operators on the -direction spin states. Chalk one up for the ladder operators.

Next, assume that you have some spin state that is an arbitrary
combination of spin-up and spin-down:

Then, according to the expressions above, application of the spin operator will turn it into:

while the operator turns it into

And of course, since and are the eigenstates of ,

If you put the coefficients in the formula above, except for the
common factor , in little 2 2 tables,
you get the so-called Pauli spin matrices

:

You can now go further and find the eigenstates of the and operators in terms of the
eigenstates and of the operator. You
can use the techniques of linear algebra, or you can guess. For
example, if you guess 1,

so 1 is an eigenstate of with eigenvalue , call it a ,

spin-right, state. To normalize the state, you still need to divide by :

Similarly, you can guess the other eigenstates, and come up with:

Note that the square magnitudes of the coefficients of the states are all one half, giving a 50/50 chance of finding the -momentum up or down. Since the choice of the axis system is arbitrary, this can be generalized to mean that if the spin in a given direction has an definite value, then there will be a 50/50 chance of the spin in any orthogonal direction turning out to be or .

You might wonder about the choice of normalization factors in the spin states (12.16). For example, why not leave out the common factor in the , (negative spin, or spin-left), state? The reason is to ensure that the -direction ladder operator and the -direction one , as obtained by cyclic permutation of the ones for , produce real, positive multiplication factors. This allows relations valid in the -direction (like the expressions for triplet and singlet states) to also apply in the and directions. In addition, with this choice, if you do a simple change in the labeling of the axes, from to or , the form of the Pauli spin matrices remains unchanged. The and states of positive -, respectively -momentum were chosen a different way: if you rotate the axis system 90 around the or axis, these are the spin-up states along the new -axis, the -axis or -axis in the system you are looking at now, {D.69}.