Applying to Spin 1/2 Systems

In the previous sections, we introduced the main concepts of the state vector formalism in quantum mechanics. This included state vectors, Hilbert spaces, observables, eigenstates, Hermitian operators, commutators, the uncertainty principle, tensor products, and more. If one has made it this far, one deserves commendation. These are some of the most abstract and counterintuitive concepts in physics, and the mathematics can be challenging. Now, we will apply these concepts to a specific system: spin-1/2 particles.

Before reading this, one should have read the Stern Gerlach Experiment section. The SG experiment is very nice as an introduction to quantum mechanics; it is so obviously different from the classical prediction, and it also provides motivation for a specifically complex vector space.

Lastly, this will only be the first part of the discussion on spin-1/2 systems. In fact, this is pretty much just a brief introduction to the analysis of such systems. After discussing more topics relating to quantum dynamics, we will fully flesh out our theory of angular momentum.

Table of Contents

• Table of Contents
• The Basics
  • Matrix Representations
• Observables and Probabilities
• Transformation of Spin States

The Basics

We have seen that the spin of the electron on a silver atom is quantized. This was shown through the fact that the electron could only be deflected in two directions when passing through a magnetic field.

This calls, then, for a two-dimensional Hilbert space to describe the state of the electron. The eigenbasis of this space are the spin-up and spin-down states, denoted by and respectively, or and for short.

The spin operators are the operators that measure the spin of the electron. As eigenvectors, the spin states will simply scale by a factor when acted upon by the spin operators. In our case, the factor happens to be :

Recall that the completeness relation states:

We can apply this to the spin states:

Furthermore, recall that we can write an operator as a sum of projection operators:

We can similarly apply this to the spin operator :

Matrix Representations

The spin states and are often represented as column vectors:

This comes simply from the fact that they act as the basis vectors of the Hilbert space.

The spin operator can be represented as a matrix acting on these vectors. We can find its components by using Equation :

Observables and Probabilities

We can now discuss the probabilities of the different spin states of the electron. Recall that, from the Born rule, the probability of measuring a state in an eigenstate is given by:

For the spin states, we can write the probability of measuring the electron in the spin-up state as:

We now analyze the different sequential SG experiments as we did in the previous section.

1. z-z

   The electron is first passed through a -SG apparatus, then through another -SG apparatus.

   Suppose after the first SG apparatus, the electron has collapsed to the state . The probability of measuring the electron in the spin-up state after the second SG apparatus is:

   Conversely, the probability of measuring the electron in the spin-down state is:

   In other words, the electron will always be measured in the spin-up state in the second -SG apparatus. This aligns with what was observed in the SG experiment.

2. z-x

   The electron is first passed through a -SG apparatus, then through an -SG apparatus.

   The key is that and do not commute, and as such, do not share a common eigenbasis. This means that after obtaining a definite state from the first -SG apparatus, the electron would be in a superposition of the -SG eigenstates.

3. z-x-z

   This was the most interesting case; even after collapsing to a definite state after the first -SG apparatus, the electron would still be in a superposition of the -SG eigenstates after the -SG apparatus.

   We now know that after passing through -SG, the electron will collapse to , which is a superposition of because and do not commute. In other words:

   This is why the -SG apparatus will see both spin-up and spin-down states. Since the probabilities are equal, and since the Born rule requires that , we have:

   which, if you remember, was similar to what we postulated in the SG experiment section from the analogy with the polarixation of light. is a phase factor that does not affect the magnitude of the coefficients. Since and are orthogonal, must be:

   The choice of coefficient for is arbitrary by convention.

For the case, we can first write the spin states in terms of the states:

The key is that performing sequential x-y and y-x SG experiments will yield the following:

We already have expressions for and in terms of . We can plug these in:

Distributing the inner product, we get:

Rearranging gives:

This equation is satisfied when . What this means is that for and , the relative phase difference is , and at least one of the coefficients is complex.

For example, we can use the convention to set or . Then, , and we have:

The operators are then:

We note that this is very similar to the Jones vectors that we discussed in the section on the Stern-Gerlach experiment:

Jones Vector	Spin State

Additionally, if one works it out, it turns out that the spin operators follow the following commutation relations:

where is the Levi-Civita symbol, and there is an implied sum over . is if is an even permutation of , if it is an odd permutation, and otherwise. , , and if any of the indices are repeated.

Transformation of Spin States

Let us now explore how we can transform between different spin states.

Just like with the Jones vectors, rotations between the spin states are members of the group. This is to make sure that the magnitudes of the coefficients are preserved () and there is one unique transformation for each rotation ( - special; with all the transformation that correspond to a rotation, choose the one that has a determinant of ). To see why we want the determinant to be , we need to consider the Born rule.

The probability of measuring a state in an eigenstate is given by . If was transformed by a phase shift to , then the probability of measuring in would be . In other words, the phase shift does not affect the probabilities. This is analogous to the transformation of the Jones vectors. Global phase shifts do not change the polarization of light; they only change the overall phase of the wave.

The set of Jones vectors can be represented as a sphere called the Poincaré sphere. Similarly, the set of spin states can be represented as a sphere called the Bloch sphere.

Notice that in the Bloch sphere, and are apart, yet in physical space, they are orthogonal. In other words, there is an angle-doubling effect when mapping the Bloch sphere to physical space. This makes the spin state ket a spinor.

I claimed that both the Jones vectors and spin states are spinors, but we have not yet defined what a spinor is. What we have seen so far is that a spinor does not care about its phase and that it transforms using a specific set of matrices. This leads us to a preliminary definition:

A spinor is a column vector that (1) has phase ambiguity and (2) transforms using matrices.

I use the word "preliminary" because this definition is not complete. It is analogous to the definition of a vector as a quantity with magnitude and direction - it is "somewhat" correct, but it is not the full story.