story

I feel like this is a common question asked by students that have just reached the spin chapter of their undergraduate quantum mechanics text (this is me). I’ve been using Shankar (2E) which has been great, but mysterious footnotes appear to be pointing to something more fundamental under the surface. Quick internet searches have shown that what I’m looking for lies in Lie algebras (;. I haven’t posted in a while and thought that a blog post would be a good way to organize my thoughts.

In this post, I’m going to try to describe my intuitive understanding. I will then try to relate the intuition to the Lie algebra formalism. Because of this, this post will lack rigor, but that can be found elsewhere.

first attempt

rotations in \(\mathbb{R}^3\)

Firstly, let’s deal with the familiar vector space \(\mathbb{R}^3\). Rotations in \(\mathbb{R}^3\) can be parameterized by three continious variables and in general can be represented by a composition of three \(3\times 3\) matrices.

For simplicity, let’s deal with the typical rotation matrices. For example, a matrix representing a rotation by angle \(\theta\) about the \(x\)-axis will be denoted \(R_x(\theta)\).

What makes these matrices rotation matrices? Let’s look at the commutation relationship of these matrices.

Carrying out the matrix multiplication

\[ R_x(\theta)R_y(\phi)-R_y(\phi)R_x(\theta) = \begin{bmatrix} 0 & -\sin\theta\sin\phi & \sin\phi-\cos\theta\sin\phi \\ sin\theta\sin\phi & 0 & \sin\theta-\cos\phi\sin\theta \\ sin\phi-\cos\theta\sin\phi & \sin\theta-\cos\phi\sin\theta & 0 \\ \end{bmatrix} \]

Let’s approximate this to the second order, assuming infinitesimal rotations \(\theta=\epsilon_x\) and \(\phi=\epsilon_y\)

\[ R_x(\epsilon_x)R_y(\epsilon_y)-R_y(\epsilon_y)R_x(\epsilon_x) = \begin{bmatrix} 0 & -\epsilon_x\epsilon_y & 0 \\ \epsilon_x\epsilon_y & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} = R_x(\epsilon_z) - I \]

Now, let’s also approximate the matrices on the left to the first order. For example,

\[ R_x(\epsilon_x) = I + \epsilon_x \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\ \end{bmatrix} \equiv I + \epsilon_xT_x \]

where \(T_x\) is called the generator of infinitesimal rotations about the \(x\)-axis.

Any general rotation can be written as product of infinitesimal ones.

Combining the above two equations, and repeating for permutations of \(x\), \(y\) and \(z\) results in the following commutation relations.

\[ \left[T_i,T_j\right] = \sum_k \varepsilon_{ijk}T_k \]

where \(\varepsilon_{ijk}\) is the Levi-Civita symbol.

The above commutation relationship captures the essence of rotations in \(\mathbb{R}^3\).

representation as quantum operators

Let’s try to find the quantum operators for spin.

The angular momentum operators, whether orbital or spin, are generators of rotations. This statement should make some intuitive sense, and the proof can be easily found elsewhere.

Because of this, what we have derived for the generators of rotations in \(\mathbb{R}^3\) should also hold for our spin angular momentum operators.

We will take out a factor of \(2i\) by convention. This can easily be done since we can define the new generators to absorb this constant. We now have

\[ \left[\sigma_i,\sigma_j\right] = \sum_k 2i\varepsilon_{ijk}\sigma_k \]

In addition to this, we require the generators to be Hermitian. This condition (along with the factor of \(i\) in front) is necessary to ensure the unitarity of the resulting quantum rotation operator.

Let’s assume that we’ve found such a set of generators (a few more conditions are needed by convention to fully fix these generators). Sprinkling in some dimension with carefully placed \(\hbar\)’s turn these dimensionless generators into the spin operators we’re seeking.

\[ S_k = \frac{\hbar}{2} \sigma_k \]

Pauli matrices

There is one problem remaining that is preventing us from concretely expressing these generators. What are the dimensions of our generator matrices? To answer this, we will need to study a real world particle, and introduce some experimental facts. Let’s stick with the electron. The Stern-Gerlach experiment indicates that an electron has two observable spin states. A condition which is known as spin-\(1/2\) (this is not important for this discussion, but should be clear if you’ve read up on the topic).

Because of this, we will use the following representation to describe the wavefunction of an electron

\[ |\Psi\rangle = \begin{bmatrix} |\Psi_+\rangle \\ |\Psi_-\rangle \\ \end{bmatrix} \]

where \(|\Psi_+\rangle\) and \(|\Psi_-\rangle\) represent the component of the wavefunction with definite spin of \(s_z=+\hbar/2\) and \(s_z=-\hbar/2\) respectively (\(z\) is chosen out of convention).

Based on this representation, it is clear that

\[ S_z = \frac{\hbar}{2} \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} = \frac{\hbar}{2}\sigma_z \]

Now all that’s left is to find \(S_x\) and \(S_y\) that satisfy the commutation relation. The Pauli matrices \(\sigma_x\), \(\sigma_y\) and \(\sigma_z\) (up to a factor of \(\hbar/2\)) are just such a set of matrices!

eigenfunctions of spin-\(1/2\)

Now what about measuring spin in an arbitrary direction? Multiply by a unit vector \(\hat{n}=n_x\hat{i}+n_y\hat{j}+n_z\hat{k}\).

\[ S_\hat{n} = n_xS_x + n_yS_y + n_zS_z \]

in the usual spherical coordinates, this results in

\[ S_\hat{n} = \begin{bmatrix} \cos\theta & \sin\theta e^{-i\phi} \\ \sin\theta e^{i\phi} & -\cos\theta \\ \end{bmatrix} \]

the eigenfunctions of which are \( |\hat{n},+\rangle = \begin{bmatrix} \cos(\theta/2) e^{-i\phi/2} \\ \sin(\theta/2) e^{i\phi/2} \\ \end{bmatrix} \) and \( |\hat{n},-\rangle = \begin{bmatrix} -\sin(\theta/2) e^{-i\phi/2} \\ \cos(\theta/2) e^{i\phi/2} \\ \end{bmatrix} \)

Now it can easily be seen that it takes a rotation of \(2\pi\) introduces a minus sign to the eigenkets and it takes a rotation of \(4\pi\) in order to return the kets to their original position.

higher spin

The experimental fact that the spin angular momentum of electrons takes one of two values resulted in the above representation of the spin operators as \(2\times 2\) matrices.

The same commutation rules apply for deriving spin operators for higher spin particles. The only difference is a different matrix representation is needed.

As an example, for spin-\(1\) particles,

\[ |\Psi\rangle = \begin{bmatrix} |\Psi_+\rangle \\ |\Psi_0\rangle \\ |\Psi_-\rangle \\ \end{bmatrix} \]

The spin operators for spin-\(1\) particles are \( \frac{2}{\hbar} S_x = \sqrt{2} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} \) , \( \frac{2}{\hbar} S_y = \sqrt{2} \begin{bmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \\ \end{bmatrix} \) and \( \frac{2}{\hbar} S_z = 2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix} \) it can be verified that these matrices satisfy the commutation relation.

Notice that there are not enough constraints in the commutation relations to uniquely specify these spin operators. Any set of matrices that satisfy these commutation relationships would produce the same physics – this is simply a common representation.

Using the same logic, we can derive an expression for the spinor eigenfunctions. I will not because it is too much effort.

Lie algebra formalism

Pronounced “lee” (disappointing, I know).

I will not try to derive the results above using this formalism. This section is meant to informally relate the above ideas to Lie algebras and groups.

Lie group

The notion of a Lie group is actually quite intuitive. Loosely, it’s a group whose elements are parameterized by continious and differentiable variable(s).

The set of rotation matrices in \(\mathbb{R}^3\) (should be simple to verify they satisfy the group axioms) form the Lie group known as \(O(3)\). The set of proper rotations being the subgroup \(SO(3)\). One possible parameterization being the usual \(\theta_x\), \(\theta_y\) and \(\theta_z\), for example.

These groups are often used to describe symmetries, for example \(SO(3)\) can be seen as the symmetry group that preserves the Euclidean metric.

Similarly, the set of \(2\times2\) unitary matrices with determinant \(1\) is one realization of \(SU(2)\). In fact, this is the form that \(SU(2)\) takes in the basis introduced for describing spin-\(1/2\) particles. In this basis, \(SU(2)\) is generated by the Pauli matrices. Note that this is just one realization of the group \(SU(2)\). It takes a different form when working with higher spin particles as shown above.

Lie algebra

A Lie algebras can be defined as the vector space formed by the infinitesimal generators of a Lie group. In other words, a Lie algebra is the tangent space of a Lie group near the identity. Because of the relation with a Lie group, they come with a bilinear operator (the commutator) which makes the tangent vector space a Lie algebra.

The Lie algebras corresponding to \(SO(3)\) and \(SU(2)\) are \(\mathfrak{so}(3)\) and \(\mathfrak{su}(2)\) respectively. It turns out that the Lie algebras \(\mathfrak{so}(3)\) and \(\mathfrak{su}(2)\) are isomorphic.

Just as we can define a Lie algebra from a Lie group, Lie’s third theorem states that we can go the other way – from Lie algebra to Lie group (with a few mathematical caveats).

Since the Lie algebras are isomorphic, the associated Lie groups \(SO(3)\) and \(SU(2)\) are related up to a covering. We have seen that \(SU(2)\) double covers \(SO(3)\) – that is, there is a 2-to-1 surjective mapping between the two groups.

summary

To summarize, first, we derived the commutation relations for the familiar infinitesimal generators of rotations in \(\mathbb{R}^3\). By virtue of the fact that angular momentum is the generator of rotations, we argued that they must have the same commutation relationships. The spin operators could then be derived, and the spin eigenfunctions found. This showed explicitly the 2-to-1 mapping between the wavefunction spinor and the angular momentum vector.

From the perspective of Lie algebras, equating the commutation relations of the infinitesimal generators of rotations implied an isomorphic underlying Lie algebra. This implies that the associated Lie groups are isomorphic up to a covering. It turns out that \(SU(2)\) double covers \(SO(3)\) resulting in spinors which have a 2-to-1 mapping to angular momentum vectors.