Rotation for spin > 1/2

Question

For single spin 1/2 particle, we can use three Pauli matrices as generators to do rotation on Bloch sphere to get any state we want in the Hilbert space.

However, for spin greater than 1/2, I try to use three SU(2) generators to do rotation, and I can not find a rotation which rotates $S_z$ to $S_z-1$ states (for example, in spin 1 case, I can't do a rotation $S_z=1$ to $S_z=0$; in spin 3/2 case, I can't do a rotation from $S_z=3/2$ to $S_z=1/2$). Therefore I am wondering what generators should we add to get a unitary operation that transforms $S_z$ to $S_z-1$. Also, what is the topology for these higher spin Hilbert space (It does not seem to be a sphere since can't do rotation to get every state in Hilbert space)?

Answer 1

In general, for any spin, you can always construct the raising and lowering operators $S_\pm = S_x \pm i S_y$ which take a state with azimuthal angular momentum quantum number $m$ and change it to $m \pm 1$ (or annihilate the state if $|m|=s$, where the eigenvalue of $S^2$ is $s(s+1)$). So, to answer the first part of your question, just use $S_\pm$.

The Hilbert spaces become increasingly complicated for increasing spin, and in my experience it is not really useful to try to work with the analogue of the Bloch sphere for spins larger than 1/2.

To see why, let's recall why we get a sphere for spin 1/2. The state in general is a complex superposition of two eigenstates, $|\psi \rangle = \alpha |\uparrow \rangle + \beta | \downarrow \rangle$, where $\alpha$ and $\beta$ are complex numbers. Naively you would need a four dimensional space to describe the state (two for the real and imaginary parts of $\alpha$, and two for $\beta$). However, there is a normalization condition $|\alpha|^2 + |\beta|^2=1$ which knocks us down to three dimensions, and then furthermore the state is only defined up to an overall phase. Combining those two constraints, we are left with only two parameters describing the state, which we can identify with coordinates on the Bloch sphere.

Now consider spin 1, the next simplest case. You have three eigenstates and therefore 3 complex parameters. There are the same two conditions (normalization and modding out by an overall phase). So, we are left with four free parameters that we will identify with coordinates on a four dimensional space. It is of course possible to work out mathematically exactly what this space is, but I think this is mostly of mathematical interest and won't give you a nice intuitive picture like a Bloch sphere. (Unless you are significantly better than me at visualizing four dimensional spaces with non-trivial topology :))

Answer 2

I am not sure I understand your problem. The topology is continuously connected rotations around the identity. Let's fix to standard physics notation.

You want, e.g., a rotation $U|11\rangle=|10\rangle$ $$ \begin{pmatrix} 0 \\ 1\\ 0 \end{pmatrix} = U \begin{pmatrix} 1 \\ 0\\ 0 \end{pmatrix}. $$ Look at this as a Cartesian transformation (instead of the spherical one it is). From elementary geometry you know how to rotate $\hat x$ to $\hat y$: By a π/2 rotation around the $\hat z$ axis, orthogonal, hence also unitary, $$ U= R_z(\pi/2) = \begin{pmatrix} 0 & - 1 & 0 \\[3pt] 1 & 0 & 0\\[3pt] 0 & 0 & 1\\ \end{pmatrix} . $$

Using the Rodrigues rotation formula, you can always produce the exponent in the Lie algebra that achieves this. Here, it is $$ R=\exp(\pi K/2 ) =1\!\!1 + K - K^2 ,\\ K\equiv \begin{pmatrix} 0 & - 1 & 0 \\[3pt] 1 & 0 & 0\\[3pt] 0 & 0 & 0\\ \end{pmatrix}. $$

You should be able to do the $s=3/2$ case, etc... (I don't have a systematic solution.)

For systematic connections between unitary rotations and hermitian Lie algebra (spin) generators, see, e.g. , this ref.