# Direct measurement of $\mathbf S^2$

For concreteness, consider two spin-1/2 particles with spin vectors $$\mathbf S_i=(S_i^x,S_i^y, S_i^z)$$. The total Hilbertspace is four-dimensional, and as a basis we can take the (simultaneous) eigenvectors of the operators $$\mathbf S^2=(\mathbf S_1 + \mathbf S_2)^2$$ and $$\mathbf S^z=\mathbf S_1^z+\mathbf S_2^z$$. The quantum numbers associated with these basis states are the total spin $$s$$ and the magnetic quantum number $$m$$.

The operator $$\mathbf S^2$$ is an observable, and therefore it should be possible to actually perform the measurement that projects onto its eigenvectors. How can this be done?

Note that the conjugate observable $$\mathbf S_z$$ can (famously) be directly measured by doing a Stern-Gerlach experiment. If the systems composed of the two spin-1/2 particles come flying in in a random state, we will see three different measurement outcomes occurring (corresponding to the three possible values of the magnetic quantum number). Thus we can conclude that the total spin of the systems must be 1, as was noted an answer here. This is, however, not what I mean.

Note also that a Stern-Gerlach does not discriminate the triplet state with $$m=0$$ and the singlet state with $$m=0$$. Both states have the same value for $$m$$, but a different value for $$s$$, and $$s$$ is what I would like to measure directly.

The technique for measuring an observable depends greatly on the specific system in question. I will offer a way to measure $S^2$ for a particular example system - namely, the Hydrogen atom.

Consider what we normally think of as the ground state of the Hydrogen atom: $|n, l, m_l \rangle = |1, 0, 0\rangle$. Of course, the electron also has a spin: $s_e = 1/2$, and so does the single proton that makes up the nucleus: $s_p = 1/2$. The hyperfine interaction between the nuclear spin $I = s_p = 1/2$ and the electronic spin $s_e = 1/2$ yields a different eigenbasis for the total Hamiltonian that is in terms of the total spin $F = s_e + I = s_e + s_p$ (considering only the $l=0$ orbital).

By addition of angular momentum rules, $F$ can take on a value of 0 or 1, and these two states (singlet or triplet, respectively) are split in energy by the hyperfine splitting (corresponding to 21 cm radiation). So the problem now of measuring $F^2$ is reduced to identifying which of two energy levels the atom sits in. Of course, to make a pure projective measurement of the $F^2$ operator is nontrivial.

Here's the problem: the $F=0$ singlet has just one level: $|F = 0, m_F = 0\rangle$. On the other hand, at a different energy is the $F=1$ triplet, which has three states: $|F=1, m_F=-1\rangle, |F=1, m_F = 0\rangle, |F=1, m_F = 1\rangle$. The Hydrogen atom can therefore be in a superposition of all of these states: $$|\psi\rangle = a|0, 0\rangle + b|1, -1\rangle + c|1, 0\rangle + d|1, 1\rangle$$

If we measure $F^2$ and find $F = 0$ (which happens with probability $|a|^2$), then we project the state $|\psi\rangle \to |0, 0\rangle$.

Alternatively, if we measure $F^2$ and find $F=1$ (probability of this happening being $1 - |a|^2$), then we project into the state $$|\psi\rangle \to \frac{1}{|b|^2 + |c|^2 + |d|^2} (b|1, -1\rangle + c|1, 0\rangle + d|1, 1\rangle$$

So in the context of the Hydrogen atom, here is one way to make such a projective measurement: tune a laser to the frequency corresponding to the energy difference between the $|0, 0\rangle$ state (in $|n, l, m_l\rangle = |1, 0, 0\rangle$) and some higher energy level (such as a level with $n = 2$). Apply the laser for a short time to the Hydrogen atom and detect very carefully if the Hydrogen atom spontaneously emits any photon.

If it does emit a photon, then reset the atom to the ground state $|n,l,m_l\rangle = |1, 0, 0\rangle$ and $|F, m_F\rangle = |0, 0\rangle$.

If it does not emit a photon, then you have now projected the state of the atom into just the portion of states with $F = 1$.

Combining both possible outcomes, the final result is now equivalent to an ideal projective measurement of $F^2$ on the state $|\psi\rangle$.

Note: this is not really a feasible experiment, since single spontaneously emitted photons are very difficult to detect - but it gets across an idea for how these types of measurements can actually be made in practice.