I recently came across a claim that the Schrödinger equation only describes spin-1/2 particles.
Is this true?
I realize that the question may be ill-posed as some would consider the general Schrödinger equation to be
$$ i\hbar \frac{\partial}{\partial t} |\psi\rangle = H|\psi\rangle $$
which is tautological if an arbitrary Hamiltonian is allowed. I assume what was meant by "Schrödinger equation" is essentially a quantized version of the Newtonian relation $T = \frac{p^2}{2m}$.
I know the Dirac equation can be reduced to such a form if we take the nonrelativistic limit and assume there is no magnetic field, and besides the Schrödinger equation does predict the gross line spectrum of hydrogen. So the claim that the Schrödinger equation (approximately) describes spin-1/2 particles would seem to be justified.
Now, I often see it claimed that the Schrödinger equation is essentially the non-relativistic limit of the Klein–Gordon equation, which Schrödinger published instead of the Klein–Gordon equation because the latter predicted the hydrogen atom fine structure incorrectly. So is it not the case that the Schrödinger equation is also valid as a non-relativistic approximation for the behaviour of spin-0 particles?
For spin 1, obviously the Schrödinger equation cannot describe the photon, which is always relativistic. However it seems incorrect to me that the Schrödinger equation should not be able to describe, say, a deuteron.
Best Answer
As per Rob's suggestion, I decided to make this an answer.
(Addendum: I've been meditating on this very topic for some time, and have been directed to some interesting literature referenced on Streater's webpage. As per Rococo's comment, I've updated my answer, but kept the old version for posterity.)
The Answer
To talk about "spin-1/2 particles", we need the spin-statistics theorem.
The spin-statistics theorem doesn't hold for non-relativistic QM. Or more precisely, the "naive" spin-statistics theorem doesn't hold, and if we try to contrive one...it's nothing like what we'd expect.
In brief, the reason is: spin-statistics theorem depends critically on microlocality (i.e., the commutator of spacelike separated conjugate operators vanishes identically). This property holds relativistically, but not non-relativistically. (The other proofs similarly do not hold, since Lorentz invariance fails.)
But there is a trick around this. Just take a relativistic spin-1/2 particle, then take the nonrelativistic limit.
But does the Schrodinger Equation Hold?
Now, the question originally asked is: does the Schrodinger equation hold for spin-1/2 particles? To talk about spin-1/2 particles, we really are working with representations of the Lorentz group. A nonrelativistic spin-1/2 particle is obtained by taking the nonrelativistic limit (i.e., the $c\to\infty$ limit) of the Dirac equation, which is the Pauli equation:
$$\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi \right] |\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle$$
where $\boldsymbol{\sigma}$ are the Pauli matrices, $\mathbf{A}$ an external vector potential, $\phi$ an external electric potential, and $q$ the electric charge of the particle. But observe when we "turn off" electromagnetism (setting $\mathbf{A}=\phi=q=0$) we recover
$$\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot\mathbf{p})^2 \right] |\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle$$
Uh, I leave it as an exercise for the reader to show the left hand side of this equation is $(1/2m) \mathbf{p}^{2}\otimes\boldsymbol{1}_{2}$ where $\boldsymbol{1}_{2}$ is the 2-by-2 identity matrix.
References
For more thorough reviews on this matter, I can heartily refer the reader to:
The (Old) Answer
The "vanilla" Schrodinger's equation (from non-relativistic QM) does not describe a spin-1/2 particle.
The plain, old Schrodinger's equation describes a non-relativistic spin-0 field.
Case Studies
If we pretend the wave function is a classical field (which happens all the time during the "second quantization" procedure), then it turns out to describe a spin-0 field. See Brian Hatfield's Quantum Field Theory of Point Particles and Strings, specifically chapter 2 --- on "Second Quantization".
But wait, there's more! If we consider other non-relativistic fields and attempt quantizing, e.g. the Newton Cartan theory of gravity, we also get spin-0 boson! For this result (specific to quantizing Newtonian gravity), see:
The Reason
Well, this should not surprise us, since the Schrodinger equation is the nonrelativistic limit to the Klein-Gordon equation. And we should recall the Klein-Gordon equation describes spin-0 bosons!
The nonrelativistic limit $c\to\infty$ should not affect the spin of the particles involved. (That's why the Pauli model is the nonrelativistic limit of the Dirac equation!)