# Equation describing particle with spin higher than 1/2

Relativistic scalar particle might be described by Klein-Gordon equation, spin 1/2 particle
is described by dirac equation. I wonder what type of equation describes particles with
spin higher than 1/2. Can we describe composite particles (e.g. nuclei of atoms) using
equation for elementary particle with appropriate spin (e.g. particle consisting of two
spin 1/2 particles may have spin 0, does it mean that we can describe it by means of
Klein-Gordon equation)?

Solutions for both the Klein Gordon atom and the Dirac atom can be found in Schiff "Quantum Mechanics" (second edition), pages 322 and 337.
The binding energy of a spin zero particle in the hydrogen atom (Klein Gordon equation) is to order α4 (excluding reduced mass and QED corrections)

$$E =\frac{m_ec^2 \alpha^2} {2n^2}[1+\frac{\alpha^2} {n^2}(\frac{n}{\ell+\frac{1}{2}} - \frac{3}{4} )]$$

The binding energy of a spin 1/2 particle in the hydrogen atom (Dirac equation) is to order α4 (excluding reduced mass and QED corrections)

$$E =\frac{m_ec^2 \alpha^2} {2n^2}[1+\frac{\alpha^2} {n^2}(\frac{n}{j+\frac{1}{2}} - \frac{3}{4} )]$$

where j = l ± 1/2, with 0 ≤ l ≤ n-1

The accuracy of the Dirac equation has been tested many times. The Klein Gordon equation has been tested to a few parts per million by measuring x-rays of atomic transitions of negative pions in pionic atoms. The most precise pion mass measurements are in fact based on measurements of atomic transitions in pionic atoms.

The above equations imply that the solution for a spin-1 particle might be using the Klein Gordon solution with j = l or l ±1.

Bob S

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I see that not elementary particle of spin 0 is might be successfully described by
Klein-Gordon equation (if I remember correctly pion has spin 0).

I'm still curious what in case of particles with higher spin (elementary or not elementary). Such particles have inner degrees of freedom their wavefunction is
a vector at every point. I wonder what type of equation these vector wavefunction
has to obey in case of spin higher then 1/2. Let's consider for example particle
$$\rho^-$$ in an external electrostatic field (described classically) created
by positevly charged massive nucleus (atom consisting of $$\rho^-$$ instead
of $$e^-$$; I don't know if these example might be realized in nature).
What equation describes the $$\rho^-$$ (it has spin 1)?

martinbn
These are probably the two most famous equations, but there are more (infinitely many) relativistic wave equations. Some have names for example Proca equation (spin 1), Rarita-Schwinger equations (half-integer spin), Massive Tensor Fields equations (spin 2), Bargmann-Wigner, Gelfan-Yaglom ...

Ok, I see. I thought that generalization of Dirac and Klein-Gordon equation for higher
spin is more straightforward... How people found these equations?

martinbn
They are all related to induced representations of the Poincare group.

You can look at chapter 21 of "Theory of group representations and applications"- A. O. Barut, R. Raczka.