# Spherical Harmonics and the Salpeter Eq $i\hbar\frac{\partial\Psi}{\partial t}=\left(E_0\sqrt{1+L_0^2\Delta}+\frac{kZ}{r}\right)\Psi(r,\theta,\phi,t)$

Recall the spinless Salpeter equation

$$i\hbar \frac{\partial \Psi}{\partial t} = \left(E_0\sqrt{1-L_0^2 \Delta}+\frac{kZ}{r}\right) \Psi(r, \theta, \phi, t)$$

where $$E_0 = mc^2$$, $$L_0 = \frac{\hbar}{mc}$$, $$k=\frac{1}{4\pi\epsilon_0}$$, and $$Z$$ is the proton number.

The Laplacian expands as

$$\Delta=\frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r} - \frac{1}{\hbar^2} \frac{L^2}{r^2}$$

so that choosing a wave function of the form

$$\Psi(x,t) = R(r) T(t) Y_{lm}(\theta, \phi)$$

expresses the original equation as

$$i\hbar R(r) \dot T(t) = T(t) \left(E_0 \sqrt{1+L_0^2 \left(\frac{l(l+1)}{r^2} - \frac{2}{r} \frac{\partial}{\partial r} - \frac{\partial^2}{\partial r^2}\right)} + \frac{kZ}{r}\right) R(r)$$

since we treat $$\sqrt{1+L_0^2 \Delta^2}$$ as a linear operator in the context of Fourier transforms.

Is this work correct? Does this radial differential equation already have a solution? If so, is there a name for it in literature? If anyone could recommend papers, I would be greatly indebted.