# What kind of potentials can be used in Schrödinger's equation?

I have a couple of questions about what kind of potentials can be used in Schrödinger's equation:

1. How about the potential from a magnetic field? Isn't Dirac's equation more appropriate in that case, because it will take spin into account? If I choose to ignore spin and use Schrödinger's with a magnetic potential, do I still get useful results?

2. How about gravitational potential? What happens if I use e.g., $mgh$ in Schrödinger's equation? Do I still get useful results for e.g, neutrons? Or do people normally don't do that, because gravitation is negligible compared to electric fields?

In the Schrödinger equation you can introduce, in principle, whatever form of potential you like. All the question is whether it allows a physical solution.

About a particle in the magnetic field you can very well use the Schrödinger equation in which you introduce the interaction term $mB$, where $m$ is the magnetic dipole of the particle and $B$ the magnetic field.

If you ignore the spin, you ignore the magnetic dipole, s.t. what interacts with the magnetic field?

You can use $mgh$ in the Schrödinger equation if you describe free falling of particle in absence of other fields. There are also experiments of this type (see Esslinger in the arXiv quant-ph). As to movement in constant fields, the Schrödinger eq. becomes an Airy equation.

To include magnetic field into non-relativistic Schrödinger's equation, you can consider Pauli's phenomenological equation, which is a non-relativistic approximation of Dirac equation. It includes both spin and vector potential. If you drop spin-dependent term from it, you'll get an equation for charged spinless particle in magnetic field. For a spinless particle you'll then only change kinetic energy term compared to usual Schrödinger's equation, so that

$$\hat T=\frac1{2m}(\hat{\vec p}-\vec A)^2,$$

where $\vec A$ is electromagnetic vector potential.

Gravitational potential is usually added just as $mgh$ term in usual potential energy operator, i.e. for a particle in electric field with gravitational field you'll get potential energy $$U(\vec r)=q\phi_e(\vec r)+m\phi_g(\vec r).$$