Question

In the non-relativistic limit, the Schrodinger equation for the hydrogen atom can be solved using reduced mass techniques to account for the motion of both the electron and proton. I am wondering if a similar thing can be done with the Dirac equation.Wikipediasuggests so as the energy levels are written with the reduced mass, but I cannot find any discussion of this point, and most books just work with a single electron moving in a fixed Coulomb potential. I thought the Dirac equation was just for a single particle, so is it possible to add a Dirac Hamiltonian for the proton and solve the combined system?

Answer 1

The Schrödinger equation is used to describe the quantum-mechanical behaviors of subatomic particles like electrons, employing a wavefunction to represent probability amplitudes.

Step-by-step explanation:

When it comes to the motion of subatomic particles like electrons within an atom, classical physics equations are inadequate due to phenomena such as Heisenberg's uncertainty principle and the wave nature of these particles. Instead, the quantum-mechanical model is needed, utilizing wave mechanics as introduced by Erwin Schrödinger in the Schrödinger equation. This equation accounts for the wave-like behavior of electrons as probability amplitudes represented by the wavefunction (psi, y). To address the complexity of a two-particle system, such as a proton and an electron in a hydrogen atom, the approach taken can be analogous to the non-relativistic limit using the reduced mass technique in classical quantum mechanics. For the Dirac equation, which accommodates relativistic effects and spinor fields for particles like electrons and positrons, a similar approach can be considered.

Indeed, the Dirac equation usually describes a single free particle, but for systems such as the hydrogen atom, it can be extended to include the interaction with an electromagnetic field, like a Coulomb potential, which represents the effect of the proton. While solving the combined system of a Dirac Hamiltonian for both the electron and the proton may not be as straightforward, some techniques attempt to include the proton's motion. This involves using an effective potential or incorporating the reduced mass of the system, akin to the treatment within the non-relativistic Schrödinger equation.