Hydrogen

Quantum mechanics got a kick start in 1926 with Schrödinger’s wave equation for the Hydrogen one electron, with the wave function \Psi as a mimimiser over H^1(\Re^3) of the total energy

  • TE(\psi )=\frac{1}{2}\int_{\Re^3}\vert\nabla \psi (x)\vert^2dx+\int_{\Re^3}W(x)\psi^2(x)dx

where W(x)=-\frac{1}{\vert x\vert}, under the normalisation

  • \int_{\Re^3}\psi^2(x)dx=1.

The condition of minimality with respect to variations of \psi takes the form of an eigenvalue problem

  • H\Psi = E\Psi \quad\mbox{in }\Re^3

for the Hamiltonian

  • H=-\frac{1}{2}\Delta+W

with E a minimal eigenvalue and

  • TE(\Psi )=\int_{\Re^3}\Psi H\Psi dx = E\int_{\Re^3}\Psi^2(x)dx=E.

The ground state \Psi can be computed analytically and is given by

  • \Psi (x)=\frac{1}{\sqrt{\pi}}\exp(-\vert x\vert ), \quad E(\Psi )=-\frac{1}{2}.

The total energy to be mimimised is the sum of potential kernel energy

  • \int_{\Re^3}W(x)\Psi^2(x)dx

and what is normally referred to as kinetic energy, bur rather has the form of elastic compression energy:

  • \frac{1}{2}\int_{\Re^3}\vert\nabla \Psi (x)\vert^2dx

The model thus involves two types of energy of a distributed charge density; potential energy and elastic energy and as such has the form of a classical model of continuum mechanics.

The whole conundrum of quantum mechanics is how to generalise the model to more than one electron. stdQM gives one answer and realQM another.