# 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.