Reading Nature's Book of Atom Physics

The Schrödinger equation is supposed to describe all of (non-relativistic) atom physics. It should thus be possible to read the blue-print or genom of atoms and molecules from Schrödinger's equation if everything is there, right? 

Let us then take a look at Schrödinger's equation for the ground state of the Hydrogen atom with one negative electron around a positive proton, for which the solution can be identified as the function $\Psi (x)$ with $\Psi (x)^2$ representing electron charge density in 3d space with coordinate $x$ with minimal total energy 

• $E_{tot}=E_{kin} + E_{pot}$

as sum of kinetic energy 

• $E_{kin}=\frac{1}{2}\int\vert\nabla\Psi (x)\vert^2dx$ 

and Coulomb potential energy 

1. $E_{pot}=-\int\frac{\Psi (x)^2}{\vert x\vert}dx$

under the side condition of unit electronic charge 

• $\int\Psi (x)^2dx = 1$. 

We see that kinetic energy is an integral measure of the spatial variation of $\Psi (x)$,  and that potential energy is an integral measure of charge density $\Psi (x)^2$ weighted by the kernel potential $-\frac{1}{\vert x\vert}$ representing the proton placed at the origin $x=0$. The minimum represents best compromise of being close to the kernel while being spread out.

The solution $\Psi (x)$ is referred to as a wave function and as such represents a standing wave in $x$ independent of time, and has the simple analytical form (in atomic units):

• $\Psi (x)=\frac{1}{\sqrt{\pi}}\exp(-\vert x\vert )$.                    

We now take a step back and note:

1. The potential energy is classical Coulomb potential energy for a negative charge density around a positive unit point charge. 
2. The kinetic energy has the form of elastic strain energy in classical continuum mechanics. 

We can thus read the blue-print for the ground state of a Hydrogen atom as a charge density $\Psi (x)$ with minimal total energy in classical continuum mechanical terms. This is the full story displayed in full detail to us in very familiar terms from classical continuum mechanics without any mystery. 

We have now read Nature's book for the ground state of the Hydrogen atom. RealQM takes you further to atoms and molecules in the same spirit. Go to read!