Physical Quantum Mechanics 6: Interpretation of Ground State

Ground state wave function of Hydrogen (surface plot of 2d section) as minimizer of potential energy under Laplacian regularization.

The ground state wave function $\psi (x)$ of a Hydrogen atom is the minimizer of the (normalized) energy cost functional

• $E(\psi ) =\int \frac{1}{2}\vert\nabla\psi\vert^2dx -\int\frac{\psi^2(x)}{\vert x\vert}dx$,

under the normalization

• $\int\psi^2dx =1$.                                  (1)

The ground state wave function thus emerges as the minimizer of potential energy  

• $PE(\psi ) = -\int\frac{\psi^2(x)}{\vert x\vert}dx$

under Laplacian regularization expressed by the cost functional

• $RE(\psi )=\int\frac{1}{2}\vert\nabla\psi\vert^2dx$.

From the form of the potential energy $PE(\psi )$ we understand that the physical meaning of $\psi^2(x)$ must be charge distribution with (1) setting the total charge. This is a key element of the new formulation of Schrödinger's equation as a real-valued second order equation , which we are exploring in this sequence of posts.

Minimization of energy distributes the electronic charge around the proton kernel with grading monitored by the Laplacian regularization. There is nothing left to chance in this process. All Hydrogen ground states look the same, as well as excited states emerging as stationary points of $E(\psi )$. 

We compare with the standard formulation as a complex-valued first order equation, in which $\vert\psi\vert^2$ is not interpreted as charge distribution but as a probability distribution of particle position.   

The key point is that charge distribution has a direct real physical meaning in connection to a potential in a classical continuum mechanical sense, while a probability distribution of particle position has no direct real physical meaning, and thus (probably) is meaningless.