How to Make Schrödinger's Equation Physically Meaningful + Computable

The derivation of Schrödinger's equation as the basic mathematical model of quantum mechanics is hidden in mystery: The idea is somehow to start considering a classical Hamiltonian $H(q,p)$ as the total energy equal to the sum of kinetic and potential energy:

• $H(q,p)=\frac{p^2}{2m} + V(q)$,

where $q(t)$ is position and $p=m\dot q= m\frac{dq}{dt}$ momentum of a moving particle of mass $m$, and make the formal ad hoc substitution with $\bar h =\frac{h}{2\pi}$ and $h$ Planck's constant:

• $p = -i\bar h\nabla$ with formally $\frac{p^2}{2m} = - \frac{\bar h^2}{2m}\nabla^2 = - \frac{\bar h^2} {2m}\Delta$, 

to get Schrödinger's equation in time dependent form 

• $ i\bar h\frac{\partial\psi}{\partial t}=H\psi $,

with now $H$ a differential operator acting on a wave function $\psi (x,t)$ with $x$ a space coordinate and $t$ time, given by 

• $H\psi \equiv -\frac{\bar h^2}{2m}\Delta \psi + V\psi$,

where now $V(x)$ acts as a given potential function. As a time independent eigenvalue problem Schrödinger's equation then takes the form:

•  $-\frac{\bar h^2}{2m}\Delta \psi + V\psi = E\psi$,

with $E$ an eigenvalue, as a stationary value for the total energy

• $K(\psi ) + W(\psi )\equiv\frac{\bar h^2}{2m}\int\vert\nabla\psi\vert^2\, dx +\int V\psi^2\, dx$, 

as the sum of kinetic energy $K(\psi )$ and potential energy $W(\psi )$, under the normalization $\int\psi^2\, dx = 1$.  The ground state then corresponds to minimal total energy,

We see that the total energy $K(\psi ) + W(\psi)$ can be seen as smoothed version of $H(q,p)$ with

• $V(q)$ replaced by $\int V\psi^2\, dx$,
• $\frac{p^2}{2m}=\frac{m\dot q^2}{2}$ replaced by  $\frac{\bar h^2}{2m}\int\vert\nabla\psi\vert^2\, dx$,

and Schrödinger's equation as expressing stationarity of the total energy as an analog the classical equations of motion expressing stationarity of the Hamiltonian $H(p,q)$ under variations of the path $q(t)$. 

We conclude that Schrödinger's equation for a one electron system can be seen as a smoothed version of the equation of motion for a classical particle acted upon by a potential force, with Planck's constant serving as a smoothing parameter. 

Similarly it is natural to consider smoothed versions of classical many-particle systems as quantum mechanical models resembling Hartree variants of Schrödinger's equation for many-electrons systems, that is quantum mechanics as smoothed particle mechanics, thereby (maybe) reducing some of the mystery of Schrödinger's equation and opening to computable quantum mechanical models.

We see Schrödinger's equation arising from a Hamiltonian as total energy kinetic energy + potential energy, rather that from a Lagrangian as kinetic energy - potential energy. The reason is a confusing terminology with $K(\psi )$ named kinetic energy even though it does not involve time differentiation, while it more naturally should occur in a Lagrangian as a form of the potential energy like elastic energy in classical mechanics.