Physical Quantum Mechanics 3: Back to Continuum Mechanics

We are exploring a second order alternative formulation of Schrödinger's wave equation as the basic model of quantum mechanics, which for a Hydrogen atom takes the form

• $\ddot\psi +H^2\psi =0$,        (1) 

for all $(x,t)$, where $\psi =\psi (x,t)$ is a real-valued function of a 3d space coordinate $x$ and time coordinate $t$, $\dot\psi =\frac{\partial\psi}{\partial t}$, and $H=-\frac{h^2}{2m}\Delta +V$ is the standard Hamiltonian with $\Delta$ the Laplacian differential operator with respect to $x$, $h$ Planck's constant, $m$ the mass of the electron, and $V=V(x)=-\frac{1}{\vert x\vert}$ is the (normalized) kernel potential.

We observe that (1) upon multiplication by $m$ takes the form

• $m\ddot\psi +(-\frac{h^2}{2}\Delta +mV)^2\psi =0$    (2) 

with the electron mass $m$ appearing as if $m\dot\psi$ represents momentum and (2) expresses force balance according to Newton's 2nd law.  We observe that the scaling of the Laplacian conforms with an interpretation as space regularization. 

We further observe that in the limit with $h=0$, (2) decouples into a set of ordinary differential equations indexed by $x$:

• $\ddot\psi (x,t)+\frac{1}{\vert x\vert^2}\psi (x,t) =0$,   

which reflects Newton's 2nd law with a gravitational force scaling with $\frac{1}{\vert x\vert^2}$.

In the case $V=0$, (2) reduces to (with normalization)

• $m\ddot\psi +\Delta^2\psi =0$,

which can be viewed as a model of a vibrating elastic solid. 

We thus find that it is possible to interprete the atomic model (2) in classical continuum mechanical terms. Of particular interest is then the conserved quantities of (2), and of course the physical meaning of the wave function, which is not the standard one with $\vert\psi\vert^2$ a particle position probability, to which we return in the next post.