In the recent sequence of posts on The Radiating Atom 1-11, I have been led to a formulation of Schrödinger's equation as the basic equation of quantum mechanics, as a scalar second order wave equation in terms of a real-valued wave function $\psi (x,t)$ of space-time $(x,t)$ of the form (for a one-electron atom/ion to start with):
- $\ddot\psi (x,t) + H^2\psi (x,t) = 0$, for all $(x,t)$, (1)
where the dot denotes differentation with respect to time and thus $\ddot\psi =\frac{\partial^2\psi}{\partial t^2}$, and $H$ is a Hamiltonian. We have observed that (1) is closely related to the standard formulation of Schrödinger's equation in complex form (normalizing to $h=1$)
We have argued that (1) lends itself better to physical interpretation and extension to radiation than (2), and we recalled that (1) was the original starting point for Schrödinger in 1926.
We have observed that solutions of (1) satisfy conservation of
- $i\dot\psi \pm H\psi =0$ (2)
We have argued that (1) lends itself better to physical interpretation and extension to radiation than (2), and we recalled that (1) was the original starting point for Schrödinger in 1926.
We have observed that solutions of (1) satisfy conservation of
- total charge = $\frac{1}{2}\int (\psi^2+(H^{-1}\dot\psi )^2dx$
- total atomic energy = $\frac{1}{2}\int (\psi H\psi+\dot\psi H^{-1}\dot\psi )^2dx$
- total oscillator energy = $\frac{1}{2}\int (H\psi )^2+\dot\psi^2)dx$,
as results of multiplication of (1) by $H^{-2}\dot\psi$, $H^{-1}\dot\psi$ and $\dot\psi$, respectively, and integrating in space.
We have seen that (1) naturally extends to radiation and forcing in the form:
- $\ddot\psi + H^2\psi -\gamma\dddot\psi = f$,
with $f=f(x,t)$ scalar forcing and and $\gamma\ge 0$ a small radiation coefficient, and we have observed the following basic energy balance in the case of near resonant forcing:
- total outgoing radiation = $\int \gamma\ddot\psi^2dxdt\approx\int f^2dxdt$ = total incoming radiation.
We have argued that (1) can be interpreted as a force balance with the wave function $\psi$ as a form of "scalar virtual displacement" connecting to classical mechanics with the Hamiltonian involving "internal elastic forces" connected to the presence of the Laplacian and to elastic spring forces from kernel potential, and $\gamma\dddot\psi$ connecting to the Abraham-Lorentz radiation recoil force. We have then noted that the wave function $\psi$ as "scalar virtual displacement" is given a physical realization as 3d local charge displacement.
The wave function of $\psi$ of (1) can thus be given a deterministic physical meaning, as an alternative to the standard interpretation of the wave function of (2) as probabilistic particle position.
Physical conservation of total charge in (1) will then replace unphysical conservation of total probability in (2).
We thus have compared with (2), which is viewed to be an ad hoc model without physical interpretation; if (1) has a physical meaning, it does not follow that (2) as a "square-root of (1)" must have a physical meaning.
We expect that (1) to extend in a natural way to the case of several atoms as a system of one-electron equations expressing force balance of a collection of wave functions depending on $(x,t)$, which is computable. We compare with the standard extension of (2) into a complex equation for a wave function depending on $3N$ spatial coordinates for $N$ electrons, which leads to an uncomputable model.
In short, there is evidence that (1) may offer a better foundation of quantum mechanics than the standard (2), in accordance with the original thoughts of Schrödinger, which unfortunately became muddled by the later declared success of (2) in the Copenhagen Interpretation by Bohr and Heisenberg.
In short: (1) appears to be deterministic, physical and computable, while (2) appears to be probabilistic, unphysical and uncomputable. Further study will show if the expectations for (1) can be met.
The wave function of $\psi$ of (1) can thus be given a deterministic physical meaning, as an alternative to the standard interpretation of the wave function of (2) as probabilistic particle position.
Physical conservation of total charge in (1) will then replace unphysical conservation of total probability in (2).
We thus have compared with (2), which is viewed to be an ad hoc model without physical interpretation; if (1) has a physical meaning, it does not follow that (2) as a "square-root of (1)" must have a physical meaning.
We expect that (1) to extend in a natural way to the case of several atoms as a system of one-electron equations expressing force balance of a collection of wave functions depending on $(x,t)$, which is computable. We compare with the standard extension of (2) into a complex equation for a wave function depending on $3N$ spatial coordinates for $N$ electrons, which leads to an uncomputable model.
In short, there is evidence that (1) may offer a better foundation of quantum mechanics than the standard (2), in accordance with the original thoughts of Schrödinger, which unfortunately became muddled by the later declared success of (2) in the Copenhagen Interpretation by Bohr and Heisenberg.
In short: (1) appears to be deterministic, physical and computable, while (2) appears to be probabilistic, unphysical and uncomputable. Further study will show if the expectations for (1) can be met.
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