onsdag 14 januari 2015

The Radiating Atom 11: Connection to String Theory

Michio Kaku: In string theory, all particles are vibrations on a tiny rubber band; physics is the harmonies on the string; chemistry is the melodies we play on vibrating strings; the universe is a symphony of strings, and the 'Mind of God' is cosmic music resonating in 11-dimensional hyperspace.

We have been led to a alternative formulation of Schrödinger's equation as a second order wave equation in terms of a real-valued wave function $\psi =\psi (x,t)$ depending on a space coordinate $x$ and time $t$ (here for a one electron atom or ion):
  • $\frac{\partial^2\psi}{\partial t^2} + H^2\psi =0$,          (1)
where $H = -\frac{1}{2}\Delta + V(x)$ is a Hamiltonian with $\Delta$ the Laplacian and $V(x)$ a potential (with $V(x)=-\frac{1}{\vert x\vert}$ in the basic case of the Hydrogen atom and normalizing to atomic units).  

In the model case of one space dimension and $V(x) = 0$,  Schrödinger's equation (1) takes the form
  • $\frac{\partial^2\psi}{\partial t^2} + \frac{1}{4}\frac{\partial^4\psi}{\partial x^4} =0$,  
which can be interpreted as a model of a vibrating thin elastic beam. We compare with the basic model of a vibrating elastic string:
  • $\frac{\partial^2\psi}{\partial t^2} - \frac{\partial^2\psi}{\partial x^2} =0$,  
This is also the basic model of string theory (cf. (3.31) here or here) as modern fundamental physics supposedly describing a subatomic world on Planck scales of $10^{-35}$ m.

We thus find a second order in time wave equation to be a basic model of physics on all scales, from macroscopic, over atomic to extremely subatomic scales. 

We know that a macroscopic wave equation expresses a force balance, a balance of 
  • inertial forces proportional to $\frac{\partial^2\psi}{\partial t^2}$ with $\psi (x,t)$ interpreted as a displacement at position $x$ at time $t$,
  • elastic forces proportional to $\frac{\partial^2\psi}{\partial x^2}$ or $\frac{\partial^4\psi}{\partial x^4}$. 
It is natural to interpret also an atomic and subatomic wave equation as a force balance. This opens to interpret Schrödinger's equation in the form (1) as a force balance. In the previous post we saw that this naturally opens to extension to atomic radiative absorption and emission under forcing.  We compare with the standard complex first oder form of Schrödinger's equation $i\frac{\partial\psi}{\partial t}+H\psi =0$,  for which a physical interpretation is missing.

We know that a wave equation can be viewed to express in mathematical terms stationarity of an action integral as an integral in space and time of a Lagrangian $L(\psi )$, which for an elastic string takes the form
  • $L(\psi )=\frac{1}{2}(\frac{\partial^2\psi}{\partial t^2})^2-\frac{1}{2}(\frac{\partial^2\psi}{\partial x^2})^2$.
Force balance can thus in mathematical terms be viewed to express stationarity of an action integral, but unfortunately physicists have become so impressed by this mathematical equivalence as to elevate stationarity of action to be the basic principle of physics, and then so before force balance. But stationarity of an action integral is not physics, only mathematics, because there is no physical process computing action integrals and finding stationarity, while force balance is the essence of physics, as expressed by Newton's law in terms of intertial force.  

Unfortunately, confusion of physics with mathematics has led physicists to search for Lagrangians irrespective of possible lack of physical meaning, rather than seeking wave equations expressing force balance. The formulation of Schrödinger's equation as the second order wave equation (1) is a step in the other direction towards physical meaning, understanding that mathematics is not always physics.

We are then led to interpret the wave function $\psi (x,t)$ in (1) as a displacement in space at  position $x$ and time $t$, and thus $\frac{\partial\psi}{\partial t}=\dot\psi$ as a displacement velocity. From the force balance (1) then follows conservation of the following physical entities
  1. total charge = $\frac{1}{2}\int (\psi^2+(H^{-1}\dot\psi )^2dx$
  2. total atomic energy = $\frac{1}{2}\int (\psi H\psi+\dot\psi H^{-1}\dot\psi )^2dx$ 
  3. total oscillator energy = $\frac{1}{2}\int (H\psi )^2+\dot\psi^2)dx$,    
from multiplication of (1) by $H^{-2}\dot\psi$, $H^{-1}\dot\psi$ and $\dot\psi$, respectively, and integrating in space. Note that 3. naturally connects to radiation scaling with $\nu^4$ with $\nu$ frequency, with $f\dot\psi$ representing work by forcing $f$ acting on displacement velocity $\dot\psi$, with (1) extended to forcing and radiation as in the previous post.

We are thus led interpret charge as a form of energy, and the wave function $\psi$ as a measure of charge displacement, like the displacement of an elastic string, or rather a 3d elastic body.

A true physicist would probably say that (1) is no good since since it is not Lorentz invariant and thus not relativistically correct. But is this true? Well, the speed of light is $c = 3\times 10^{18}$ m/s and, a typical frequency may be $10^{15}$ Hz, the radius of an atom typically smaller than $3\times 10^{10}$ m and so the speed $v=\vert\dot\psi\vert$ will satisfy $\frac{v}{c}\le 10^{-3}$ and thus relativistic effects appear to be very small, if any.

PS Note that in this setting there is no reason to interprete $\psi^2(x,t)$ as a probability of the presence at $(x,t)$ of the electron as particle, since $\psi^2(x,t)$ is not conserved, but instead it is $\psi^2 +(H^{-1}\dot\psi )^2$ as charge, which is conserved. The scientific gamble of viewing atom physics as microscopic roulette physics with inevitable major losses, as in the textbook version of quantum mechanics, can thus possibly be avoided, and much be gained. 

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