fredag 20 februari 2015

Physical Quantum Mechanics 9: Big Lie More Convincing

In this sequence we argue that a second order real-valued form of Schrödinger's equation:
  • $\ddot\psi + H^2\psi =0$       (1) 
may be to prefer before the standard first order complex-valued form:
  • $i\dot\psi + H\psi =0$,            (2)
where $H$ is a Hamiltonian depending on a space variable $x$, the dot signifies differentiation with respect to time $t$, and $\psi =\psi (x,t)$ is a wave function. 

This is because (1) can be given a physical interpretation as a force balance, while the interpretation of (2) has baffled physicists since it was introduced by Schrödinger in 1926.  

Formally, (2) appears as the "square-root" of (1) and it is not strange that if (2) has a physical meaning then (1) as a "square-root" may lack physical meaning. 

The non-physical aspect of (2), first formulated for the Hydrogen atom with one electron and $x$ a 3d space variable, in the extension to an atom with $N>1$ electrons the space variable, is expanded to $3N$ dimensions with a 3d independent space variable for each electron. 

The standard Schrödinger wave function for an atom with $N$ electrons thus depends on $3N$ space variables, which makes direct physical interpretation impossible, and the only interpretation that physicists could come up with was in terms of a probability distribution, without physical meaning. 

This made Schrödinger very unhappy, as well as Einstein. But the newly born so promising modern physics could not be allowed to die in its infancy and so following the strong leadership by Born-Bohr-Heisenberg, the non-physical aspect of the standard Schrödinger equation was turned from catastrophe into a virtue as an expression of a deep mystical uncertain stochastic nature of atomistic physics beyond any form of human comprehension, yet discovered by clever physicists as something very new and modern and very Big.  

In this process, the non-physical aspect of (2) was helpful: If (2) already for a Hydrogen atom with one 3d space variable was deeply mystical as a "square-root" without physical interpretation, expansion to non-physical multi-d $3N$ space variables was just an expansion of the mystery and as such could only be more functional following a well-known device: The great masses (of physicists) will be more easily convinced by a Big Lie than a small one.

To the non-physical aspect of (2) could then be added non-computbality as an equation in $3N$ space dimensions asking for impossible a $googol=10^{100}$ flops already for small $N$. But it did not matter that (2) was uncomputable, since (2) anyway was unphysical and as such of no scientific interest and value, although very Big.

On the other hand, sticking to physics with (1) as a physical force balance, an atom with $N>1$ electrons may naturally be described as a system of $N$ wave functions each one depending on a 3d space variable, which can be given a direct physical meaning including extensions to radiation, and is computable as a system in 3d.

One may compare with another Big Lie, that of dangerous global warming by back radiation evidenced by a pyrgeometer from human emission of CO2, which is threating to send Western civilization back to stone-age. Physicists in charge of the basic physics of global climate including radiative heat transfer in the atmosphere, do not tell the truth to politicians and the people. One Big Lie thus appears to be compatible with another Big Lie and even demand it. The reckoning in the history of science to be written will be harsh, even if as of now nobody seems to care.

Another thing is that questioning a Big Lie may not be a small thing and may draw a big cost. But if Humpty Dumpty falls, then the Fall may be great.

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