On the Discretization of Energy Levels

Question

We consider a system of "n" particles whose total energy E and net momentum $\vec{P}$ are fixed are fixed.There no net force on the system(assumed)

$$\Sigma \epsilon_i= E$$ $$\Sigma\vec{p_i}=\vec{P}$$ For an individual particle its momentnum and energy remain constant for the time $\tau$,the relaxation time(average time between successive collisions----a constant). That's an extra constraint for each particle [Radiational energy density at some point is assumed to be constant for some physically small time interval]

All that seem to indicate that classical theories favor the discretization of energy levels in the equilibrium state. It is important to note that for each particle to satisfy on the mass shell condition we should have: $$\epsilon_i^2-p_i^2=m_0^2c^4$$ The above equation is frame invariant.Any theoretical speculation considering energy and momentum independent of each other should correspond to what we understand by "off the mass shell" situation An orbiting electron in an atom ,so far as the classical theories are concerned , should radiate energy. But it can also receive energy form other particles.The net radiation from a block of iron at constant temperature is zero.This should favor the discretization of the electronic orbits in the atoms in the expected manner.

[For a cluster of charged particles one should take into account the electromagnetic potential energy of the system( a closed system ) due to the presence of charges and currents.This energy for the system should be considered constant for our model]

Should we use intuition(commonsense classical theories ) to interpret QM with the understanding that the results of observation should not change in view of the system interacting with the measuring instrument and that QM will be no less useful to human activity?

[Some points to examine have been placed on the following link:

http://independent.academia.edu/AnamitraPalit/Papers/1892195/Fourier_Transforms_in_QM_Gravitons_and_Otherons_

The document may be downloaded though you get a message that conversion is going on]]

Answer 1

This is nonsense--- the "quantization of energy" you are referring to in classical theories is not a quantization at all, it is equipartition. It is only true that independent degrees of freedom have an average energy which is roughly quantized in classical mechanics.

You can't make a classical equilibrium between a field and an atom, because the field always has infinitely many degrees of freedom, and the atom finitely many. So the atom goes to the classical ground state, the electrons sit on top of the nucleus, and the field jitters thermally infinitesimally (because the finite energy is all sucked to the smallest wavelength modes).

This is called the ultraviolet catastrophe, and it dooms attempt to explain quantum behavior from a classical theory.

The simplest argument against a modified classical mechanics reproducing quantum mechanics is that quantum mechanics introduces a dimensional constant $\hbar$ which is not there in the classical limit. So if you start with the classical limit, you have to explain what sets the scale $\hbar$. This was debated in the early quarter of the twentieth century, and these ideas do not work.