The Radiating Atom 2: Those Damn Quantum Jumps

If we are going to have to put up with those damn quantum jumps, I am sorry I ever had anything to do with quantum theory.

Schrödinger formulated the Schrödinger equation as the foundation of quantum mechanics in 1926, but his equation was then hijacked by Bohr, Born and Heisenberg, who gave it a meaning as statistics of discrete energy quanta, which Schrödinger could not accept and forced him out of business.

Schrödinger returned to the  in 1952 in his article Are There Quantum Jumps? seeking to resurrect quantum mechanics as wave mechanics resonances without any need of particles and discrete energy quanta or light quanta (photons). Schrödinger's view was present in the previous post considering interference resonance in superposition (linear combination with (real say) coefficients $c_1$ and $c_2$)

• $\psi (x,t) = c_1\psi_1(x,t)+c_2\psi_2(x,t)$

of two eigen-states $\psi_1(x,t)=\exp(i\nu_1t)\phi_1(x)$ and $\psi_2(x,t)=\exp(i\nu_2t)\phi_2(x)$ satisfying Schrödinger's equation

• $ih\frac{\partial\psi_j}{\partial t} + H\psi_j = 0$  for $j=1,2$,

where $H\phi_1=E_1\phi_1$  and $H\phi_2=E_2\phi_2$ with $E_1=h\nu_1$ and $E_2=h\nu_2$ and $H$ is the Hamiltonian operator acting with respect to a space coordinate $x$, thus with $\phi_1$ and $\phi_2$ eigen-functions of the Hamiltonian with eigen-values $E_1$ and $E_2$ and corresponding frequencies $\nu_1$ and $\nu_2$ (with $\nu_2 > \nu_1$).

Introducing

• $\rho (x,t) = \vert\psi (x,t)\vert^2 =  \psi (x,t)\overline{\psi (x,t)}$,

as a measure of electronic charge distribution, direct computation shows that  

• $\rho (x,t) = c_1^2+c_2^2 + 2c_1c_2\cos((\nu_2 -\nu_1)t)$.

We see that if either $c_1=0$ or $c_2=0$, then the electronic charge distribution $\rho$ is constant in time and thus does not generate any electromagnetic radiation. An atom in a simple eigen-state such as the ground state does not radiate.

On the other hand, in real superposition with if $c_1c_2 > 0$, the electronic charge varies in time with frequency $\nu_2-\nu_1$, and thus generates electromagnetic radiation according to the Abraham-Lorentz law or Larmor formula stating that radiation power is proportional to the square of charge acceleration.

This means that an electron in true superposition of two states of different eigenstates of different frequencies, must radiate and thus needs external forcing to persist. This is what happens in emission/absorption spectrography with a hot/cold gas emitting/absorbing light of specific frequencies.

This phenomena of interference in superposition is the (sincere and true Schrödinger) rational of the Einstein-Planck's relation

• $h\nu = E$      

with $E=h\nu_2 - h\nu_2$ by Bohr-Heisenberg-Born instead viewed as a difference in "energy" between two states, and $h\nu$ a so-called "quantum of energy" supposedly being emitted/absorbed when an electron "jumps" between two eigen-states.

Schrödinger's main point is that there is no need to introduce any concept of "energy quanta" and electron "jump" to give the relation $h\nu = E = h\nu_2 -h\nu_1$ a meaning, because its (sincere and true) meaning is that the frequency $\nu$ emitted from superposition is simply equal to the difference $\nu_2 -\nu_1$, that is a beat frequency. This is highly remarkable and gives strong support to Schrödinger's view.

But without energy quanta the quantum mechanics of Bohr-Heisenberg-Born has no meaning and that is why Schrödinger left the field in dismay.

It remains to continue from where Schrödinger ended in 1952 (or 1927). My idea is then to extend the analysis in Mathematical Physics of Blackbody Radiation (proving Planck's radiation law using finite precision wave mechanics without the statistics of energy quanta used by Planck in his proof)  to atom physics following the (Vedanta) spirit of Schrödinger.