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The Fundamental Problem

October 14, 2011

Start with the simple Bohr model of a single electron orbiting a single proton. Let’s ignore the issue of classical orbit decay due to the constantly accelerated electron radiating energy away in an electromagnetic wave; assume that there are “stable”, non-radiating orbits. Let’s also ignore angular momentum and focus on a electron falling from S2 to S1, emitting a photon in the process.

The S2 orbit is at 2.119 x 10^-10 m, where the proton’s electric field is 3.207 x 10^10 V/m and the electron’s orbital frequency is 8.216 x 10^14 Hz. The electron falls to the S1 orbit at 5.297 x 10^-11 m, where the electric field is 5.131 x 10^11 V/m and the orbital frequency is 6.573 x 10^15 Hz. In the process of falling from S2 to S1, the electron emits a photon with an energy of 10.204 eV, corresponding to a frequency of 2.467 x 10^15 Hz.

So far, so good. The emitted photon’s frequency is neatly tucked in between the two orbital frequencies (about 29% of the way from S2 to S1). We get a nice physical picture of an electron falling from one non-radiating orbit to a second non-radiating one, emitting an em wave with an intermediate frequency. Classically, we would say that this intermediate frequency is some sort of mean or average value since the frequency would increase from S2 to S1, i.e. the photon would be a “chirped” em wave with an amplitude that goes from zero to some maximum then back to zero again.

Now for the fundamental problem. The emitted photon flies off into space and interacts with an isolated, stationary electron that happens to be sitting along the axis of the emitting hydrogen atom, so that the incident em wave is circularly polarized (CP). What happens? Well, QM says that all the photon energy is absorbed by the electron and then re-emitted. Let’s calculate how strong the electric field needs to be for an isolated electron to have absorbed all that energy. The CP wave forces the electron into a circular orbit at the frequency of the photon. If we calculate the electron’s orbital kinetic energy and equate that to the photon’s energy, we get an orbital radius of 1.222 x 10^-10 m and an electric field of 1.669 x 10^11 V/m. This is an impossibly high value for the electric field of an em wave and, worse, according to QM it is independent of the distance between the hydrogen atom and electron.

That is the fundamental problem with quantum mechanics and the reason why QM will never have an adequate physical model.

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