It's because the generally accepted value of Planck's constant is only half the correct value.

The quantity h-bar in the Schroedinger equation is actually h-bar/2. That's because it represents the spin of the electron, not of the photon, as implicitly assumed by de Broglie and carried forward by Schroedinger, and everyone else.

This fact has been overlooked because it was not recognized that the photon frequency is related to its energy not as E = h nu, but as E = h nu /(1 + v_l/c), where c is the limiting velocity of relativity theory, and v_l is the velocity of the photon, which is only very slightly less than c.

Although I derived that the energy of a composite photon is half the accepted value several months ago, and posted it about here previously, I didn't realize the implications until a few days ago, when I realized that the modulating factor on the Coulomb-like magnetic force between zitter particles satisfies the Schroedinger equation for a free particle in one spatial dimension. Except, the hbar must be replaced by hbar/2. This makes the energy difference between atomic energy levels half as large as previously thought, consistent with the photon energy being half the accepted value. However, simply replacing the hbar in the SE by hbar/2 leads obviously also to an angular momentum spectrum in multiples of hbar/2 rather than hbar. This cannot be, and the photon energy being hbar/2 can only be correct if, in addition to hbar being twice the currently-accepted value, hbar remains the quantum unit of angular momentum. So, the form of the SE has to change from its current form.

The modulation factor that I derived is equivalent to the Schroedinger wavefunction in one dimension, but not in three dimensions. There is a preferred dimension defined by the null four-displacement between the zitter particles. It is essentially the relativistic Doppler effect along the radial direction that gives rise to the modulating factor that can be equated with the de Broglie wave if hbar is replaced by hbar/2. (De Broglie was reasoning by analogy with the photon without knowing of the intrinsic angular momentum difference of the electron. Had he known of the difference, he would have expected the hbar/2.) In the cross-radial direction, though, the modulation goes over to a de Broglie wavelength (rather than half it for radial motion), as I showed in my FOOP paper. So it seems plausible that the Schroedinger-like equation based on the magnetic force modulation could lead to an energy spectrum in terms of multiples of hbar/2 but an angular momentum spectrum that agrees with observation, but that is work to do at the moment.

Sorry there are some typos I'll fix in the next revision. Current is v4: https://arxiv.org/abs/1609.04446

## Thursday, January 12, 2017

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