^{1}

^{2}

^{*}

In a recent publication in this journal one of us introduced the concept of “half quanta” and used it to provide a new insight into the origin and nature of the presence of dark energy in the cosmos. We investigate in the present note the possibility that, in spite of this success, the concept of half quanta should be regarded to be an illegitimate intruder in the realm of modern Quantum Physics.

In 1900 or thereabouts, Berlin theoretician Max Planck accidentally uncovered the fact that motion in nature occurs in the form of elements each carrying the same amount, the same quantity―the same quantum―of dynamical action. He assigned to this element the symbol b later changed to h.

While in England some thirteen years later (in 1913) Danish physicist Niels Bohr published his celebrated paper “On the Constitution of Atoms and Molecules” [_{0} thereby introducing―not necessarily on purpose but certainly in effect―a cacophony in the nascent theory of Quantum physics: physicists had suddenly not one but two competing elements of action to contend with, Pkanck’s h and Bohr’s M_{0} for rotational motion.

A decade or so later, Cambridge physicist Paul Adrien Dirac aggravated the situation when he substituted his own symbol, the graphic (h-bar) ħ. to designate the Bohr angular momentum “quantum element” M_{0}, whereby the symbol became known as the “Dirac constant”.

Whatever Dirac’s intimate motivation might have been when he took this initiative, it backfired on him: the “Dirac constant” soon became known and is commonly known nowadays as the Reduced Planckconstant.

In a recent publication [

The situation is quite simple. Consider a 1-D quantum harmonic oscillator oscillating with vibrational frequency ν. Instead of ν one can just as well consider the angular frequency w given by

Keeping in mind that the reduced Planck constant is related to the “ordinary” Planck constant by the equation

one obtains readily for the 1-D quantum harmonic oscillator ground-state energy (n = 0):

Thus Equation (3b) yields for the ground state (n = 0), the “half quanta” E_{0} = ½ħw. This does not happen with Equation (3a).

In [

In [

in which d =1/ν measures a time duration.

A sharp difference thus opposes the two points of view we are briefly examining in this note. The difference is expressed forcefully in [

Not so in [

The considerations presented in [

Readers of this note are cordially invited to contribute their thoughts on how to reconcile the apparently irreconcilable results we have presented and to suggest how best to address this vexing question: is the Dirac h-bar symbol ħ a legitimate and useful innovation in Quantum Physics, or should it be mercilessly discarded henceforth as a meaninglessly disturbing intruder? (

We wish to express our gratitude to Ms. Han Xu (Hellen), JMP Editorial Board Assistant, for her valuable help and kind advice in properly preparing this note for publication in this journal.

Auffray, J.-P. and El Naschie, M.S. (2016) Revisiting the Half Quanta Defining Quantum Step Mechanics. Journal of Modern Physics, 7, 1949-1952. http://dx.doi.org/10.4236/jmp.2016.714172