Revised Theory of Matter Waves
The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt.
- Bertrand Russell
According to Louis de Broglie, when a micro-particle (or object in our macro world) moves, it is accompanied by probability waves. The probability waves decide the destination of particle. In case of macro-objects, the role of probability waves is insignificant, but in case of micro-particles, you cannot ignore the role of probability waves. The wavelength of probability waves is given by the equation:
l = h/p (1)
where l = wavelength of probability waves, h = Planck’s constant, and p = momentum of particle. de Broglie also assumed that following equation holds true for a moving particle:
m’c2 = hn (2)
where m’ = relativistic mass of particle, c = velocity of light, h = Planck’s constant, n = frequency of probability waves. Equation (2) also provides relation between classical energy and quantum energy of a particle. LHS of equation (2) represents the classical energy of a particle and RHS of equation (2) represents the quantum energy of a particle. Equation (1) is verified for all particles. Equation (2) is verified for photons only.
2. An Enigmatic Equality
An enigmatic equality [Ref. 1] is observed in case of an electron in the ground state of hydrogen atom, which can be stated as follows:
where m = rest mass of electron, h = Planck’s constant, v = velocity of electron in the ground state of hydrogen atom, n = orbital frequency of electron in the ground state of hydrogen atom. LHS of equation (3) represents the kinetic energy of electron, and RHS of equation (3), however, represents an enigmatic term. Equation (3) can be proved easily but cannot be interepreted, using existing theories.
But why should we try to interpret this equality? Because, history of science tells us so! In science, whenever such equality exists, there is always some physical principle lying behind it rather than that equality being a mere coincidence. For example, since the times of Newton it was believed that equality between inertial mass and gravitational mass of a matter is a mere coincidence, but Einstein interpreted that equality using his famous "principle of equivalence." Notice one more example from other discipline. The salinity of our blood is almost same as that of seawater. This is, however, not a coincidence. Millions of years ago ancestors of human beings were sea-animals, and their body fluids were derived from the seawater; and that’s why the said equality. Revised theory of matter waves offers satisfactory explanation to the enigmatic equality stated in equation (3).
3. Revised Theory Offers Solution
According to revised theory [Ref 2], the classical energy of an elementary particle is related to its quantum energy by the following expression:
m’c2 – mc2 = s.h.n (4)
where, m’ = relativistic mass of an elementary particle, m = rest mass of an elementary particle, c = velocity of light, s = spin number of an elementary particle, h = Planck’s constant, n = phase frequency of probability waves. Combining equation (4) with the standard wave equation v = n.l, we get the following expression for l (the wavelength of probability waves):
l = s.h.v / (m’c2 – mc2) (5)
where v is the phase velocity of probability waves. In revised theory, the phase velocity of probability waves is same as the group velocity of probability waves, which in turn is equal to the particle velocity; therefore, hereafter we identify the “v” with the particle velocity. In case of photons, putting s = 1, m = 0, and v = c, equation (5) yields the familiar relation, as follows:
l = 1.h.c / (m’c2 – 0) = h / (m’c) = h / p
where p = momentum of photon; and equation (4) also yields the familiar relation, as follows:
m’c2 – 0 = 1.h.n, therefore, m’c2 = hn
In case of nonrelativistic electrons, using the substitutions: s = 1/2 and m’c2 – mc2 = (1/2).m.v2, equation (5) yields the familiar relation, as follows:
l = (1/2).h.v / (1/2).m.v2 = h / mv = h / p
where, p = momentum of nonrelativistic electrons; and equation (4), however, yields the relation which is not familiar but certainly a welcome, as follows:
(1/2).m.v2 = (1/2).h.n (6)
In equation (6), n is the phase frequency of the probability waves associated with an electron. For first orbit of hydrogen atom, circumference of orbit = wavelength of probability waves, therefore, orbital frequency of electron = phase frequency of probability waves. And this completes the physical interpretation of otherwise enigmatic equality stated in equation (3). Equations (4) and (5) are stated for elementary particles, but also hold good for composite particles. However, in case of composite particles the "s" stands for the common spin number of constituent particles (and not the total spin number of the composite particle). As all matter in the universe is composed of spin (1/2) quarks and spin (1/2) leptons, the behavior of composite particles is essentially same as electron. See the references given below for details.
4. Bell’s Cautious Reaction
As expected, revised theory instantly came under the charge of blasphemy, and despite the sound and consistent theoretical framework, scientists strongly resented it with religious zeal. Theoretical Physicist John Bell (Bell’s inequality fame) wrote to author in a personal communication [Ref. 3]:
.... it takes a great courage to start all over again at this stage .... Does your theory like that of Dirac give the correct hydrogen spectrum?
The trouble is that predictions of revised theory hardly differ from the predictions of established theories, and hence, it gives correct hydrogen spectrum, among many other things. At ultra-relativistic velocities wavelengths according to existing and revised theories differ widely. What is needed is that some experimentalist should take the readings without any prejudice; and that is very difficult. Because, theory guides the experimental results, or more correctly, theory dictates the experimental results. As it is well known that phenomenon of wave particle duality was experimentally observed by some scientists much before the discovery of wave particle duality by Louis de Broglie. But there was no theory to guide those results and hence they were ignored. Results of Davisson and Germer were not ignored because they came after Louis de Broglie proposed the theory of wave particle duality. Now Louis de Brogile’s theory is established so firmly that no experimentalist has courage to defy it.
1. Chavan, S., "Uncertain Life of Schrödinger’s Cat," Bull. of IAPT, 14 (10), 318-326 (1997).
2. Chavan, S., "The Theory of Matter Waves Under Review," Indian Journal of Theoretical Physics, 38 (1), 51-58 (1990).
3. Bell. J., Personal communication with the author (March 1, 1990).
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