“QUANTUM
MECHANICS” AS THE
MECHANICS OF THE TIME REGION The preliminary results of a critical study of the Wave Mechanics carried
out in the light of the knowledge of the Reciprocal System of theory have
been reported earlier. One corollary is that the theorists “assumption that the wave associated with the moving particle is spatially co-extensive with the particle is wrong since the former exists in the equivalent space, not in the extension space of the conventional spatial reference system. The Uncertainty Principle stems from the theorists” practice of resorting to wave packets. It has further been shown that the probability connotation of the wave
function arises from the two facts that the wave is existent in the three-dimensional
temporal manifold, and that locations in the three-dimensional temporal
manifold are only randomly connected to locations in the three-dimensional
spatial manifold. The Calculations based on the From the above study it became abundantly clear that the critics’comments that the small-scale world is not intrinsically rational, and that the Quantum theory cannot be understood intuitively were wrongly founded. What was really missing was the knowledge of the existence and characteristics of the Time Region, the region inside the natural unit of space, where only motion in time is possible. Since our knowledge of the Reciprocal System helped straighten some of the conceptual kinks of the Wave Mechanics and has indicated that its original basis has been rightly (though unconsciously) founded, an attempt has been made to inquire into its mathematical aspects in order to see whether they are valid in the light of our understanding of the Reciprocal System. The results of this inquiry are reported in this article. *Where Do We Stand**The Wave Equation**Radiation Waves**Matter Waves**States of Negative Energy**The Step Function**Explanation of the Negative Energy States**The Potential Energy Barrier**The Potential Energy Well**Origin of the Pauli Exclusion Principle**The Spin**Indistinguishability**The Two Types of Reference Points*- Potentials in the Time Region
*Dimensional Relations across the Regions**Potentials in the Time-space Region**Potentials in the One-dimensional Zone of the Time Region**Potentials in the Three-dimensional Zone of the Time Region*
*Conclusions*- The Schödinger equations were found to be valid general rules for the exploration of the wave functions in the various situations.
- In the time-space region, speed can be vectorial (that is,
*directional*in the context of the three-dimensional spatial reference frame), whereas inverse speed (like, energy) is scalar. In the Time Region, speed is found to be scalar, whereas inverse speed is*directional*—*directional in the three-dimensional temporal reference frame.*Variables of the latter type, therefore, could take on inherently negative values and be represented by complex numbers or quaternions as the case may be. - The penetration of the wave associated with particle into the regions of negative kinetic energy resulting from potential energy barriers is found to be a genuine Time Region phenomenon.
- In a similar vein, it is found that the occurrence of a well-type potential energy function in the Time Region leads to the limiting of possible values of total energy to a discrete set.
- Such an important empirical law as Pauli exclusion principle, which has no theoretical explanation in the context of the conventional theory, could easily be understood form the knowledge of the positive and negative reference points brought to light by the Reciprocal System.
- Reasoning from the principles of the Reciprocal System the possible potential energy functions of the Time Region relevant to atomic systems are surmised. While they evince a close qualitative resemblance to the empirically found potentials, detailed further study needs to be carried out to see if they lead to the correct prediction of the properties pertaining to spectroscopy, radioactivity and the scattering experiments.
Before proceeding further it would be desirable to take a stock of the atomic situation form the point of view of the Reciprocal System. Firstly, Larson Secondly, he argues that there is no electrical force either, involved in the atomic structure. This, therefore, leaves gravitation and the space-time progression as the only two motions (forces) that operate inside the Time Region with, of course, the appropriate modifications peculiar to the Time Region introduced into them. Under these circumstances the question of a “nuclear” force does not arise at all. But it is perfectly legitimate to inquire what forces (motions) are encountered by a particle as it approaches the vicinity of an atom, and indeed, as it enters the very atom itself. Equally important is to inquire into the mechanics of the converse process of the emission of a particle by the atom. The most fundamental starting point for the mathematical treatment
in the Quantum Mechanics is the wave equation. The wave equations in
the quantum theory govern the wave functions associated with the particles,
and correspond to Newton’s laws of classical mechanics. From our
earlier study we have seen that changing from the particle picture to
the wave picture is a legitimate strategy that needs to adopted on entering
the Time Region, as it is tantamount to shifting from the conventional
three-dimensional spatial reference frame of the Since it is always possible to constitute a wave of any shape by superposing different sinusoidal waves of appropriate wavelengths and frequencies, we shall limit our discussion to these elementary sinusoidal waves. The relation between the wave number k and the wavelength l on the one hand, and that between the angular frequency w and frequency n on the other, are as follows
The wave speed u is given by
The general functional forms of sinusoidal waves are
and in complex exponential form (see Appendix I)
where the imaginary unit i is defined by i
Let us derive the governing equation for the wave propagating at constant speed, like that of radiation. First we note the relation between the momentum p of the wave and the wave number k, and the energy E and its angular frequency w,
where h is Planck’s constant h divided by 2p. From the energy-momentum relationship of the wave, p
Assuming the simplest wave form, that of a sine wave, we write the wave function in complex exponential form as
where A is an arbitrary constant. For such a function,
That is, taking the derivative with respect to x is equivalent to multiplying by ik, and taking the derivative with respect to time t is equivalent to multiplying by -iw. Thus
Substituting these in the last of Eq.(6) we obtain
which is exactly the wave equation we are seeking (see Appendix II).
At the instance of his mentor Peter Debye, Erwin Schrödinger made a detailed study of the wave hypothesis advocated in 1924 by de Broglie. Schrodinger noted that the energy-momentum relationship of a free particle (not acted by forces) of mass m
leads to the wave number-angular frequency relation
From Eqs. (2) and (12) we see that the wave speed in this case is given by
Therefore the speed of the matter waves is not constant like that of the radiation waves, but is a function of the wave number k. Eq. (12) could be rearranged as -(h Multiplying both sides by Y, we can at once see from Eqs. (8) and (9) that
which is the governing equation for the wave associated with the free particle that we are looking for. This is the Schrödinger equation for the free particle. It is the equation in the Time Region which corresponds to Newton’s first law of the time-space region. In order to include
The corresponding wave number-frequency relation, associating frequency with the total energy, is h Adopting Eqs. (8) and (9) as before, we arrive at the Schrödinger wave equation with interaction present
This corresponds in the Time Region to Newton’s second law in the time-space region. As can be seen from the foregoing derivations, nothing against the principles of the Reciprocal System has been introduced so far. Hence the Schrodinger equations can be admitted as legitimate governing principles for arriving at the possible wave functions of an hypothetical particle of mass m traversing the Time Region, with or without potential energy functions as the case may be. We may note in the passing that often considerable mathematical dexterity is required in solving these differential equations, though computer-oriented numerical methods are fast replacing closed-form solutions. Any wave corresponding to a state of definite energy E has a definite frequency w = E/h. Therefore from Eq. (7) we can write
where y (x) is a function of space
variable only. Inserting the above into Eq. (16) and dividing out the
factor e
which is referred to as the It is instructive to see what the solutions of Schrödinger equation turn out to be. Firstly, in any region of constant potential energy V, we see that the solution of Eq. (18) is a sinusoidal function,
(E-V) being the kinetic energy. In Fig. 1(a) we picture a step-function potential energy, which is
constant at V An interesting case occurs when the potential energy V in any region
is greater than the total energy E. Here the kinetic energy, E-V, becomes
The sign of the exponent is so chosen as to see that y tends to zero for large x. Fig. 2 illustrates this case: in the region x> 0 we see that E is less than the potential energy. The wave function is sinusoidal in the region of positive kinetic energy and is exponential in the region of negative kinetic energy. Both functions join smoothly at x=0 with a first order continuity. The penetration of the wave function into the region of negative kinetic energy has no classical analog and is purely a phenomenon of the Time Region. When we turn to the Reciprocal System for an explanation of the possibility
of the existence of negative energy states, what we find is as follows.
In the time-space region, that is, in the context of the three-dimensional
spatial reference frame, speed (space/time) is Moreover the possibility that even potential energy (being an inverse speed) could be “directional” in the three-dimensional time, and hence be represented by complex numbers in the Time Region, cannot be overlooked. Indeed the Quantum theorists find it necessary to adopt the complex potential V+iW in place of V in scattering theory. Here the wave number k becomes complex and is written as k+iq. b of Eq. (20) becomes b = i(k + iq) = -q + ik, and we have
We can at once see that this is the wave function of a travelling wave of whose amplitude decreases as it advances, and therefore represents a beam of particles some of which are getting absorbed. An interesting situation arises when two regions of positive kinetic
energy occur separated by a The last case of interest we wish to consider is that of a potential
w The so-called But first we must recognize a point that we have been emphasizing, In a Paper entitled In connection with a class of elementary particles, we know that
any two individual particles (say, two electrons) are absolutely alike.
In the time-space region, the fact that two particles are identical
presents no complications since they can be kept distinguished by
their respective locations. But in the quantum phenomena, because
of the [y (1,2)] This can be satisfied in two ways,
The first type of wave functions are referred to as the Now the empirical finding is that the wave functions of particles
like protons and neutrons which are known to have half-integral spin
(½ h ) are antisymmetrical, and those of particles with integral
spin (like the photons) are symmetrical. The most fundamental statement
of Pauli exclusion principle goes somewhat like this: “Any permissible
wave function for a system of spin-½ particles must be antisymmetric
with respect to interchanging of all coordinates (space and spin)
of any pair of particles.” But enunciating a principle is quite
different from explaining its origin, and the fact is that no theoretical
explanation has been found for this empirical finding. One author
writes: “For reasons that are not clearly understood, for electrons,
protons, neutrons, and all other spin-½ particles, the
From the Reciprocal System we have now the explanation. Let us recall
that in the universe of motion there are two types of reference frames—the
conventional, stationary three-dimensional spatial reference frame
(or its cosmic analog, the three-dimensional temporal reference frame)
and the moving natural reference frame. We also have two kinds of
objects, those having independent motion like the gravitating particles
and those having no independent motion of their own and hence are
stationary in the natural reference frame, like the photons and those
particles having As already remarked a unit of one-dimensional rotation carries unit spin (½ h ). The resultant spin of a two-dimensional rotation with unit spin in each dimension is 1x1 = 1 (that is, ½ h ) or is 1x(-1) = -1 (that is, -½ h ). On the other hand, the resultant spin of a birotation (like the photon) is 1+1 = 2 (that is, h ) or 1-1 = 0. Since gravitation arises out of the two-dimensional rotation, we can see that a gravitating particle carries spin-½. Thus the wave function of spin-½ particles turns out to be antisymmtric. On the other hand, the In the context of the three-dimensional reference frame, photons
appear to move Finally it might be of interest to explore the nature and type of the potential energy functions V (see Eq. (15)), in the Time Region. In view of the maiden nature of the investigation and the insufficient time available, the results reported in this section may have to be treated as tentative.
Discussing the effect of the inversion of space and time at the unit
level on the dimensions of inter-regional relations, Larson In an earlier article Looking back, we can now easily see why the quantum theorists required
At this stage of our study we have only two scalar motions (forces) to consider: the space-time progression and gravitation. In the outside region (the time-space region), the forces due to the space-time progression and gravitation are respectively given by
where all the quantities concerned are in the natural units, the K’s are positive constants and r the distance factor. Suffix G refers to gravitation, P to space-time progression and O to outside region. From the definition of potential, F = -¶ V/ ¶ r, we obtain the expressions for the corresponding potentials due to the space-time progression and gravitation, in the outside region respectively as
The potential due to the space-time progression is repulsive while that due to gravitation is attractive as can be seen.
Potential energy being inverse speed, the expressions for the potentials in the one-dimensional zone of the Time Region would be the second power expressions of the corresponding ones in the time-space region (Section 5.1). Consequently the space-time progression and gravitational potentials in this zone could be written as
with suffix 1 referring to the one-dimensional zone. We can at once
verify that gravitation is repulsive and the space-time progression
attractive in this region. In addition there could be a constant term
K
The values of K It is instructive to see what the expressions for the corresponding
Larson
where K is a function of the several atomic rotations. These expressions
can be seen to differ from Eqs. (27) above. But whether we take Eqs.
(27) or Eqs. (28), the force equilibrium equation, F The Time Region potential Eq.(26) results in a potential well and therefore the solutions of Schrödinger’s Eq. (18) yield a set of discrete energy levels for the atomic system (see Section 3.4). It remains to be verified whether these truly correspond to the values inferred from the spectroscopic data. Turning now to the potentials in the three-dimensional zone, following our earlier analysis of the dimensional situation (Section 5.1), we adopt the fourth power expressions of the corresponding outside region (that is, the time-space region) quantities from Eqs. (24)
with suffix 3 denoting the three-dimensional zone. We know that the space-time progression acts away from unit space.
In the time-space region away from unit is also away from zero (the
origin of the conventional spatial reference frame), whereas in the
Time Region (that is, in less than unit space) away from unit is r where A is the atomic weight. Expressing this in the natural units
as r
Adding a constant term K
We note that this corresponds to what the conventional Quantum theorists
would call the nuclear potential. Our study indicates that Eq. (31)
bears a remarkably close qualitative resemblance to the potentials
arrived at through the scattering experiments. An unexpected feature
of the experimental data analysis was the occurrence of a
Let us summarize the highlights. Having resolved
the riddle of the wave-particle duality in an earlier article On the whole there seems to be a References - Nehru K.V.K., “The Wave Mechanics in the
Light of the Reciprocal System,”
*R eciprocity,*Vol. XXII, No. 2, Autumn 1993, p. 8 - Larson D.B.,
*The Case Against the Nuclear Atom,*North Pacific Pub., Oregon, USA, 1963 - Nehru K.V.K., “The Law of Conservation of Direction,” Reciprocity,Vol. XVIII, No. 3, Autumn 1989, p. 3
- Nehru K.V.K., “On the Nature of Rotation and Birotation,” Reciprocity, Vol. XX, No. 1, Spring 1991, p.8
- Larson D.B.,
*Basic Properties of Matter,*International Society of Unified Science, Utah, USA, 1988, pp. 102-3 - Nehru K.V.K., “The Photon as Birotation,”
*Reciprocity,*Vol. XXV, No. 3, Winter 1996-97, pp. 11-16 - Cohen B.L.,
*Concepts of Nuclear Physics,*Tata McGraw Hill, India, 1971, p. 38 - Larson D.B.,
*Nothing But Motion,*North Pacific Publishers, Oregon, USA, 1979, pp. 141-2, 165-7 *Ibid.,*p. 155- Larson D.B.,
*Basic Properties of Matter,**op. cit.,*p. 8
Appendix I: Euler’s Relations Often calculations are facilitated by adopting exponential functions
with e e which directly follow from the series expansions of these functions. A number containing sin a = (e cos a = (e represents a birotation. Appendix II: The General Equation of a Constant Speed Wave Let a wave of arbitrary but unchanging shape be traveling in the X-direction
of the stationary reference frame X-Y at a constant speed u. This wave
appears stationary in a reference frame X x If the wave shape in the co-moving frame is given by y y = f(x - u.t) (ii) By the chain rule for derivatives we have ¶ y/¶ x = (dy/dx ¶ y/¶ t = (dy/dx Therefore the relation between the two derivatives is
Similarly for a wave traveling in the -X direction we obtain
Now a repeated application of the above procedure yields
which is the governing equation of the wave function; and it is the same for waves traveling in either direction of the X-axis.
—
Reciprocity, Volume XXIV, Number 1, Spring
1995, p. 1; Revised Feb. 1998 |