Download particularized wave equations and their parameters

General Theory of Matter
J. Šuráň
9. PARTICULARIZED WAVE EQUATIONS AND THEIR PARAMETERS
A. The electron theory and the spin of particles
In the previous chapter, we discussed certain periodic features of the elements,
which we called their cross-over group properties. They reflect periodicities in the
proton-meson (and proton-proton) structure of the elements′ nuclei.
Other, finer periodicities of the elements are revealed in the detailed structure of
the Periodic table of the elements itself. (See its scheme in Chapter 7, A, c. Other more
refined schemes of this table, reflecting some even subtler chemical properties of the
elements, were also elaborated upon. The above scheme is, however, sufficiently
illustrative for our purpose.) These periodicities reflect cyclic chemical properties of the
elements, which were first recognized empirically (in a course outline) by the Russian
chemist Dmitri Ivanovich Mendeleyev, towards the end of the 19th century.
If the elements are arranged according to their increasing atomic numbers and
arranged so as to produce the scheme of the above Periodic table, similar chemical
properties occur in groups (which form its vertical columns and are marked by Roman
numerals), and change, more or less continuously, in periods (following in horizontal
lines marked by Arabic numerals).
The electron theory successfully explained these cyclic chemical properties as
consequences of the structure of the electron shells and subshells surrounding the
nucleus of each element. As some of its results will also be required in our further
deductions, we shall briefly review it below.
It is based on the time-independent (stationary) Schrödinger wave equation
∇2Ψ + (2me/ℏ2)(E – V)Ψ = 0,
(128)
which is a modified form of our wave equation (21) (its other modifications are KleinGordon and Dirac equations), and which may be obtained from the well-known
considerations involving the wave function Ψ of a free particle and certain constraints
involving energies. Symbols me, E, and V denote the electron mass, the total, and the
potential energies of the particle, respectively. ℏ = h/2π, where h is Planck′s constant.
An orbital (a state) of each electron in the above-mentioned electron
configurations is characterized by four quantum numbers: the principal quantum
number n = 1, 2, 3 ... (specifying its principal energy states En = –n−2(mee4/32π2ε2oℏ2),
with e being the electron′s charge, and εo, the "vacuum" dielectric constant), the orbital
quantum number l = 0, 1, 2, ..., n – 1 (characterizing the magnitude of the electron′s
orbital angular momentum L = [l(l + 1)]1/2.ℏ), the magnetic quantum number ml = 0,
±1, ±2, ..., ±l (specifying the magnitude of the vector of the electron′s orbital angular
momentum Lz = ml . ℏ projected onto the direction of the applied external magnetic field
z), and the (intrinsic) spin magnetic quantum number ms = ±1/2. For each particular
combination of the four quantum numbers there then exists the corresponding wave
86
General Theory of Matter
J. Šuráň
function (eigenfunction) representing a particular (quantized) solution of the
Schrödinger wave equation.
The first three quantum numbers were derived from solutions of the above timeindependent Schrödinger wave equation for an electron constrained in the electrostatic
field of the proton. (Such solutions are obtained most conveniently with the wave
function expressed in the form Ψ(φ,ϑ, r) = Φ(φ)θ(ϑ)R(r), where φ, ϑ, and r are spherical
coordinates, and Φ(φ), θ(ϑ), and R(r) are functions of only the respective single variable.
Upon separation of variables, solutions of θ(ϑ) are expressed by means of Legendre
polynomials, and those of R(r) by means of Laguerre polynomials.)
The last quantum number ―the spin magnetic quantum number― was first
postulated (in 1925), on empirical grounds, by two Dutch physicists, S. A. Goudsmit and
G. H. Uhlenbeck, in order to explain certain aspects connected with the fine structure of
spectral lines, and the anomalous Zeeman effect of multiple spectral lines, observed in
spectra of some elements in particular. These two physicists suggested that the electron
has an intrinsic angular momentum (spin), and with this, an associated magnetic
moment, which successfully explained the relevant phenomena.
Later (in 1928) the spin aspect of the electron was also derived mathematically by
P. A. M. Dirac from his relativistically formulated electron wave equation. The
magnitude of the electron′s spin (i.e., of its intrinsic angular momentum) is: S = [s(s +
1)]1/2.ℏ, where the quantum number s has a value of only 1/2. (The relation for S is
formally the same as that for its orbital angular momentum L.) Similarly, as the vector of
the orbital angular momentum has 2l + 1 possible orientations in a magnetic field, the
vector of the spin may have 2s + 1 orientations. As s = 1/2, the magnetic spin quantum
number ms may have only two values: ±1/2. The vector component of the electron′s
magnetic spin in the direction of the magnetic field z then is Sz = ms.ℏ = ±(1/2)ℏ.
That the orbital state of an electron requires four quantum numbers for its
characterization is also apparent from the time-dependent Schrödinger wave equation
(ℏ2/2me) ∇2Ψ – (ℏ/i)∂Ψ/∂t – V.Ψ = 0,
(129)
if we assume its periodic solution of the type Ψ(x,y,z,t) = eiνtψ(x,y,z) =
eiνtψ1(x)ψ2(y)ψ3(z). Here ψ1, ψ2, ψ3 are auxiliary wave functions, and we suppose them to
depend only on the particular space-like variable; and ν designates a frequency.
The above four quantum numbers determine certain discrete states of the
electrons in the shells and subshells of atoms. It should be remarked that in a rigorous
treatment the mass me of an electron in equation (128) is substituted for by a reduced
mass m′e to account for the influence of the proton′s mass ≈ 1836 me. As these discrete
configurations are being successively filled up with electrons around more complex
nuclei, the sequence of the elements arises concordantly with their growing atomic
number.
87
General Theory of Matter
J. Šuráň
The key rule to occupancy of the shells and subshells by an electron is the
exclusion principle formulated empirically (in 1925) by the Austrian physicist, Wolfgang
Pauli. This states that "no two electrons in an atom may exist in the same quantum
state." In other words, each electron in an atom must be characterized by a differing set
of quantum numbers n, l, ml , ms.
This principle is fully concordant with our theory which, in fact, requires the
uniqueness of solutions. Uniqueness demands distinguishable states (see also Chapter 5,
A, c, Remark 2), otherwise the respective system could not be stable. Also, PV forbids two
particles to occupy the same region of space.
In its mathematical form, the exclusion principle leads to the conclusion that the
electron configuration in the electrostatic field of an atom is expressed by an
antisymmetric wave function (represented by the product of wave functions of individual
electrons Ψ = ψ1.ψ2.ψ3...), and that it changes its sign if a pair of electrons is exchanged
in the configuration (i.e., when their coordinates are interchanged, then Ψ = –
[ψ1.ψ2.ψ3...] ).
It has also been established that this mathematical rule generally holds for
particles having half-values of the spin. Such particles are called fermions; while
particles having a zero or integral spin, and called bosons, obey a symmetric wave
function (not changing its sign when some particles are interchanged in a common field).
Bosons, consequently, do not satisfy the Pauli exclusion principle. The above
antisymmetric and symmetric wave functions obviously reflect antisymmetric and
symmetric properties of space fields which arise in the domains of these two types of
particles ―being antisymmetric with the fermion fields, and symmetric with the boson
fields.
According to the above classification, fermions include all leptons, quarks, and
baryons, whereas bosons are all scalar and vector mesons, intermediary particles of the
weak interaction W+, W−, Zo, and some other particles.
If we compare this division of particles with our theory, we observe that fermions
are particles which may form stable systems, whereas bosons comprise intermediary
particles. To the former, therefore, belong all the basic (ordinary) particles and their
colourless aggregates (baryons), while the latter are products of interactions
accompanying ordinary particles in their particular subspaces (particles of the strong
and weak interaction) and domains (mesons of various kinds).
Bosons, then, existing in conjunction with the basic particles (and being a kind of
secondary particles), cannot themselves form stable systems (to which the Pauli
exclusion principle refers). Hence, in congruence with what was stated above, they
cannot obey the exclusion principle. The existence of intermediary particles and their
boson character enables us to tap their energy.
88
General Theory of Matter
J. Šuráň
We have thus outlined the main features of the electron theory, which also simply
and elegantly explained the once somewhat mysterious progression of the elements of
the closing Group VIII of the Periodic table (inert gases) with atomic numbers Z = 2, 10,
18, 36, 54 and 86.
Noting its success ―the electron wave theory also elucidated a number of
phenomena connected with electrical, magnetic and optical properties of the elements,
as well as ionic, valence and some other molecular bonds― it might seem that this
approach could now be extended in a similar fashion to other subsystems of atoms; i.e.,
to those comprising protons and neutrons, and, eventually, to their still more basic
subunits. Having the necessary wave equations, their integration would, theoretically,
determine the structure of these subsystems, too.
There is, however, a great difference between applying the wave equation of an
electron (that is, the Schrödinger wave equation) in the electrostatic field of a proton,
and applying wave equations of other particles. The electrostatic field is relatively simply
defined by a relation V = –e2/4πεor, giving the potential energy V of charge –e separated
by a distance r from charge +e; while the other particles have their interaction fields
defined by their intermediary particles. These are also described by wave equations and
should be solved simultaneously with wave equations of the basic particles. We have
remarked earlier that (exact) simultaneous solutions of several such equations present
formidable problems, even if restricted to the 2U-space field. In substance inverse
mathematical operations are incomparably more complicated than solutions of direct
problems, and we shall not deal with this task, considering it to be beyond the intended
scope of this work.
It is also necessary to remark that the general form of our wave and interaction
equations is not suitable for specialized solutions. For such solutions these equations
have to be adequately reformulated (in an analogous manner as our wave equation (21)
was modified into the form of the Schrödinger wave equation for the electron, for
instance) to obtain meaningful physical results. For the physically reformulated
equations we shall use the term particularized wave and interaction equations.
B. Masses of particles
In a general theory the problem of the masses of particles also appears. In the
electron theory, and in other applications of quantum mechanics, their masses are
substituted for by their experimental values. In our theory these might, eventually, be
deducible from their wave equations, although it may be difficult to achieve. The basic
problem is that masses of particles are commonly expressed in units derived from their
macroscopic effects mediated by the gravitational interaction; and they should be
derived from effects exclusively involving particle interactions. (A similar problem
89
General Theory of Matter
J. Šuráň
appears with the magnitude of charges which is likewise derived from their macroscopic
effects.)
The guiding idea that, in principle, this might be possible, follows from the
theoretical equality of the masses of particles of the same kind being manifestations of
the same homogeneous space-time medium. They are characterized by the same wave
equation, and homogeneity of the space-time warrants that their physical properties be
identical. Owing to PV, a particle (wave) should encompass a maximum of mass
(-energy), being conserved within limits given by the uncertainty principle (∆E≥ℏ/∆t).
This means that the limiting value of the mass-energy of the particle is necessarily
somehow constrained.
To get a little more insight into the question, we shall make use of the mechanical
analogy of a vibrating string and an electron, which have wave equations of essentially
the same form. In solutions of the wave equation of a vibrating string (bound at both
ends and of perfect elasticity) there appears a relation
(130)
(T/µ)1/2 = v,
between parameter T denoting tension of the string element, mass µ of its unit length
(density), and the wave (phase) velocity v. Putting v = c for space-time waves, (130)
becomes
(T(spt)/µ)1/2 = c,
(131)
and
T(spt) = µc2,
(132)
where T(spt) may analogously be interpreted as the "tension of the space-time." Eq. (132)
is, in fact, the energy-mass equivalence relation of special relativity, if we consider the
work done by (constant) force T(spt) over a unit distance. Multiplying both sides of (132)
by the unit distance, we get ðE(spt) = ðm.c2, for a unit element of mass ðm, and for the
whole mass m,
(133)
E = E(spt) = ∑ðE´(spt) = ∑ðm.c2 = m.c2,
which is the famous Einstein energy-mass equivalence relation. ∑ means a sum (or an
integral) of all mass elements (corresponding to their sum over the whole length of the
string).
We may, in line with our theory, (reasonably) assume that an electron should also
have, apart from the earlier-mentioned intrinsic spin angular momentum, an intrinsic
structure. Then, in analogy with the vibrating string (based on the common general form
of their wave equations), we may likewise expect that its intrinsic mass-energy Ee should
also be the sum of some unit energies.
We may imagine that every particle, the electron included, occupying some finite domain of space,
is in reality a "ball of space waves," although in the Schrödinger wave equation this is represented only by a
single mass parameter. Because of this, the Schrödinger wave equation can thus describe only some mean
energy states of the particle (and not single electron waves of which the "electron wave ball" must be
composed) and eventually, their probability density –proportional to the product |Ψ.Ψ*| of the wave
function Ψ and the complex conjugate wave function Ψ* (and normalized accordingly, so that the
90
General Theory of Matter
J. Šuráň
respective probability be between zero and unity). This, is now a generally accepted interpretation,
suggested (in 1926) by the German physicist, Max Born, which also forms an important principle of
quantum statistics.
It appears to be a cogent principle which refers to all kinds of particles and their states defined by
their respective wave equations. A particle (as will be explained in more detail later below) may be thought
of as being a form of intrinsic oscillations of the complex vectors ξ1, ξ2 of our space. These oscillations
would be described by the wave functions p and q appearing in our wave equations (as well as the wave
function Ψ in the Schrödinger wave equation).
Expressing generally a complex vector, ξ, by means of an exponential function of e, it has the
known form
ξ = r.eiθ.
In this relation, r designates vector´s modulus and θ an angular argument. (In the vector´s complex plane,
r and θ would be its polar coordinates, and θ an angle subtended by the real coordinate axis and the vector
ξ.)
Owing to PV and the homogeneity of our space, a particle would occupy a quasi-spherical region of
space. Then (but for uncertainties due to the uncertainty principle of Heisenberg), r ~ const., and a
measure with which a particle manifests its presence at its external boundary (to an observer), would be
proportionate to (4π)r2. If Ψ = ξ, r2 = |Ψ.Ψ*|, which is the above relation indicating a probability density
(statistical weight) with which the particle may be localized within a wave field.
The resultant (but for the uncertainty due to the uncertainty principle), constant
sum, giving the mass-energy of the electron, would equal its intrinsic energy. This
implies that some (quantized) states of energy should exist in the intrinsic domain of the
particle contributing to this value. What kind of wave equation should these obey?
What may help us here is, again, an analogy with the energy density of the
vibrating string, which could have been chosen in arbitrarily small units ―quanta.
However, as is also the case with the string, these could not have been smaller than some
of the smallest quanta that are permissible by the quantum theory. Suppose that µ in
(131) and (132) are some such quanta of the energy density. In the case of the electron,
these, characterizing its intrinsic energy density, should be Planck′s quanta h.
Now, realizing that energy states of a harmonic oscillator are quantized in
multiples of h.ν, we are led to the assumption that the intrinsic wave function of an
electron corresponds to this form of oscillations.
Harmonic oscillations are also generally described by the wave equation (128). If
the potential energy V in equation (128) is represented by a relation Vx = (½) Λx x2, with
Λx being a constant, equation (128) describes harmonic oscillations of a one-dimensional
harmonic oscillator (depending on only variable x). Its corresponding energy states are:
Ex = (nx + 1/2)h.ν, with nx = 0, 1, 2, ...
(134)
As for a free (unexcited) particle, its relevant matter-energy state should
correspond to the lowest energy state of the harmonic oscillator E0, nx = 0, and
E0x = (1/2)h.ν.
(135
In compliance with PVI, we are equally justified in supposing an intrinsic energy
state of the electron corresponding to Vy = (½) Λy y2, where Λy is a constant related to
91
General Theory of Matter
J. Šuráň
variable y. The energy level E0y, corresponding to ny = 0, is given by (135), and the total
lowest intrinsic energy level of a free electron E0e then is
E0e = E0x + E0y = h.ν,
(136)
which is Planck′s renowned quantum law.
It might seem that a third energy state of the electron, Eez, should also be taken
into consideration with nz = 0. Then the total electron energy E0e would correspond to
(3/2)h.ν. But this lowest total energy state is excluded by the Pauli exclusion principle,
for this is one of the two energy states of the electron, corresponding to quantum
numbers nx = ny = 1, which are nearest to the electron′s lowest (ground) state.
It is not too surprising that we have been able to obtain Planck′s relation (136)
from the Schrödinger wave equation solved for a harmonic oscillator, because Planck′s
relation (and also the de Broglie wavelength λ = h/p, where p is the linear momentum of
the particle) was used in its derivation. In our procedure, however, we have also been
able to interpret another aspect it describes: the intrinsic structure of the particle.
In the treatment of the harmonic oscillator we have thrice applied a wave
equation of only one variable, instead of equation (128) of all the three (space-like)
variables. Such a treatment was permissible because their sum corresponds to the more
general equation (128). The three particular solutions thus represent a solution of wave
equation (128) resolved into three independent harmonic oscillations.
Consequently, there are only two possible, physically significant intrinsic
directions of a free electron which characterize its internal oscillations at the ground
state. These obviously refer to the two space-like components (e.g., x and y in subspace
A) of the complex vector ξ1 of our space. Relation (136) should similarly refer to other
particles of a 2U-space field.
The two significant directions of intrinsic oscillations of the electron also explain
the existence of the only possible direction of the particle′s spin. This (also owing to PV)
then refers to the third space-like component (z, in our example) of the complex vector
ξ2, and with reference to the complex vector ξ1 having two space-like components (x, y in
our example), can be characterized by only two antiparallel directions. As there is no
energy level associated with the spin with a free electron at its ground state, this can
manifest itself only in an externally applied electromagnetic field.
Then it is quantized proportionately to spin magnetic quantum numbers ms =
±1/2. However, in this case (in congruence with our previous considerations) it cannot
attain a zero value.
It is also interesting to note the existence of two possible directions of polarization
for (identical) photons (obeying Fermi-Dirac statistics) in photon gas emitted as blackbody radiation, and also, generally, with electromagnetic radiation. Furthermore, as is
shown in the theory of specific heats of solids, quantum states of electrons forming
"electron gas" display the same behaviour as "photon gas." This correspondence,
92
General Theory of Matter
J. Šuráň
likewise, cannot be incidental; it is, evidently, a consequence of the two prominent
intrinsic directions existing with electrons and photons.
Another remarkable correspondence with the theory of specific heats is between
the form for expression of the energy-mass relation in Einstein and Planck formulae (i.e.,
in (133) and (136)), and that for the mean energy of thermal oscillations in solids in
thermal equilibrium with its surroundings. The latter expression is
E = k.T,
(137)
where k is Boltzmann′s constant, and T is the absolute temperature. Although in the
derivation of the latter relation only a one-dimensional harmonic oscillator was taken
into consideration, the resulting expression (137), nevertheless, has the same general
form as the two-parameter expressions (133) and (136), each of which is also the product
of a constant and a varying quantity: mass m, frequency ν, and absolute temperature T
(in formulae (133), (136) and (137) respectively).
The resulting equation (137) is interpreted as describing energy states of a onedimensional harmonic oscillator with two degrees of freedom corresponding to two types
of energy ―potential and kinetic― which is always (1/2)k.T.
There, however, may be a deeper inner correlation to the black body radiation. It
is noticeable that in a system of particles (as is known from statistical mechanics),
thermal equilibrium corresponds to particles′ most probable distribution. Because the
wave and interaction equations (as derived in our theory) also relate to most probable
states, these and thermal equilibrium states would tend toward mutual convergence.
This aspect may also be reflected in the above correspondences.
Further, there is a question of the intrinsic energy and mass and of a
particularized form of wave equations, of a neutrino and antineutrino and, eventually, of
other particles. The general form of the wave equation of these particles in subspace A is
given by (23) (and (24) for the conjugate particle). With the same considerations that led
to the Schrödinger wave equation for the electron, the particularized time-independent
wave equations for these two particles are:
±∂2Ψ/∂x2 ± ∂2Ψ/∂y2 ∓ ∂2Ψ/∂z2 ∓ (2|mν|/ℏ2)(E – V)Ψ = 0,
(138)1,2
with |mν| being the absolute value of the mass of the respective particle or antiparticle.
Following the deduction applied earlier to an electron, intrinsic energy states of a
neutrino (and antineutrino) would likewise correspond to three states of a harmonic
oscillator. Regarding the upper signs of equation (138), and considering (134), E0x =
–(1/2)h.ν, and E0y = –(1/2)h.ν, for nx = 0, and ny = 0, respectively. The total energy
of the particle at its ground state then is
E0 = –h.ν,
93
(139)
General Theory of Matter
J. Šuráň
since E0z = (1/2)h.ν (which corresponds to nz = 0), is excluded by the exclusion principle.
The latter state would contribute to its total energy of –(1/2)h.ν, which is one of the
states E0x and E0y .
Because E0 is negative, the particle corresponding to the upper signs in (138) (i.e.,
equation (138)1) should be an antineutrino. Then E0 ≡ E0 ~ν , which is in accord with our
previous interpretation of the wave equation (23) (see Chapter 5, A, c). The lower signs
in equation (138) (i.e., equation (138)2) then refer to a neutrino, having total energy
E0ν = h.ν. (Frequency ν of the latter two particles is in reality different from that of the
electron, and a different symbol should be used for their characteristic frequency
―adding a suffix to ν, for example. We have, however, omitted this for simplicity.)
To the total energy E0 = ∓h.ν of an antineutrino and neutrino there corresponds
a mass of ∓m = ∓h.ν/c2. As (according to our theory) these two particles are generated in
a common subspace ―A, B, or C― along with an electron, quarks and intermediary
particles (and eventually their antiparticles), they should likewise possess a rest mass.
According to our theory, they cannot be massless particles, as is sometimes assumed,
even though their rest mass should be very small, as some of the most recent
experiments indicate.
C. Particularized wave equations in a 2U-space field
We have given particularized wave equations for an electron, a neutrino and an
antineutrino. For completeness with respect to leptons, we shall here add the
particularized time-dependent wave equations for the latter two particles, in terms of the
parameters of the Schrödinger wave equation:
(ℏ2/2|mν|)(±∂2Ψ/∂x2 ± ∂2Ψ/∂y2 ∓ ∂2Ψ/∂z2) ± (ℏ/i)∂Ψ/∂t ± V.Ψ = 0,
(140)1,2
where the upper signs refer to an antineutrino, and the lower ones to a neutrino.
Both wave equations (138) and (140) relate to subspace A. For their
transformation to the other two subspaces B and C, of the other two colours, the
symbolic matrices (14) can be used (or regard given to their form in Appendix I).
Particularized wave equations for conjugate leptons, corresponding to general wave
equations (22) and (24) of the reference domain, are supersymmetrical.
In order to obtain particularized wave equations for quarks and intermediary
particles, apart from the form of their general equations, we also have to take account of
certain additional relations. The first will be derived from a wave function of the form
A.exp[–(i/ℏ)(Et – S.p)] ≡Ψ.
(141)
Relation (141) is a representation of the wave function of a free particle with total
energy E and linear momentum vector p. A is a constant (which may be regarded as an
amplitude) and S is a space-like vector of coordinates x, y and z, with S.p being a scalar
product.
94
General Theory of Matter
J. Šuráň
As is evident, we identify equation (141) of harmonic oscillations with wave
function Ψ of our previously particularized wave equations, and we have earlier
expounded reasons (see section B above) justifying such an identification. For a review,
we shall now give their systematic summary once again here: (a)we have ascertained that
the intrinsic wave functions of an electron, and neutrino (and of their antiparticles)
generally obey an equation of harmonic oscillations of this type; (b)this wave function was
also used, in addition to the Einstein energy-mass equivalence relation and the de
Broglie wavelength λ = h/p (mentioned earlier), in the derivation of the time-dependent
Schrödinger wave equation (129); (c)homogeneity of our space-time warrants its
universal applicability; and also (d)it suits PVII.
In usual derivations of, for instance, the time-dependent Schrödinger wave equation ―instead of
the scalar product in (141)― an ordinary product of linear momentum and a space variable, say x, is used.
The derived time-dependent wave equation then has the form: (ℏ2/2m)∂2Ψ/∂x2 – (ℏ/i)∂Ψ/∂t – VΨ = 0.
In a subsequent step, the equation is generalized to an equation in all three space-like variables, obtaining
the form (129). The logical nuance of inconsistency, which we see in the latter step of the derivation, is in
no way reasoned: the generalized wave equation is considered a kind of new postulate. In our theory,
however, neither the Schrödinger wave equation, nor any other wave equation derived here (nor any, which
might eventually be derived) appear as additional postulates.
We recall that our general equation of the type (129) was derived as a sum of two component wave
equations (of (19)1 and (19)2 in subspace A of the reference domain, for example ―see Chapter 5, A, a), of
which one always contained (only) two space-like variables, and was a kind of Laplace type equation.
According to our theory, the above-mentioned generalization is fully justified.
In a slightly varied interpretation, equation (141) may also be thought of as resolved into three
component (wave) equations. These will always involve only one of the three space-like variables. Each of
these equations will thus be representative of a certain subspace (e.g., that involving z of subspace A). Their
subsequent generalization ―considering that in each subspace there is also a parallel Laplace equation
involving the other two space-like variables― will then always lead to an identical wave equation in all the
three space-like variables in the particular subspaces.
From (141) we obtain a relation:
∂Ψ/∂t = –(i/ℏ)EΨ.Ψ.
(142)
A similar relation, replacing wave function Ψ with wave function Ω, which is associated
with conjugate particles of the imaginary domain, is:
∂Ω/∂t = –(i/ℏ)EΩ.Ω.
(143)
To obtain further particularized wave equations, we also need to newly define the
constant ҝ from our general wave equations. This may be done via its calibration vis-àvis wave equation (128) and the corresponding general wave equation (21).
If we consider (128) to be a wave equation of a particle with general mass m,
obeying wave function Ψ (with correspondingly changed parameters) defined by (141),
then from (141),
∂2Ψ/∂t2 = –(E2/ℏ2).Ψ.
(144)
Comparing further forms of equations (21) and (128), and considering the last relation
(144),
95
General Theory of Matter
J. Šuráň
ҝ2.∂2Ψ/∂t2 = – ҝ2.(E2/ℏ2).Ψ = (2m/ℏ2)(E – V)Ψ, and
ҝ = ±(i/E)[2m(E – V)]1/2.
(145)
Then, with respect to general wave equations (26) and (27) for quarks, and
relations (142), (144), and (145) (considering in the latter the upper sign only), we have
∂2Ψ/∂x2 – ∂2Ψ/∂y2 ± ∂2Ψ/∂z2 ∓ (2mΨ/ℏ2)(EΨ – VΨ)Ψ – 2{∂2Ω/∂x∂y ± (1/ℏ)[2mΩ(EΩ –
VΩ)]1/2.∂Ω/∂z} = 0,
(146)1,2
which are time-independent particularized wave equations for a d quark (equation (146)1
with the upper signs) and a u quark (equation (146)2 with the lower signs) of the real
domain. mΨ and mΩ denote a quark′s mass relevant to wave functions Ψ and Ω.
Time-independent particularized wave equations for conjugate d and u quarks of
the imaginary domain are obtained analogously from their general wave equations (29)
and (30), but with regard to (143), instead of (142):
∂2Ω/∂x2 – ∂2Ω/∂y2 ± ∂2Ω/∂z2 ∓ (2mΩ/ℏ2)(EΩ – VΩ)Ω + 2{∂2Ψ/∂x∂y ± (1/ℏ)[2mΨ(EΨ –
VΨ)]1/2.∂Ψ/∂z} = 0.
(147)1,2
The upper signs in particularized wave equation (147) refer to a conjugate d quark
(equation (147)1), and the lower signs refer to a conjugate u quark (equation (147)2).
Particularized time-independent wave equations of the weak interaction follow
from general wave equations (34), (35), and (42). Applying relevant partial derivatives to
relations (142), eventually to (143), and with respect to (145), the time-independent
particularized wave equations of the weak interaction are:
∂2Ψ/∂x∂z + (1/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂y = 0,
(148)
∂2Ψ/∂y∂z – (1/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂x = 0,
(149)
and
∂2Ψ/∂x∂z – (i/ℏ)EΨ.∂Ψ/∂y = ∂2Ω/∂y∂z + (i/ℏ)EΩ.∂Ω/∂x.
(150)
They represent intermediary particles W+, W−, and Zo, respectively. The
particularized wave equations of the weak interaction in the imaginary domain,
transforming from general equations (36) and (37), have a form supersymmetrical to
equations (148) and (149), and that corresponding to (43) has the time-independent
form:
∂2Ω/∂x∂z – (i/ℏ)EΩ.∂Ω/∂y = –[∂2Ψ/∂y∂z + (i/ℏ)EΨ.∂Ψ/∂x].
(151)
Particularized wave equations of the strong interaction in the real domain
transform from general wave equations (38)1, (38)2, (39)1, and (39)2. Applying, as before,
partial derivatives to relations (142) or (143), and considering (145), the timeindependent form of the particularized wave equations of the strong interaction in the
real domain is:
∂2Ψ/∂x∂z – (1/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂y – (2/ℏ)[2mΩ(EΩ – VΩ)]1/2.∂Ω/∂x = 0,
(152)
∂2Ψ/∂y∂z + (1/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂x + 2.∂2Ω/∂x∂z = 0,
(153)
96
General Theory of Matter
J. Šuráň
∂2Ψ/∂y∂z + (1/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂x – (2/ℏ)[2mΩ(EΩ – VΩ)]1/2.∂Ω/∂y = 0,
(154)
(155)
∂2Ψ/∂x∂z – (1/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂y – 2.∂2Ω/∂y∂z = 0.
Particularized time-independent wave equations (152), (153), (154), and (155)
represent 4 gluons of the real domain. Particularized time-independent wave equations
of the strong interaction in the imaginary domain representing 4 gluons of this domain
are:
∂2Ω/∂x∂z – (1/ℏ)[2mΩ(EΩ – VΩ)]1/2.∂Ω/∂y + (2/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂x = 0,
(156)
∂2Ω/∂y∂z + (1/ℏ)[2mΩ(EΩ – VΩ)]1/2.∂Ω/∂x – 2.∂2Ψ/∂x∂z = 0,
(157)
∂2Ω/∂y∂z + (1/ℏ)[2mΩ(EΩ – VΩ)]1/2.∂Ω/∂x + (2/ℏ)[2mΨ(EΨ – VΨ)]1/2.∂Ψ/∂y = 0,
(158)
(159)
∂2Ω/∂x∂z – (1/ℏ)[2mΩ(EΩ – VΩ)]1/2.∂Ω/∂y + 2.∂2Ψ/∂y∂z = 0.
Each of the derived time-independent wave equations is characterized by its
individual mass and energy parameters m, E, and V. They should be marked by a
different notation peculiar to the type of the particle. We have omitted such a distinction
for simplicity. In the equations a distinction is made, however, between parameters of
the real and the imaginary domain, which would hold in a space-time continuum which
had different intrinsic properties in each of these two domains.
As we may assume that our Riemannian complex 2-space forms a homogeneous
space-time continuum in the particular domains, then with regard to the
supersymmetrical form of wave equations of the real and imaginary domain, the
following identity (or eventually linear) relations will obviously hold: mΨ = mΩ, EΨ = EΩ,
and VΨ = VΩ between mass and energy parameters of the respective wave equations.
All the derived particularized time-independent wave equations refer to subspace
A and the reference space domain (of aroma a). For their transformation into subspaces
of colours B and C, symbolic matrices (14) (or wave equations in Appendix I) apply; and
for transformation to space domains of aromas α, β and γ (where relevant), the
respective wave equations in Chapter 5, A, d should be referred to.
Similarly, the particularized wave equations could be derived for combined
particles from their general wave equations (for baryons, protons, neutrons and mesons)
of a 2U-space field; and in procedures analogous to those of a 2U-space field, the timeindependent and time-dependent particularized wave equations could also be derived in
3U and 4U-space fields. The latter would be obtained by applying higher order (3rd and
4th) derivatives of equation (141) to the particular general wave equations in these two
space fields, with respect to wave functions Ψ and Ω, while considering relation (145).
We shall, however, not derive these equations here.
97
General Theory of Matter
J. Šuráň
Remark 4: If we substitute(case (I)) relation –(ℏ/iΨ)∂Ψ/∂t for E into the third terms of the timeindependent wave equations (128) for the electron, and (138) for a neutrino or an antineutrino, we get their
respective time-dependent wave equations (129) and (140).
Conversely(case (II)): substituting relation (142) (or relation (143) in the imaginary domain) into
the second term of the time-dependent Schrödinger wave equation (129) for the electron, we get the timeindependent Schrödinger wave equation (128); and, similarly, such a substitution into the second term of
the time-dependent wave equation (140), for a neutrino or an antineutrino, leads to the time-independent
form of the particularized wave equations (138) for these two particles.
These transformation rules may be generalized to apply to other derived particularized wave
equations as well.
Case (I): Using operator relations
E.(.) =
–[(ℏ/i)(.)]∂./∂t
R4(1)
R4(2)
and
(1/ℏ)∂./∂s = (1/E)∂2./∂s∂t,
2
with each of the symbols (.), ∂., and ∂ ., referring to wave functions Ψ or Ω and relevant terms of the
particularized time-independent wave equations, the time-dependent particularized wave equations will be
obtained. s designates a space-like variable.
Case (II): Applying an operator relation
∂./∂t = –(i/ℏ)E.(.),
R4(3)
and R4(2) in an inverse order (substituting now its right side for the left) on a particularized timedependent wave equation, we obtain its time-independent form. These transformations apply equally to
the imaginary domain, where the wave function Ψ is substituted for Ω, and correspondingly, in wave
equations with mixed wave functions.
By the above transformations, we can always obtain the respective time-independent or timedependent particularized wave equation. For example, to convert quark time-independent wave equation
(146) into a time-dependent one (case (I)), both operator relations R4(1) and R4(2) apply ― R4(1) to the
fourth term of its imprint, and R4(2) to the second term of its trace; while in the analogous conversion of
an equation of the strong interaction, say, (152), only operator relation R4(2) is applicable (to both of its
terms); and likewise only R4(2) applies to the time-independent wave equation (148) of the weak
interaction (to its second term).
98