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UDC 531.145
Dynamical Methods of Investigations in Application to the Schrodinger Particle
Yu. A. Rylov
Institute for Problems in Mechanics, Russian Academy of Sciences 101-1 ,Vernadskii Ave., Moscow, 119526, Russia
Quantum systems are dynamic systems restricted by the principles of quantum mechanics (linearity of dynamic equations, linear transformation of the wave function etc.). One suggests to investigate the quantum systems simply as dynamic systems, ignoring the quantum principles and constraints imposed by them. Such dynamic methods of investigation appear to be more adequate and effective, than the conventional quantum methods of investigation. Using these methods, which ignore the quantum principles, one can overcome the principal problem of quantum field theory: join of nonrelativistic quantum principles with the relativity principles. Investigation of dynamic system SS, described by the Schrodinger equation, admits one to prove that the Copenhagen interpretation is incompatible with quantum mechanics formalism. Besides, it is shown that sometimes the application of quantum principles leads to incorrect results.
Introduction
One uses new dynamical methods of investigation, which ignore the quantum principles. Elimination of the quantum principles admits one to eliminate the principal problem of QFT: join of the relativity principles with the principles of quantum mechanics. After this elimination the relativistic quantum theory is constructed on the basis of relativistic dynamics only, whereas the conventional approach uses in addition the linearity of dynamic equations. This linearity is based on a special gauge of the wave function. A use of such an artificial property of the wave function for construction of the relativistic quantum theory seems to be unjustified, although the linearity is very convenient, because the linear equations are simple for solution. We show in the simple example of Schrodinger particle SS (dynamic system described by the Schrodinger equation), that the difference between the quantum system and the classical one is purely dynamical, and the quantum system may be described without a reference to quantum principles.
A fundamental physical theory Tf must be a logical structure. It means, that the theory contains a few fundamental propositions. All predictions of the theory Tf as well as other propositions Pi intermediate between the fundamental propositions Pf and experimental data De are deduced from the fundamental propositions Pf by means of logical reasonings and mathematical calculations. Practical derivation of predictions and explanations from the fundamental propositions Pf may be difficult and complicated, because of long reasonings and complicated calculations. In this case one uses a set of intermediate propositions Pi, which are valid in some region of physical phenomena. Practical application of the intermediate propositions Pi as some physical theory Tc may appear to be more effective and simpler, than a use of the fundamental theory Tf.
In general, the intermediate propositions Pi are to be corollaries of the fundamental propositions Pf. However, if it is impossible to discover the true fundamental propositions Pf, we may guess intermediate propositions Pi and consider them as a curtailed physical theory Tc. The set of prescriptions Pi is chosen in such a way to explain some set of experimental data. The curtailed theory Tc is not a logical structure, it is simply a list of prescriptions, which are not connected logically between themselves. The logical structure appears, only if we discover and add the fundamental propositions Pf, which generate this list of prescriptions Pi under some conditions. In
this case the intermediate propositions Vi may be eliminated from the formulation of the fundamental theory T, because the propositions Vi are corollaries of fundamental propositions Vf.
Thus, we may use the curtailed physical theory Tc instead of the fundamental physical theory T. For instance, the axiomatic thermodynamics is a curtailed theory with respect to kinetic theory, which may be considered to be a fundamental theory. The conventional quantum mechanics is a curtailed theory, whereas the corresponding fundamental theory is not known yet. Such an application of T is possible only under those conditions (for instance, in the nonrelativistic case), when the intermediate propositions Vi are valid. However, extension of the curtailed theory Tc to other conditions (for instance, to the relativistical case) may appear to be impossible, because the intermediate propositions Vi are only corollaries of the fundamental propositions Vf in the nonrelativistic case. Under another conditions (for instance, in the rela-tivistical case) the fundamental propositions Vf may generate another intermediate propositions Vf, which does not coincide with Vi and cannot be obtained from Vi, because the set of Vi is simply a list of prescriptions (a list of corollaries of Vf), but not a logical structure.
1. Artificiality of conventional method of description
The action for the Schrodinger particle Ss has the form
Лз [Ф] = J{j {ГМ - M* • Ф) - w*}d
(1)
where ^ = ^ (t, x) is a complex wave function. The action carries out the complete description of the Schrodinger particle Ss, because it generates the dynamic equation
ti2
iHdotp =--V2ip
2m
(2)
and corresponding canonical quantities: the 4-current jk and the energy-momentum tensor Tk
jk = {p, j} = {w, - • vo} • (3)
Connection between the particle and the wave function is described by the relations.
(F (x, p)> = В J U [ф*F (x, p) dx
p = —itiV, В = / ф*ф dx
-i
(4)
which define the mean value (F (x, p)) of any function F (x, p) of position x and momentum p. We shall refer to these relations together with the restrictions imposed on its applications as the quantum principles, because von Neumann [1] has shown, that all proposition of quantum mechanics can be deduced from relations of this type.
Setting h = 0 in the action (1), we hope to obtain a description of classical particle. Instead, the description disappears, and in this sense the description (1) is artificial.
2. Natural description of the Schrodinger particle
To obtain the natural description of the Schrodinger particle, we produce the change of variables (another gauge of the wave function phase)
ф ^ : ф
b = const = 0
(5)
4
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in the action (1). We obtain
^ib b2 - ffldoVb ~ don ■ Vb) ~ +
b2 ,2 h2
2m 2m
This change of variables leads to the replacement h ^ b and to appearance of two nonlinear terms which compensate each other, if b = h.
The dynamic equation becomes to be nonlinear, if b2 = h2
b2 h2 - b2 / (Vp)2 Vp\
= -—V2^ - —- Mp- + 2V— 7
2m 8m \ P2 P /
ib 2m
However, the description becomes to be natural in the sense, that after setting h =0, the action As ] turns into the action
^ib b2 b2 - - • - + — (v |^|)2} dt dx, (9)
which describes the statistical ensemble E [Sd] of free classical particles Sd. The action AE[Sd] for this statistical ensemble can be represented in the form
' m /dxx 2
^[5d]M = y 2" Idf J did€' (10)
where x = x (t, £) is a 3-vector function of independent variables t, £ = {£i,£2,£3}. The variables (Lagrangian coordinates) £ label particles Sd of the statistical ensemble E [Sd]. The statistical ensemble E [Sd] is a dynamical system of the hydrodynamical type. One can show that the dynamic system, described by the action AScl [#&] (9) is a partial case (irrotational flow) of the dynamic system E [Sd] [2].
Connection between the Schrodinger equation and hydrodynamical description is well known [3,4]. But a connection between the description in terms of wave function and the hydrodynamic description was one-way. One can transit from the Schrodinger equation to the hydrodynamic equations, but one cannot transit from hydrodynamic equations to the description in terms of the wave function, because one needs to integrate hydrodynamic equations. Indeed, the Schrodinger equation consists of two real first order equations for the density p and the phase <p, whereas the system of the hydrodynamical equations consists of four first order equations for the density p and for the velocity v. To obtain four hydrodynamic equations one needs to take gradient of the equation for the phase On the contrary, if we transit from the hydrodynamic description to the description in terms of the wave function, we are to integrate hydrodynamic equations. In the general case this integration was not known for a long time.
Change of variables, leading from the action AE[Sd] [x] to the action AScl contains integration (see [2] or mathematical appendices to papers [5,6]). The constant b in the action AScl is an arbitrary constant of integration (gauge constant). Arbitrary integration functions are "hidden" inside the wave function Thus, the limit of Schrodinger particle (6) at h ^ 0 is a statistical ensemble E [Sd], but not an individual particle Sd. It means, that the wave function describes a statistical ensemble of particles, but not an individual particle, and Copenhagen interpretation, where the wave function describes an individual particle, is incompatible with the quantum mechanics formalism.
A use of Copenhagen interpretation does not generate any problems, until we consider mathematical formalism of QM, because in the framework of this formalism
we have the only object of investigation. It is of no importance what is the name of the investigated object. But at the consideration of the measurement we have two different kinds of measurement:
1) Individual measurement (S-measurement), which is produced over the individual stochastic particle Sst.
2) Massive measurement (M-measurement), which is produced over the statistical ensemble E [Sst], or over the statistical average particle (<S).
3. Statistical ensemble as a starting point of quantum theory
One can show, that the Schrodinger particle Ss is a special case of the statistical ensemble E [Sst] of stochastically moving free particles Sst, i.e. the action (6) for Ss can be obtained from the action for the statistical ensemble E [Sst] by means of a proper change of variables [2].
The statistical ensemble E [Sst] of stochastic particles Sst is a dynamic system, described by the action
[Set] [x udf] = j{
m ~2
dxY dt)
m
ti
+ ~2~Udf " 2
Vud^ dt d£,
(11)
where udf = udf (t, x) is the diffusion velocity, describing the mean value of the stochastic component of the velocity, whereas 0 describes the regular component of the particle velocity, and x = x (t, £) is a 3-vector function of independent variables t, £ = {£i,£2,£3}. The variables £ label stochastic particles Sst, constituting the statistical ensemble. The operator V is defined in the space of coordinates x. Dynamic equations have the form
SA
E[Sst]
Sx
d^x 'dt2
m 2
-m—— + V ( -irUjf - ^Vudf
2
0,
S
■£[Sst]
Su
df
mpudf + - Vp
-i
д (x1, x2, x3) д
Resolving (13) with respect to udf in the form
д
д (x1, x2, x3)
udf
ti
(12)
(13)
(14)
(15)
and eliminating udf from equation (12), we obtain the dynamic equations of the hy-drodynamical type
m
dSc
di2
—VU (p, Vp), U (p, Vp)
IL
8m
(Vp)'
P2
2
v2p
(16)
Hydrodynamic equations (16) may be written in terms of the wave function [2,5,6]. The proper change of the variables together with integration turns the action (11) for E [Sst] into the action containing the quantum constant h and an arbitrary integration constant b.
The wave function is not a specific quantum quantity. The wave function is a method of description of any ideal fluid. Quantum and classical dynamic systems distinguish dynamically (by the form of their actions). The form of description (in
0
p
terms of wave function, or in terms of position and momentum of the particle) is of no importance.
Considering the statistical ensemble of stochastic particles as a starting point of the quantum mechanics, one does not need quantum principles , because the interpretation is carried out directly via the particles of the statistical ensemble. Besides, the quantum theory turns into a consistent statistical theory, where there are two sorts of particles: (1) individual stochastic particle Sst and (2) statistical average particle (Sst), which is the statistical ensemble, normalized to one particle. Formalism of quantum mechanics deals only with the statistical average particle (Sst). All predictions of QM relate only to (Sst). In accordance with two sorts of particles we have two sorts of measurements: (1) S-measurement produced over Sst and (2) M-measurement produced over (Sst). The two kinds of measurements have different properties and one may not confuse them (see for details [7]).
The Copenhagen interpretation meets the difficulties, when it tries to test predictions of mathematical formalism in single experiments. For instance, there exists the problem of the mechanism of the wave function reduction at a single experiment. Another problem concerns the two-slit experiment. How can an individual particle pass through two slits at once? The physical journals publish discussions concerning problems of quantum measurements. For instance, such a discussion was declared in 2002 by the journal Uspekhi Fizicheskich Nauk. These problems cannot be solved in the framework of the Copenhagen interpretation, which does not distinguish between the individual particle S and the statistically average particle (S). Confusion of two different objects, having different properties generates difficulties and paradoxes. The wave function does not describe the state of individual particle S, and it is meaningless to ask, how the wave function changes at a single measurement (S-measurement). At the massive experiment (M-measurement) we obtain a distribution F (R') of the measured quantity R, but not a single value R' of the measured quantity. At such a situation it is useless to ask, how the obtained result R' influences on the the state of the statistical ensemble (statistical average particle). Finally, we may define the third type of measurement (SM-measurement): the massive measurement of the quantity R leading to a definite value R' of the measured quantity R. The SM-measurement is the M-measurement leading to a distribution F (R'), accompanied by a selection of those particles, where result of S-measurement is R'. Uniting all particles with the measured value R' in one statistical ensemble Er , we can put the question about the wave function of Er . Of course, the wave function of Er does not coincide, in general, with the initial wave function and this change of the wave function is considered as a reduction of the wave function. The origin of the reduction is quite transparent. It is the selection, which is produced to obtain the same value R' of the measured quantity for all particles of the statistical ensemble. Thus, the problems of reduction are conditioned by the confusion of concepts of the individual particle S and the statistical average particle (S), which takes place at the Copenhagen interpretation (see for details [7]).
As concerns the particle, passing through two slits simultaneously, it is a reasonable property of the statistical average object. It is a pure statistical property, which has nothing to do with quantum properties. Individual particle S can pass either through one slit, or through another, whereas the statistical average particle (S) can pass through both slits simultaneously. (Compare, individual person is either a man, or a woman, whereas the statistical average person is a hermaphrodite (half-man half-woman), and there are no quantum mechanical properties here).
Using statistical ensemble as a starting point, we can test validity the quantum principles. The quantum principles or their conventional interpretation appear to be wrong in some cases. For instance, the momentum distribution w (p) = |(p|^)| is in reality a distribution over mean momenta (p). Let us illustrate the difference between the two distribution in the example of ideal gas, whose state may be described by the wave function. In the case, when the gas moves with the constant velocity u, the wave function has the form
^ (x) = Aie
|mux
Ai = const,
(17)
©-
4
—©
4
where m is the mass of a molecule. Corresponding momentum distribution has the form
w (p) = |(p|^)|2 = (p - mu), A = const. (18)
This distribution does not coincides with the Maxwell momentum distribution
2
F (x, p)dp
1
(2nmkT )3/2
exp
(p — mu)2 2mkT
dp,
(19)
which depends on the gas temperature kT.
Let us divide the space into small cells V and calculate the mean momentum (p)i of molecule in each cell Vi. In the given case we obtain in the ith cell (p)^ = mu. All mean momenta (p)^ form the mean momenta distribution (18). The mean momenta (p)i correlate with the position x» of the cell Vi, and mutual distribution over position x and momentum (p) appears to be impossible. More detail consideration one can find in [8].
Description of "classical particle" SScl = E [Sd] in the "quantum language" is realized by the action.
- - dor ■ r - + (V I^D2} didx' (2°)
where we use the quantum constant h instead of arbitrary constant b.
^ , , ,V2P (Vp)2
ihdo^p + —vV - — 2 -
2m 8m \ p p2
ф = 0, p =
(21)
Description of "quantum particle" SS = E [Sst] in the "classical language" is realized by equations
dp dx p
— = - VU (p, Vp), -77 = — , dt dt m
(22)
where U is defined by the second relation (16).
Dynamic equations (22) are the partial differential equations, because p contains derivatives with respect to £a, a = 1, 2, 3.
Description of "classical particle" SScl = E [Sd] in the "classical language" has the form
£ = 0, ^ = (23)
dt dt m
where x = x (t, £), p = p (t, £). Dynamic equations (23) are ordinary differential equations.
Describing quantum system S and corresponding classical system Scl in the classical language, we recognize that dynamic equations for the quantum system S are partial differential equations, whereas dynamic equations for its classical approximation Scl are ordinary differential equations.
The difference between the ordinary and partial differential equations is dynamical (but not quantum). This invariant difference is a foundation of the procedure of the dynamical disquantization (transition to classical approximation).
Dynamic disquantization is the procedure of projecting of derivatives onto the direction of 4-current jk
д
j jk
= 1 = 0,1,2,3.
j sjs
To obtain ordinary differential equations, one needs to project derivatives onto one direction.
The dynamic disquantization admits one to formalize the transition to the classical description. Being applied to Schrodinger particle Ss, the dynamic disquantization transform SS = E [Sst] into the statistical ensemble E [Sd] of free classical particles Sd.
The procedure of dynamic disquantization is determined by the dynamic system completely and does not refer to the quantum constant and the quantum principles.
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Concluding remarks
We have analysed application of dynamical methods to the Schrodinger particle and have obtained results, which cannot be obtained by means of conventional axiomatic methods. Analogous results are obtained at application of dynamical methods to other quantum systems. Application of dynamical methods to investigation of quantum systems admits one to eliminate quantum principles. The number of fundamental propositions is reduced essentially. The quantum theory ceases to be a list of prescriptions and becomes a logical structure.
In the conventional quantum theory the allness of the quantum constant h is explained by the quantum nature of all physical phenomena. In particular, it means that all physical fields should be quantized. Application of dynamical methods and elimination of quantum principles suppose some primordial stochastic motion of free particles with dependence of the stochasticity intensity on the particle mass. The universal character of the quantum constant and the particle motion stochasticity may be explained freely by the space-time properties, when the quantum constant is a parameter of the space-time geometry [9,10]. In this case not all physical fields have the quantum nature. Dynamic equations for the metric fields (gravitational and electromagnetic) do not contain the quantum constant h. There is no necessity to quantize these fields.
It is well known that the electromagnetic field is absorbed and emitted in the form of quanta of energy hu. However, this fact does not mean that the electromagnetic field exists in the form of quanta. There are no experiments, which show directly or indirectly that the electromagnetic field exists in the form of quanta. (At least, such experiments are unknown for us). Absorbtion and emission of the electromagnetic field in the form of quanta may be easily explained by the quantum properties of the absorbtion devices and the emission ones. The same relates to the gravitational field. All attempts of its quantization appear to be unsuccessful.
Quantum and classical dynamic systems distinguish only dynamically, i.e. by their dynamic equations, but not by their enigmatic quantum properties. Linearity of the conventional axiomatic methods is simple and attractive, but these methods are founded on nonrelativistic quantum principles, and we cannot be sure that these methods may be applied in the relativistic case.
There is a hope that application of dynamical methods in the relativistic case will be successful. First, extension to the relativistic case is produced on the logical basis, but not by means of incident hypotheses. Second, in the relativistic case the diffusion velocity udf turns to the relativistic field, responsible for pair production, whereas neither quantum principles nor classical relativistic dynamics can describe effect of pair production, which is the principle effect of the high energy physics.
1. Neumann J. V. Mathematische Grundlagen der Quantenmechanik. — Berlin: Berlin, 1932.
2. Rylov Y. A. Spin and wave function as attributes of ideal fluid // J. Math. Phys. — Vol. 40, No 1. — 1999. — Pp. 256-278.
3. Madelung E. Quanten Theorie in hydrodynamischer Form // Z. Phys. — Vol. 40. — 1926. — Pp. 322-326.
4. Bohm D. A suggested interpretation of the quantum theory in terms of "hidden" variables // Phys. Rev. — Vol. 85. — 1952. — Pp. 166, 180.
5. Rylov Y. A. Is the Dirac particle composite?— http://arXiv.org/abs/physics/ 0410045.
6. Rylov Y. A. Is the Dirac particle completely relativistic? — http://arXiv.org/ abs/physics/0412032.
7. Rylov Y. A. Dynamics of stochastic systems and pecularities of measurements in them. — http://arXiv.org/abs/physics/0210003.
8. Rylov Y. A. Hydrodynamical interpretation of quantum mechanics: the momentum distribution. — http://arXiv.org/abs/physics/0402068.
References
9. Rylov Y. A. Non-Riemannian model of the space-time responsible for quantum effects // J. Math. Phys. — Vol. 32, No 8. — 1991. — Pp. 2092-2098.
10. Rylov Y. A. Tubular geometry construction as a reason for new revision of the space-time conception. — http://arXiv.org/abs/physics/0504031.
УДК 531.145
Динамический метод исследования в приложении к Шредингеровской частице
Квантовые системы — это динамические системы, ограниченные принципами квантовой механики (линейность динамических уравнений, линейные преобразования волновых функций и т. п.). Предлагается исследовать квантовые системы просто как динамические системы, игнорируя ограничивающие их квантовые принципы. Такой динамический метод исследований представляется более адекватным и эффективным. Используя этот метод можно преодолеть основные трудности квантовой теории поля: объединить принципы нерелятивистской квантовой механики и принцип относительности. Исследование динамической системы, описываемой уравнением Шрёдингера, позволяет показать, что Копенгагенская интерпретация несовместна с квантовомеханическими принципами. Кроме того, показано, что иногда использование квантовомеханических принципов приводит к неверным результатам.
Ю. А. Рылов
Институт проблем механики РАН, Россия, 119526, Москва, просп. Вернадского, 101-1
