Classical Quantum Theory

Transcription

1 J.P. Wesley Weiherammstrasse Blumberg, Germany Classical Quantum Theory From the extensive observations an the ieas of Newton an from classical physical optics the velocity of a quantum particle is given by w S E, where S is the Poynting vector an E the wave energy ensity. Integrating w r/t yiels the quantum particle motion along a iscrete trajectory as a function of time an initial conitions. All observables are thus precisely preicte. The wave velocity equals the classical particle velocity, where the e Broglie wavelength an the Planck frequency conitions are k p an ω p v. For boun particle motion, the space part of the wave function ψbr,tg ψbrgtbtg satisfies the usual time-inepenent Schroeinger equation, an so the usual eigenvalues are preicte. The reasons the traitional quantum theory is unable to make precise vali preictions in general an the e Broglie-Bohm interpretation are iscusse. 1. Why traitional quantum theory makes no precise vali preictions in general Apart from a few correct empirical formulas, the traitional quantum theory is generally funamentally wrong (Wesley 1961, 1965, 1983a, 1984, 1988a, 1991, 1994, 1995). The trouble all starte when Schroeinger (196) base his quantum theory on Hamilton s (1834) geometrical optics approximation instea of basing it on exact physical optics, or classical wave theory. Following extensive experimentation involving the interference an iffraction of light, Newton (1730) conclue that photons were essentially point particles that move such as to yiel wave phenomena. Newton s original iea can be implemente with the mathematical apparatus of moern wave theory, or physical optics. The geometrical optics approximation, being vali only for situations where the wavelength may be neglecte as small, cannot explain interference an iffraction, where the wavelength plays a vital role an cannot be neglecte. However, Schroeinger i recognize the nee for exact wave behavior to obtain staning-wave eigenvalues for boun atomic systems, as provie by his time inepenent Schroeinger equation. As a result, the traitional Schroeinger theory is hopelessly inconsistent. Macroscopically, only the geometrical optics approximation is suppose to be vali, where the wavelength can play no role an observations are suppose to be given by averages over many e Broglie wavelengths, while submicroscopically exact wave behavior is suppose to be true, where the amissible wavelengths etermine the esire eigenvalues. Futile attempts have been mae to try to reconcile Schroeinger s hopeless inconsistency. To fit the geometrical optics approximation, Schroeinger (196) propose a wave packet to represent a quantum particle as a huge object smeare out over thousans of e Broglie wavelengths, an so the iniviual component wavelengths coul be neglecte as small. No such wave packet has, of course, ever been observe, as it is in principle unobservable (Wesley 1961, 1983b, 1991, 1995). Such a huge wave packet cannot fit into a tiny submicroscopic atomic system; nor can it account for interference an iffraction, in agreement with the limitations of the geometrical optics approximation. Schroeinger also employe the complex representation of a wave, exp ibkx ωtg, with an irreplaceable i 1 which has a π ambiguity in phase, as require by the phase inepenent geometrical optics approximation. Born (196) introuce the mystical notion that the wave function represents an intrinsic probability amplitue, such that the precise behavior of an iniviual quantum particle cannot, even in principle, be either observe or preicte! Heisenberg (197) introuce the uncertainty principle, which can be shown (Wesley 1995) to be merely a prescription of the limitation in accuracy require that can make the geometrical optics approximation vali. It is traitionally claime that observables are given by expectation values (Schiff 1949), which are integral averages over all space that eliminate etaile preiction, in agreement with the rough geometrical optics approximation but, of course, not in agreement with actual etaile observations involving interference an the wavelength. To try to justify these futile attempts to resolve Schroeinger s original inconsistency, a huge amount of further nonsense has been generate: collapsing wave packets, the nonexistence of nonobserve states, thought APEIRON Vol., Nr., April 1995 Page 7

2 experiments in lieu of actual experiments, self interference of a single quantum particle, superposition of wave amplitues representing the superposition of physical states, the operator approach, the Hilbert-space approach, the von Neumann theorem, wave particle uality, the complementary principle, the claime violation of Bell s theorem (Wesley 1994), Schroeinger s (196) time epenent equation with its ambiguous irreplaceable i 1, etc., etc.. An aequate quantum theory must explain interference As mentione above, the Schroeinger (196) quantum theory, being base upon the geometrical optics approximation of Hamilton, cannot explain interference, where the wavelength plays a vital role an cannot be neglecte. Newton (1730) explaine his observations of the interference an iffraction of light in terms of the wavelength (equal to two of Newton s fits ). He assume that photons move such as to yiel this wave behavior. To account for all of Newton s etaile observations, a photon must be regare as essentially a point particle. The mathematical aspects of the wave behavior of light were evelope by Fresnel (186), This fiel of stuy is now calle physical optics. Although the photon aspects of light were largely neglecte or even enie in the 1800 s, it is now known that Newton was right: light consists of a flux of photons that exhibit wave behavior. An aequate quantum theory must yiel the extremely successful results preicte by physical optics, or classical wave theory, which can be erive from a prescription for the precise behavior of photons, as well as the precise behavior of phonons. Most of the empirical ata available that nees to be explaine by an aequate quantum theory is provie by the macroscopic observations of classical waves. The enumerable sets of eigenvalues provie by staning waves in boun submicroscopic atomic systems represent a relatively smaller amount of empirical information. In orer to fit all of the empirical evience available, both macroscopic an atomic, the theory presente here is base irectly upon exact classical physical optics instea of the rough geometrical optics approximation of the traitional Schroeinger theory. 3. The Wesley wave An aspect of light unknown to Newton an Fresnel an first iscovere by Planck (1901) was its quantization in energy units of E hν ω, (1) where h is Planck s constant, h π, an ω πν is the angular frequency. The propagation constant k is then also quantize in momentum units given by p Ec c ω c c, or p k. () e Broglie (193, 194) speculate that particles with a nonzero rest mass shoul also exhibit wave behavior, where the propagation constant (or wavelength λ π k ) is also given by Eq. (), the so-calle e Broglie wavelength conition. His conjecture was confirme in the laboratory. Since a classical wave is empirically efine by a flux of quantum particles, the wave velocity u an the particle velocity v must be ientical for a free-space wave, or ω u v k k. (3) From Eq. () the angular frequency then becomes, in general, for particles with zero or nonzero rest mass ω p v. (4) This result (4) is seen to be the Planck frequency conition (1), where for photons v c an p Ec c. The empirically correct free space wave for any quantum particle is then the Wesley (196, 1965, 1983c) wave, L N M Ψ sin p r vt b g O Q P. (5) For a slow particle with a nonzero rest mass, the frequency from Eq. (5) or (4) is quantize in units of twice the kinetic energy W; thus, ω W, (6) which may be contraste with the traitional theory where ω is suppose to be either the total classical energy or, inconsistently, the total energy incluing the rest energy mc. This empirically correct Wesley wave, Eq. (5), may be compare (Wesley 1965, 1983) with the physically impossible traitional e Broglie (193, 194) wave, where the frequency for a slow particle of nonzero rest mass is suppose to be mc γ mc, an the phase velocity is then suppose to be u c v > c. 4. Classical quantum particle trajectories Since quantum particles must carry the energy an momentum, an the classical Poynting vector P prescribes the energy an momentum flux in a wave, quantum particles must necessarily move along the lines of the Poynting vector to yiel wave phenomena. If the wave energy ensity E is the energy of the quantum particles that prouce the wave, then the quantum particle velocity w is given by w S, (7) E in agreement with Eq. (5) for a free quantum particle. Page 8 APEIRON Vol., Nr., April 1995

3 A classical scalar wave function Ψ satisfies the wave equation Ψ t Ψ 0, (8) u where u is the phase velocity, subject to appropriate bounary conitions. The classical Poynting vector an energy ensity for a scalar wave are then given by K S Ψ Ψ t E K L Ψ b g + b Ψ tg O NM u QP, (9) where the appropriate constant K epens upon the amplitue constant chosen for Ψ. If E ω P is to represent the quantum particle ensity an S ω G the quantum particle flux, then the constant K must be chosen as K ω. (10) k Light can be treate as a scalar wave. Polarization phenomena can be hanle with two such scalar waves (Wesley 1983e). The scalar theory given by Eqs. (7), (8), an (9) may thus be regare as perfectly general an vali for all quantum particles. Once the solution Ψ to the wave equation (8) is known as a function of position an time, the motion of a quantum particle along a iscrete trajectory as a function of time an the initial conitions are given by integrating Eq. (7), where w r t, thus, t rmrb0g, tr z wbr, tg t. (11) Once the precise quantum particle motion is known, all possible observables can be calculate an precisely preicte. For example, the force acting on a slow quantum particle of nonzero rest mass m that gives rise to the motion is given by w F m, (1) t where the velocity w is given by Eqs. (7) an (9), expresse as a function of time, using Eq. (11). Similarly, the potential that acts in a conservative situation is given by the total energy minus the kinetic energy expresse as a function of position only, using Eq. (11); thus, 0 U E mw. (13) 5. Time-average classical quantum particle trajectories For light, the time variation is harmonic an rapi, a perio being of the orer of s; most macroscopic observations of interest are time-average observations. In this case it is of value to efine a time-average quantum particle velocity w as w S E, (14) where S an E are given by averaging Eqs. (9) over a cycle. It may be reaily shown that the instantaneous trajectories given by Eq. (7) can iffer from the time-average approximation, Eq. (14), by at most a quarter wavelength, which is a negligibly small eviation for macroscopic observations (Wesley 1983f). A specific example is inicate in Fig. 1 for the classical time-average flux of coherent quantum particles passing through Young s ouble pinholes to prouce the famous ouble pinhole interference pattern (Wesley 1983g, 1984, 1991). This figure was obtaine by isplaying the slopes of the time-average Poynting vector S on a gri an sketching the trajectories in by eye so that the quantum particles emanate isotropically from each pinhole. It must be stresse that the approximation w, Eq. (14), which is excellent for macroscopic time-average quantum particle motion, is not at all sufficient for submicroscopic atomic systems, where neither the perio nor the wavelength can be regare as small. It cannot be use, for example, to escribe the motion of the electron in the hyrogen atom. Figure 1. Time-average Poynting vector or trajectories of quantum particles passing through two pinholes, the istance between pinholes being 3.6 times the wavelength. APEIRON Vol., Nr., April 1995 Page 9

4 6. Arbitrary flux an ensity of the traitional theory In contrast to the present classical quantum theory, which is base irectly upon the classical empirical evience provie by Newton an classical physical optics, the particle flux an ensity of the Schroeinger (196) theory is base upon theoretical speculation only. As a consequence it is ifficult to iscover any empirical merit for his arbitrary specification. In terms of his unfortunate improper use of complex notation involving an irreplaceable i 1, Schroeinger s (196) relativistic, particle flux an ensity (where for photons an phonons mc ω ) are given by HG c G Ψ Ψ Ψ Ψ iω i F Ψ Ψ Ψ Ψ P ω t t i, I KJ ΨΨ, (15) where the time variation is always prescribe by the time harmonic factor expb iω tg, in which ω is a constant. Clearly Schroeinger s arbitrary prescription (15) cannot agree with the correct empirically observe instantaneous flux an ensity given by the classical expressions (9) (for the appropriate choice of the constant K). The traitional prescription (15) is not a function of the time; it can, at best, only represent a steay state flow for time harmonic waves average over a perio. Classically, only pure real wave functions are amissible, an if complex notation is use, they must be reucible to a pure real result. A pure real time harmonic function f can be expresse as the real part of a complex function Ψ ψ r exp iω t ; thus, b g b g b gcosω + Re ψb gexpb b g b g b g f A r t B r sinω t r iω tg. (16) Ψ + Ψ where Ψ r A r + ib r. The time average of a prouct of two real time harmonic functions f 1 an f of the same angular frequency is then given by f f 1 Ψ Ψ b g + Ψ Ψ (17) From Eqs. (16), (17), (9), an (10) the empirically correct classical time-average quantum particle flux an ensity in correct complex notation are given by c G Ψ Ψ Ψ iω Ψ Ψ Ψ, (18) ΨΨ P +, k which may be compare with the arbitrary traitional flux an ensity, Eq. (15). The Schroeinger flux G is seen to agree with the empirically observe classical timeaverage flux G for time harmonic waves. But the Schroeinger ensity P oes not agree with the observe i classical time-average ensity P, except for the trivial case of a free particle, where Ψ expmibk r ω tgr. The fact that the Schroeinger flux happens to agree with the correct classical time-average Poynting vector for time harmonic waves oes not mean the traitional theory is thereby rescue, because: 1) The traitional flux, G, the first of Eqs. (15), is suppose to be the instantaneous flux, which oes not agree with the correct instantaneous flux given by the first of Eqs. (9) for real Ψ functions. ) Since the Schroeinger specification (15) assumes only time harmonic waves; it cannot yiel the quantum particle flow for transient waves. In contrast, the correct classical specification (9) yiels particle flow for transient waves, as well as for time harmonic waves. 3) The traitional specification (15) is assume to be vali for submicroscopic atomic systems, which le Schroeinger to make the impossible claim that boun systems are evoi of any motion what-so-ever. 4) In the Schroeinger theory G, Eq. (15), is suppose to be exact without any approximation being involve; but the timeaverage of the correct classical flux G, Eq. (9) (that equals G ), for time harmonic waves is an approximation that cannot agree with the exact instantaneous flux. 5) The traitional Copenhagen interpretation claims that the flow lines efine by Schroeinger s flux, G have no physical meaning; because quantum particles are not suppose to follow precise (or even approximate) trajectories. 7. Bohm s interpretation of the traitional theory Maelung (196), e Broglie (197) an Bohm (195, Bohm an Hiley 1993) trie to resolve Schroeinger s original inconsistency by proposing that quantum particles follow iscrete trajectories given by integrating a particle velocity w efine by p mw i ln Ψ, (19) where Ψ is the usual traitional complex wave function. Since the particle velocity must be pure real, the complex conjugate of the right sie of Eq. (19) must yiel the same particle velocity. Hence, Eq. (19) can be written in the symmetrical form w ln Ψ ln Ψ im Ψ Ψ Ψ Ψ im ΨΨ i. (0) Making the replacement mc ω, Eq. (0) is seen to be simply the particle velocity expecte from Schroeinger s flux an ensity, Eq. (15), for time harmonic solutions. In particular, w G P. (1) Page 30 APEIRON Vol., Nr., April 1995

5 Thus, the e Broglie-Bohm theory offers nothing new. It merely says that Schroeinger s arbitrary flow lines, G, shoul be taken seriously an shoul be accepte as actual quantum particle trajectories. Although Eq. (19) or (1) yiels the correct timeaverage approximate trajectories for macroscopic observations for time harmonic waves; it oes not give the correct magnitue of the particle velocity along these trajectories; because the correct particle ensity P, the secon of Eqs. (18), oes not equal P ΨΨ. Moreover, the particle velocity as prescribe by Eq. (19) or (1) cannot, contrary to Bohm s claims, yiel the correct trajectories, nor the correct velocities for submicroscopic atomic systems where the time-average approximation is not amissible. Like Schroeinger, Bohm must make the impossible claim that boun atomic systems are necessarily completely motionless. The Bohm interpretation of the traitional quantum theory, thus, merely serves to make Schroeinger s original inconsistency painfully obvious without, however, resolving it. The present classical quantum theory exhibits no such inconsistency. 8. Classical quantum theory for boun systems To satisfy bounary conitions, a staning wave solution Ψbr,tg to the wave equation (8) appropriate for boun quantum particle motion must be separable as a prouct of a function of position only Ψbrg times a function of the time only T(t); thus, b g b g b g Ψ r,t ψ r T t. () Substituting Eq. () into (8) then yiels two separate equations + ψ k ψ 0 T, + ω T 0, (3) t where the phase velocity square is given by u ω. (4) k To solve these Eqs. (3), k must be expresse as a function of position only an ω must be expresse as a function of time only. Since for classical macroscopic observations a free-traveling wave travels with the quantum particle an the phase velocity equals the particle velocity, the phase velocity for boun particle motion shoul also be taken as the classical particle velocity (ignoring possible quantum wave effects). Because energy is conserve for boun particle motion, the phase velocity square that occurs in the wave equation can be expresse in terms of the energy. For a slow particle of nonzero rest mass, using the e Broglie wavelength conition (), the propagation constant square k to be use in the first of Eqs. (3) becomes k p m E V b r g, (5) the kinetic energy being p m, an where E is the total constant energy. Here, k, Eq. (5), is seen to be expresse as a function of the position only, as esire. Combining the first of Eqs. (3) an (5) yiels the fruitful time inepenent Schroeinger equation, m b gr. (6) ψ + m E V r ψ 0 From the Planck frequency conition (4), ω to be use in the secon of Eqs. (3) becomes ω bp vg 4W btg, (7) where the kinetic energy W is to be expresse as a function of the time only. The secon of Eqs. (3) becomes t T + W t T b g. (8) 4 0 Consistent with the ifferential equation for the space part, the Schroeinger equation (6), W(t) is to be taken as the classical kinetic energy expresse as a function of the time only. For fast particles, where the momentum p an kinetic energy W are given by p mγ v, W E V mc γ, (9) where E is the total constant energy incluing the rest energy, the appropriate Eqs. (3) from () an (4) become 4, c ψ + E V ψ m c ψ W T 4 + W m c T 0 i, t i (30) where W in the secon of Eqs. (30) is to be expresse as a function of the time only. The first of these Eqs. (30) is Schroeinger s relativistic wave equation. To erive the motion for a boun quantum particle, the classical problem, inepenent of possible quantum wave effects, must first be solve. Secon, the resulting classical expressions for the kinetic energy W E V (or else E V) expresse as a function of position an also as a function of time only are then to be substitute into the appropriate ifferential equations (6) an (8) (or else Eqs. (30)). Thir, after these ifferential equations are solve, Ψ ψbrgtbt g is substitute into Eqs. (9) an (7) for the quantum particle velocity w, which inclues both classical an quantum effects. Fourth, Eq. (7) is integrate to yiel the esire quantum particle motion along iscrete trajectories as a function of time an initial conitions. The approximate time average velocity w, Eq. (14), an the approximate trajectories obtaine by integration, which are appropriate for macroscopic observations of time harmonic waves, cannot be use for boun particle APEIRON Vol., Nr., April 1995 Page 31

6 motion, because the wavelength involve cannot be regare as small. In the traitional Schroeinger theory the variation with time is assume to be nonexistent for boun particles. The function T(t) must be assume traitionally to be a constant. This peculiar iea results from Schroeinger s improper use of complex notation to try to represent pure real physical phenomena. Some explicit examples of boun particle motion prescribe by the present classical quantum theory may be foun elsewhere (Wesley 1983h, 1984, 1991). 9. Discussion The classical quantum theory presente here is able to successfully account for a vast amount of empirical evience that cannot be explaine by the traitional Schroeinger quantum theory. The classical quantum theory reaily reveals the precise motion of quantum particles along iscrete trajectories that yiel interference an transient wave phenomena. It inicates, in terms of the initial positions of photons in the incient beam, which photons are transmitte an which are reflecte at a ielectric interface (Wesley 1988), no intrinsic probabilities being involve. The iea of inherent unpreictability, which the traitional quantum theory nees to excuse its inability to make precise vali preictions, is not a feature of the precise classical quantum theory. Despite the extreme superiority of the classical quantum theory presente here, it seems to lack a truly funamental character. To a first approximation, the classical motion without quantum wave effects is erive. To a secon approximation, quantum wave effects are generate from the first approximation. It then appears that quantum (or wave) behavior may be merely a superficial effect superpose upon the classical motion. It woul seem that perhaps a thir approximation might be neee, or, of course, a truly funamental theory. The arrogant ieas of Heisenberg, Born, Dirac, Bohr, an others that the Copenhagen quantum theory represents a new funamental science or quantum mechanics is harly justifie. The observation of quantum phenomena is not new. The general ieas of how quantum particles move to prouce interference an iffraction were alreay investigate by Newton 300 years ago. The unerlying cause of the wave behavior exhibite by quantum particles remains obscure. The classical quantum theory presente here is able to show how quantum particles move to exhibit interference an wave phenomena; but it oes not say why quantum particles shoul move in this fashion. References Bohm, D., 195. Phys. Rev. 85:166,180. Bohm, D. an Hiley, B.J., The Univie Universe: An Ontological Interpretation of Quantum Theory, Routlege, New York. Born, M., 196. Zeits. Phys. 38:803. e Broglie, L., 193. Compt. Renus 177:507, 548. e Broglie, L., 194. Compt. Renus 179:39, 676. e Broglie, L., 197. Compt. Renus 184:73; 185:380. Fresnel, A., 186. Mem. Aca. Sci. 5:339. Hamilton, W.R., Phil. Trans Heisenberg, W., 197. Zeits. Phys. 43:17. Maelung, E., 196. Zeits Phys. 40:33. Newton, Sir Isaac, Opticks, 4th e., Lonon, a 195 reprint, Dover, New York. Planck, M., Ann. er Phys. 4:553. Schiff, L.I., Quantum Mechanics, McGraw-Hill, New York, pp. 4-7, 53. Schroeinger, 1., 196. Ann. er Phys. 79:361, 489, 734; 80:437; 81: 109; Naturwiss. 14:664. Wesley, J.P., Classical interpretation of quantum mechanics, Phys. Rev. 1:193. Wesley, J.P., 196. Wave mechanics with phase an particle velocities equal, Bull. Am. Phys. Soc. 7:31. Wesley, J.P., Causal quantum mechanics with phase an particle velocities equal, Il Nuovo Cimento 37:989. Wesley, J.P., 1983a Causal Quantum Theory, Benjamin Wesley, Blumberg, Germany. Wesley, J.P., 1983b. Ibi. pp , Wesley, J.P., 1983c. Ibi. pp Wesley, J.P., Ibi. pp Wesley, J.P., 1983e. Ibi. pp Wesley, J.P., 1983f. Ibi. pp Wesley, J.P., 1983g. Ibi. pp Wesley, J.P., 1983h. Ibi. pp Wesley, J.P., A resolution of the classical wave-particle problem, Foun. Phys. 14:155. Wesley, J.P., 1988a. Why the EPR paraox has been resolve in favor of Einstein, in: Microphysical Reality an Quantum Formalism, Urbino, Italy Kluwer Acaemic, Amsteram, es. van er Merwe et al., pp Wesley, J.P., 1988b. Completely eterministic reflection an refraction of photons, in: Problems in Quantum Physics, Gansk 1987, Worl Scientific, Singapore, es. Kostro, L., Posiewnik, A., Pykacz, J., an Zukowski, M., pp Wesley, J.P., Selecte Topics in Avance Funamental Physics, Benjamin Wesley, Blumberg, Germany, Chap. 8, pp Wesley, J.P., Experimental results of Aspect et al. confirm classical local causality, Phys. Essays 7:40. Wesley, J.P., Failure of the Uncertainty Principle, submitte for publication, preprints available from author. Page 3 APEIRON Vol., Nr., April 1995