
Revision of Classical Quantum Mechanics Di HUA Research Associate (ret.) of Stanford University Academician of Russian Academy of Astronautics huadi1936@gmail.com Abstract The author applies his new relativistic mechanics, developed in his previous work, to several basic issues of the classical quantum mechanics. This study shows: (1) A moving particle抯 velocity is exactly it抯 own matter wave抯 phase velocity so that the particle and its own matter wave move together never separated. (2) Bohr抯 quantum structure of the hydrogen atom must be rectified; an electron in the hydrogen atom can have only circular orbits; the electron抯 transition from an inner orbit to an outer orbit, not the other way round, causes the radiation. (3) The Compton photon scattering formula violates the principle of elastic collision and must be corrected. (4) Experiments on the _{ }photon scattering and on the diffraction of electrons with nearlight velocity may validate the new relativistic mechanics and justify the revision of the classical quantum mechanics. (5) Schrodinger抯 wave equation is incorrect and must be replaced by a new equation capable of correctly quantizing rectangular potential well and harmonic oscillator. (6) There is no such a thing as nonrelativistic wave equation; all wave equations are relativistic, but the KleinGordon equation is not a correct relativistic wave equation. (7) 揥ave packetis a misleading concept; moving particles are not wave packets. (8) Schrodinger抯 wave function of complex variables is misconceived; wave function is just a useful tool made of real variables. (9) Born抯 statistical interpretation of the wave function and Heisenberg抯 uncertainty principle are questionable. Brief Review of the New Relativistic Mechanics In a previous paper entitled 揜elativistic Mechanics Based on Variable Speed of Light the author has revealed many inconsistencies and errors in Einstein抯 theory of relativity and then developed a new relativistic mechanics. This paper applies the new relativistic mechanics to several issues of the earlyyear quantum mechanics, clears misunderstandings in classical works and thereby further justifies the new relativistic mechanics. Assuming readers are well familiar with the author抯 previous paper, we just briefly review key points of the new relativistic mechanics, which are relevant to the discussios in this paper. In the paper on 揜elativistic Mechanics Based on Variable Speed of Light the author has proven: (1) The essential difference between relativistic mechanics and nonrelativistic mechanics lies in that: a body (subject) must use the former to do relative assessment of another moving body (object) whereas the latter must be used by a body to do subjective selfassessment of its own state of motion. 揜elative assessmentis relativistic whereas 揝elfassessmentis nonrelativistic. 揜elativisticor 揘onrelativisticis not determined by the speed of relative motion. Quantum mechanics is a science, in which we (subjects) study relatively moving particles (objects). Therefore, all the quantum mechanics ought to be relativistic, no matter how slow particles move. However, Einstein抯 relativistic mechanics is incorrect and must be replaced by our new relativistic mechanics. Newtonian mechanics, although correct, is nonrelativistic and can not be used in the quantum mechanics. (2) Moving at speed _{ }, a body with static mass _{ } has moving mass _{ }. (3) The new relativistic mechanics has discovered a new physical quantity _{ }, which is the kinetic energy possessed by moving mass _{ } (shortly kinetic energy of moving mass). This physical quantity will play a significant role in the revision of classical quantum mechanics. (4) If a body with static mass _{ } is accelerated by an external force to become moving at speed _{ }, then the body has acquired kinetic energy _{ }. (5) A body抯 static mass _{ } and moving mass _{ } are equivalent to its static massenergy _{ } and moving massenergy _{ } respectively. A body抯 total energy can be expressed in two ways: Total Energy_{ }Moving massenergy _{ }_{ }Kinetic energy of moving mass _{ } _{ }_{ }_{ } or Total Energy_{ }Static massenergy _{ }_{ }Acquired kinetic energy _{ } _{ }_{ }_{ }. (6) Photons have static mass. A moving photon with static mass _{ } and velocity _{ } has moving mass _{ }, moving massenergy _{ }, kinetic energy possessed by its moving mass _{ } and total energy _{ } or _{ }. ^ A particle moving at velocity _{ } has moving mass _{ } and momentum _{ }. The particle抯 de Broglie matter wave has frequency _{ } and wavelength _{ }, where _{ } is the Planck constant. The particle抯 velocity _{ } must be the same as its matter wave抯 velocity _{ }. Otherwise, the particle would be separated from its own matter wave and the separation would be farther and farther over time. Hence, there must be _{ } or _{ }, which is exactly the physical quantity 搆inetic energy possessed by moving massdiscovered by the new relativistic mechanics. Both Newtonian mechanics and Einstein抯 mechanics do not have this physical quantity _{ }. Newtonian kinetic energy _{ } and momentum _{ } lead to _{ }. So, Newtonian mechanics is incompatible with the de Broglie matter wave theory. This is because the Newtonian mechanics is nonrelativistic whereas the quantum mechanics, with which we study relatively moving particles, must be relativistic. If Einstein抯 kinetic energy _{ } is used as _{ } in the matter wave抯 frequency formula _{ }, then _{ }, unless _{ }. However, according to Einstein抯 mechanics, no ponderable particles can move as fast as _{ }. If Einstein抯 total energy _{ } is used as _{ }, then again _{ }, unless _{ }. Obviously, Einstein抯 mechanics does not conform with the de Broglie matter wave theory either. Physicists have so far not been able to clearly explain the relationship between a particle抯 velocity _{ } and its matter wave抯 phase velocity _{ } due to the lack of a physical quantity _{ }. They explain moving particles as wave packets. The explanation is questionable (for more see .1). Only our new relativistic mechanics is able to eventually solve this old puzzle: _{ } in the matter wave抯 frequency formula _{ } is the kinetic energy _{ } possessed by a particle抯 moving mass _{ } so that the moving partcle抯 velocity and its own matter wave抯 phase velocity are always the same _{ }. This is precisely true for all kinds of particles, including photons, with any velocity from _{ } to _{ }. A moving particle抯 total energy is _{ }, in which its moving massenergy _{ } represents its corpuscularity and the kinetic energy possessed by its moving mass _{ } generates its wave property. A static particle (_{ }) does not demonstrate any wave property but preserves its corpuscularity. Particles display their wavecorpuscular duality only when they are moving. Moving at sublight speed (_{ }), a particle抯 corpuscularity overwhelms its wave property (_{ }). A particle with superlight speed (_{ }) has overwhelming wave property (_{ }). Photons and any particles moving with the speed of light (_{ }) have balanced wave corpuscular duality (_{ }). Moving at the speed _{ }, a photon抯 moving mass _{ }possesses kinetic energy _{ } and its matter wave抯 frequency is _{ }, which is exactly its electromagnetic wave抯 frequency. Thus, the new relativistic mechanics unites the electromagnetic wave of light with photon抯 matter wave. The light is both electromagnetic wave and matter wave. A photon has total energy _{ }, in which one _{ } is the photon抯 moving massenergy embodying its ponderable corpuscularity and another _{ } is the phonton抯 kinetic energy possessed by its moving mass to display its wave property. Since _{ }, so a photon抯 total energy is _{ }=_{ }, not _{ }. Interpretation of the Hydrogen Atom抯 Quantum Structure .1. Interpretation Based on the Matter Wave Theory and the New Relativistic Mechanics. According to the new relativistic mechanics, an electron (static mass _{ }) moving with constant speed _{ } has moving mass _{ }, momentum _{ } and kinetic energy of its moving mass _{ }. Its matter wave抯 frequency and wavelength are _{ } and _{ }. Its matter wave抯 velocity _{ }, so the electron always moves together with its own matter wave. Suppose the electron moves along a circular orbit (see .3.2) of radius _{ } around an hydrogen atom抯 nucleus. In order to avoid the matter wave抯 selfinterference, the length of one orbital round must be _{ } or _{ }, where _{ } is the integral number of matter waves on one orbital round and also is the orbit抯 quantum number. The electron抯 orbital period is _{ }_{ }, its orbital frequency is _{ }. Since there are _{ } matter waves on one round of orbit during an orbital period _{ }, so the matter wave抯 frequency is _{ }. Thus, _{ }, which again proves that the energy _{ } in the matter wave抯 frequency formula _{ } is exactly the physical quantity _{ } discovered by our new relativistic mecahnics. Moving with constant velocity _{ } along a circular orbit inside a hydrogen atom, an electron is under a balanced action of centrifugal force and the Coulomb forceof attraction: _{ }, where _{ } is the electron抯 charge. Thus, _{ }. The electron抯 all physical quantities are quantized: _{ }, _{ }, _{ }, _{ }, _{ }, _{ }, _{ }, _{ }. In case of _{ }, the electron has minimal orbital radius _{ }m (the Bohr radius) and maximal velocity _{ }m/s<<_{ }, so that _{ }. The less the quantum number (_{ }), the shorter the orbit抯 radius (_{ }) and the larger the electron抯 velocity and kinetic energy (_{ }，_{ }). When the electron transits from an inner orbit _{ } into an outer orbit _{ }, it releases its extra energy (_{ }) to radiate a photon of frequency _{ }. A photon抯 total energy is _{ }, not _{ } (see ). Therefore, _{ } or _{ }. This is the Balmer formula. ^ ics. It is well known that a single body or a single system alone by itself does not possess any potential energy. A body in a pair with another body has certain potential energy with regard to the other. In a pair as a whole, however, the two bodiespotential energies are mutually offset. The pair as a whole is a single system and does not have any potential energy available to be released out by the system. A hydrogen atom includes a nucleus and an electron moving around the nucleus. The electron has potential energy _{ } with regard to the nucleus, where _{ } is the distance from the electron to the nucleus and _{ } is the constant of the Coulomb attraction. The nucleus has potential energy _{ } with regard to the electron, where _{ } is the distance from the nucleus to the electron. Since _{ } and _{ }, the two potential energies inside of the hydrogen atom are mutually offset so that the hydrogen atom as a whole system does not have any potential energy available to be radiated. A hydrogen atom抯 total energy includes its massenergy, which remains constant, and the moving electron抯 kinetic energy which depends on the electron抯 velocity. The energy available for the radiation may come only from the electron抯 transition from an inner orbit, where it has larger velocity (because of shorter radius) and larger kinetic energy, into an outer orbit where it has less velocity (because of larger radius) and less kinetic energy. Classical theory contains two mistakes: (1) It assumes the Newtonian kinetic energy _{ } as a part of the energy to be radiated, which causes the electron抯 separation from its own matter wave; (2) It assumes the electron抯 potential energy _{ }, nullified by the potential energy of the nucleus, as another part of the energy to be radiated. Therefore, _{ } (_{ } for the hydrogen atom). Since _{ } and _{ }, so _{ }. Classical theory has to accept the unreasonable negative energy _{ }, which leads to a wrong direction of the electron抯 transition for the hydrogen to be able to radiate energy: The hydrogen atom radiates as the moving electron releases its energy when it transits from an outer orbit with less negative _{ } into an inner orbit with more negative _{ }. Worse, viewed from the de Broglie matter wave theory which appeared later, the moving electron抯 negative energy _{ } means it抯 matter wave has not only negative frequency _{ } but also negative phase velocity _{ } so that the electron and its matter wave would move apart in opposite directions. Moreover, the classical theory has to maintain that a photon抯 total energy is _{ }, not _{ }, which denies a photon of its waveparticle duality (see ). Only by this wrong way, Bohr抯 classical theory can obtain the Balmer formula: _{ } or _{ }. ^ ple. Bohr built the hydrogen atom抯 quantum structure in 1913 by use of his correspondence principle. Because, at that time, the de Broglie matter wave theory had not been born yet. We will first discuss Bohr抯 approach in a detailed way and then show our new approach based on the new relativistic mechanics,. ^ roach. Bohr assumes the Keplerian elliptic orbit for a charged particle to move around a nucleus and proposes a relationship _{ } between the charged particle抯 energy _{ } and its orbital frequency _{ } on an orbit with the quantum number _{ }. He also defines the orbit抯 conservative energy as _{ }, where _{ } is the charged particle抯 potential energy at a distance _{ } from the nucleus, _{ } is its mass and _{ } is its nonrelativistic Newtonian kinetic energy. The elliptic orbit requires the orbit抯 conservative energy _{ }. Because, if _{ }, then _{ }, the charged particle would move radially with regard to the nucleus; if _{ }, then the particle would escape from the nucleus along a hyperbolic path. It is well known that, given an orbit抯 conservative energy _{ }, the conservation laws of the orbit抯 angular momentum and the orbit抯 conservative energy leads to a whole family of elliptic orbits with common semimajor axis _{ }, common conservative energy _{ } and common orbital frequency _{ }. It is important to note, however, _{ } and _{ } are irrelevant to semiminor axis _{ }. Therefore, the family contains countless elliptic orbits with the same _{ } but different _{ }, including orbits with _{ }. A charged particle may move along an elliptic orbit with semiminor axis _{ } and appear absurdly close to or even collide with the nucleus! Next, Bohr assumes that the orbit抯 coservative energy _{ } and the charged particle抯 energy _{ } are the same so that the particle has negative energy _{ }! ^ _{ } only decides the orbit抯 form; it has nothing to do with the charged particle抯 energy _{ }. Further, by use of his correspondence principle, he deduces_{ } (see Appendix ) and obtains: _{ }<0. The combination of _{ } with the above equation gives: _{ } and _{ }. For the electron in the hydrogen atom, _{ } and _{ }; so he has _{ }. Bohr抯 confusing assumption of _{ }=_{ } makes the electron having negative energy _{ }. If _{ }, then_{ }, which forces him to conclude that the hydrogen atom radiates when its electron transits from an outer orbit into an inner orbit, despite the fact that the less the orbit抯 radius the larger the electron抯 velocity and kinetic energy. Bohr accepts Einstein抯 theory of photoelectric quantum, according to which a photon of frequency _{ } has total energy _{ }, not 2_{ }. Finally, he obtains the Balmer formula: _{ } or _{ }. Bohr抯 approach suffers from five shortcomings: (1) His elliptic orbit may cause the electron抯 collision with the nucleus; (2) He confuses the electron抯 energy _{ } with the elliptic orbit抯 conservative energy _{ }, which renders the electron to have negative energy _{ } because of _{ }; (3) He accepts electron抯 nonrelativistic Newtonian kinetic energy, which separates the electron from its own matter wave; (4) He deems the potential energy between the electron and the nucleus, which as a whole is mutually nullified, as releasable energy; (5) He deems that a photon抯 total energy is _{ }, not _{ }, and thus denies photon抯 waveparticle duality. The five shortcomings lead to the wrong direction of the electron抯 transition, although superficially he seems to have obtained the Balmer formula. ^ heory. Based on the matter wave theory, we can prove that, moving around a nucleus, a charged particle must have circular orbits. In polar coordinates, the equation of an ellipse is _{ }, where _{ } and _{ } are semimajor and semiminor axes respectively, _{ } is semifocal distance, _{ } is the argument. Since an elliptic orbit抯 conservative energy is _{ }, so _{ } or _{ }. According to the new relativistic mechanics, the matter wave抯 frequency is _{ }. As the velocity _{ } varies along the orbit, the matter wave抯 frequency _{ } varies accordingly: _{ }. For a whole round of an elliptic orbit, the value of the matter wave抯 frequency is: _{ }. On the other hand, for an orbit with quantum number _{ }, there must be the same integral _{ } number of matter waves forming a smooth headtail continuity on a whole round of the orbit in order to avoid the matter wave抯 selfinterference. If a particle抯 orbital period is _{ } and one round of its orbit has _{ } matter waves, then its matter wave抯 frequency is _{ }, where _{ } is the orbital frequency. However, as above mentioned , the family of elliptic orbits has common orbital frequency _{ } and common semimajor axis _{ }. Therefore, _{ } and _{ }. From these two equations, we can obtain _{ } or _{ } so that _{ } and _{ } so that _{ } or _{ } which is exactly a charged particle抯 kinematic equation of circular motion in a central Coulomb force field. This means that, moving around a nucleus, a charged particle is destined to have only circular orbits. Thus, a correct approach must first assume circular orbits instead of the Keplerian elliptic orbits. All circular orbits have zero conservative orbital energy _{ }. Being always zero, _{ } cannot be the source of energy radiated by an atom due to its electron抯 transition from one circular orbit to another circular orbit. Let抯 stress again: Orbit抯 coservative energy _{ } (_{ }, _{ } or _{ }) decides only the orbit抯 form but has nothing to do with the energy radiated by an atom. The source of the radiated energy is the electron抯 kinetic energy _{ } which varies from one circular orbit to another circular orbit. _{ } must not be confused with the orbit抯 conservative energy _{ } which is always a constant zero for all circular orbits. Let a circular orbit with energy level _{ } have radius _{ }. In Appendix , by use of the correspondence principle, we have proven _{ }. According to the new relativistic mechanics, on the other hand, the kinetic energy of a moving particle is _{ } (for _{ }) and its kinematic equation on a circular orbit of radius _{ } is _{ }. So, _{ }. Thus, the combination of _{ } with _{ } gives: _{ }>0 and _{ }. For the electron in the hydrogen atom, _{ } and _{ } so that: _{ } and _{ }. The smaller the quantum number _{ } the smaller the orbit抯 radius _{ } and, correspondingly, the larger the electron抯 velocity _{ } and kinetic energy _{ }. Thus, the hydrogen atom radiates energy when its electron transits from an inner _{ } orbit into an outer _{ } orbit (_{ }). According to the new relativistic mechanics, a photon of frequency _{ } has total energy _{ }. Finally, we obtain the Balmer formula with the correct direction of an electron抯 transition: _{ } or _{ }. To sum up, our conclusions are: (1) The hydrogen atom抯 electron can only have quantized circular orbits. Only circular orbits can match the correspondence principle with the matter wave theory. Elliptic orbits may cause the electron to break the minimal Bohr radius or even to collide with the nucleus. Historically, Schrodinger tried and failed to find out a nonexpansive wave packet of electron moving along a classical Keplerian elliptic orbit in the hydrogen atom. Our approach proves that his failure is destined by his assumption of the elliptic orbit. (2) The correspondence principle is correct and effective per se. What is wrong, besides the assumption of elliptic orbits with negative 