An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schr<span style="white-space:nowrap;">ö</span>dinger and the Klein-Gordon equatio...An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schr<span style="white-space:nowrap;">ö</span>dinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.展开更多
Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained by abduction. These equations, in contrast with the Klein-Gordon and other relativistic quantum me...Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained by abduction. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.展开更多
Some consequences, due to the existence of a pair of decoupled Schrödinger-like but relativistic quantum mechanics wave equations, are explored. It is shown that one equation directly describes the quantum st...Some consequences, due to the existence of a pair of decoupled Schrödinger-like but relativistic quantum mechanics wave equations, are explored. It is shown that one equation directly describes the quantum states of a single spin-0 particle, and the other one indirectly describes the quantum states of the corresponding antiparticle. In correspondence with the matter-antimatter symmetry, for a Coulomb potential, a charge conjugation operation transforms the second equation in the first one. However, if a particle could interact with itself (gravitationally or electrically) due to the spread of its wavefunction, the C-symmetry could be broken;therefore, matter and antimatter could be distinguished. Under these assumptions, it is deducted the impossibility of the existence of particles and antiparticles with a mass larger than the Plank mass (m<sub>P</sub>), or with the absolute value of the charge larger than the Plank charge (q<sub>P</sub>). It is proposed the existence of primordial antimatter electrical sinks. It is also suggested that all macroscopic matter objects with a mass m > m<sub>P</sub>, and all macroscopic antimatter bodies with a charge |q| > q<sub>P</sub> should not be quantum but classical objects. It is argued that these findings could explain the absence of antimatter with a complicated structure and partially explain the excess of charged matter in the known Universe.展开更多
Using a novel wave equation, which is Galileo invariant but can give precise results up to energies<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style=&...Using a novel wave equation, which is Galileo invariant but can give precise results up to energies<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> as high as </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><i><span style="font-family:Verdana;">mc</span></i></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><sup>2</sup></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">, exact quasi-relativistic quantum mechanical solutions are found for the Hydrogen atom. It is shown that the exact solutions of the Grave de Peralta equation include the relativistic correction to the non-relativistic kinetic energies calculated using the Schr</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="line-height:102%;font-family:Verdana;"><span style="white-space:nowrap;">ö</span></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">dinger equation.</span></span></span>展开更多
The basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. This is achieved by using a Schr<span style="white-space:nowra...The basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. This is achieved by using a Schr<span style="white-space:nowrap;">?</span>dinger-like equation, which describes a particle with mass and spin-0 and with the correct relativistic relation between its linear momentum and kinetic energy. Some simple but instructive free particle examples are discussed.展开更多
文摘An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schr<span style="white-space:nowrap;">ö</span>dinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.
文摘Using the Hamilton-Jacobi and the Lagrange formalisms, a pair of relativistic quantum mechanics equations are obtained by abduction. These equations, in contrast with the Klein-Gordon and other relativistic quantum mechanics equations, have no solutions with both positive and negative kinetic energies. The equation with solutions with only positive kinetic energy values describes a spin-0 particle of mass m, which is moving at relativistic speeds in a scalar potential. The wavefunctions and the energies corresponding to the associated antiparticle can be obtained by solving the other equation, which only has solutions with negative kinetic energy values.
文摘Some consequences, due to the existence of a pair of decoupled Schrödinger-like but relativistic quantum mechanics wave equations, are explored. It is shown that one equation directly describes the quantum states of a single spin-0 particle, and the other one indirectly describes the quantum states of the corresponding antiparticle. In correspondence with the matter-antimatter symmetry, for a Coulomb potential, a charge conjugation operation transforms the second equation in the first one. However, if a particle could interact with itself (gravitationally or electrically) due to the spread of its wavefunction, the C-symmetry could be broken;therefore, matter and antimatter could be distinguished. Under these assumptions, it is deducted the impossibility of the existence of particles and antiparticles with a mass larger than the Plank mass (m<sub>P</sub>), or with the absolute value of the charge larger than the Plank charge (q<sub>P</sub>). It is proposed the existence of primordial antimatter electrical sinks. It is also suggested that all macroscopic matter objects with a mass m > m<sub>P</sub>, and all macroscopic antimatter bodies with a charge |q| > q<sub>P</sub> should not be quantum but classical objects. It is argued that these findings could explain the absence of antimatter with a complicated structure and partially explain the excess of charged matter in the known Universe.
文摘Using a novel wave equation, which is Galileo invariant but can give precise results up to energies<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> as high as </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><i><span style="font-family:Verdana;">mc</span></i></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><sup>2</sup></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">, exact quasi-relativistic quantum mechanical solutions are found for the Hydrogen atom. It is shown that the exact solutions of the Grave de Peralta equation include the relativistic correction to the non-relativistic kinetic energies calculated using the Schr</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="line-height:102%;font-family:Verdana;"><span style="white-space:nowrap;">ö</span></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">dinger equation.</span></span></span>
文摘The basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. This is achieved by using a Schr<span style="white-space:nowrap;">?</span>dinger-like equation, which describes a particle with mass and spin-0 and with the correct relativistic relation between its linear momentum and kinetic energy. Some simple but instructive free particle examples are discussed.
