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Physical Review A

Interference of nonzero-rest-mass particles with time-varying energy and momentum

Frank V. Kowalski, Benjamin A. Kowalski

Published Jul 26, 2010• 2 citations
Publisher

Abstract

Solutions to the Schr\"odinger and Klein-Gordon equations for retroreflection of a nonzero-rest-mass particle from an accelerating mirror are determined using a retarded-phase method. The output of a Michelson interferometer, in which one mirror moves, reveals differences between the predictions of these two equations in the nonrelativistic limit when the energy and momentum of the reflected particle both vary with time.

DOI
10.1103/physreva.82.012110

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Solutions to the Schr"odinger and Klein-Gordon equations for retroreflection of a nonzero-rest-mass particle from an accelerating mirror are determined using a retarded-phase method. The output of a Michelson interferometer, in which one mirror moves, reveals differences between the predictions of these two equations in the nonrelativistic limit when the energy and momentum of the reflected particle both vary with time.
Adding a constant energy offset leaves classical dynamics unchanged. In quantum mechanics it changes the phase velocity of the wavefunction. The inclusion of the constant rest energy in the Klein-Gordon formulation leads to significantly higher phase velocities compared with the Schrödinger equation. The Schrödinger equation predicts an attenuation of the wavefunction along one of the paths in a Sagnac interferometer when a beamsplitter's trajectory along that path includes a segment where its speed exceeds the phase velocity of a free particle. Such an attenuation does not occur for electromagnetic waves nor for eigenstates of momentum in the Klein-Gordon equation since the speed of the beamsplitter cannot then exceed the phase velocity of the wave. This attenuation reduces the amplitude without introducing a phase shift, preserving the overall structure of the transmitted wave group. While a Klein-Gordon wave group undergoes three traversals of the beamsplitter that moved, it experiences attenuation equivalent to only a single pass, whereas the Schrödinger equation predicts the expected attenuation for three passes. This discrepancy highlights a fundamental incompatibility between the Klein-Gordon equation and the Schrödinger equation in a regime where their predictions should converge.
A system of two equations is found that has solutions which coincide with the solutions of the Klein‐Gordon equation in the rest frame. This system includes the Schrödinger equation for a free neutral spinless particle. Using the Schrödinger equation as an additional condition for solving the Klein‐Gordon equation in the rest frame leads to two Helmholtz equations. Helmholtz equations can be solved by specifying a particle model and boundary conditions. One of the Helmholtz equations leads to discreteness of the rest masses of relativistic particles.
In a previous note, we argued that exp(i Action) for a nonrelativistic particle may be varied independently in T and X, where velocity = X/T. This variation indicates that energy is not kept constant. The solution exp(i Action)= exp(i .5m XX/TT) (for this case) leads to the nonrelativistic Schrodinger equation. In this note, we consider the free particle relativistic Lagragian (1): mo sqrt(1-vv) where v=X/T. We argue that the equation describing independent fluctuations of X and T is the Klein-Gordon equation.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
Abstract This work revisits the apparent paradox that emerges from some results found in many basic textbooks which address the problem of calculating the phase velocity of free particles in the relativistic regime and its relation with the same quantity in the non-relativistic regime. Because some books use the Newtonian kinetic energy and others use the total relativistic energy (including the rest energy), the non-relativistic limit of the relativistic phase velocity differs from the Newtonian phase velocity, which can be confusing for many students. This paper proposes expressing the phase velocity by first subtracting the rest energy from the relativistic energy before imposing the usual de Broglie relations. In addition to providing the corresponding Newtonian phase velocity, it also provides a relativistic Schrödinger equation which intuitively reduces to the non-relativistic Schrödinger equation in the appropriate regime. The problem of the equivalence between the relativistic Schrödinger equation and the Klein–Gordon equation is also addressed by using arguments which can be understood by undergraduate students.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.