Reasons are given for believing that the boundary conditions at a discontinuity in the potential ( $V$ ) in wave mechanics may be ${\ensuremath{\varphi}}_{1}={\ensuremath{\varphi}}_{2}{\ensuremath{\mu}}^{\frac{1}{2}}$ and $\frac{{\ensuremath{\varphi}}_{1}d{\ensuremath{\varphi}}_{1}}{\mathrm{dx}}=\frac{{\ensuremath{\varphi}}_{2}d{\ensuremath{\varphi}}_{2}}{\mathrm{dx}}$. Here $\ensuremath{\varphi}$ is the wave function and $\ensuremath{\mu}$ the ratio of the group velocities. With these boundary conditions the waves are either completely transmitted or completely reflected according as the energy $E$ is greater or less than $V$ . The particles therefore need not be supposed to enter the region in which $V>E$ where their velocities would be imaginary and their kinetic energies negative. It is suggested that the group of waves associated with a particle should be identified with the particle and that the energy and momentum of the particle are the energy and momentum of the group of waves. When the group is stopped or absorbed it contracts to small dimensions and so becomes a particle.Reasons are given for believing that the boundary conditions at a discontinuity in the potential ($V$) in wave mechanics may be ${\ensuremath{\varphi}}_{1}={\ensuremath{\varphi}}_{2}{\ensuremath{\mu}}^{\frac{1}{2}}$ and $\frac{{\ensuremath{\varphi}}_{1}d{\ensuremath{\varphi}}_{1}}{\mathrm{dx}}=\frac{{\ensuremath{\varphi}}_{2}d{\ensuremath{\varphi}}_{2}}{\mathrm{dx}}$. Here $\ensuremath{\varphi}$ is the wave function and $\ensuremath{\mu}$ the ratio of the group velocities. With these boundary conditions the waves are either completely transmitted or completely reflected according as the energy $E$ is greater or less than $V$. The particles therefore need not be supposed to enter the region in which $V>E$ where their velocities would be imaginary and their kinetic energies negative. It is suggested that the group of waves associated with a particle should be identified with the particle and that the energy and momentum of the particle are the energy and momentum of the group of waves. When the group is stopped or absorbed it contracts to small dimensions and so becomes a particle.

Physical ReviewApr 15, 193024 citations

Propagation of Wave Packets and Concept of Group Velocity

Ajoy Kumar Ghatak, S. Lokanathan

Springer eBooksJan 1, 20040 citations

Negative Velocity Characteristics in Electromagnetism

Huang Zhi-xun

Newton mechanics is great, Newton's contribution is indelible. But from Newton's classical mechanics, the definition of velocity (=) must mean that this physical parameter is a vector, so "negative velocity" only means the opposite direction of motion, no other meaning. It is difficult to understand negative velocity within the framework of classical mechanics. However, wave is a special form of material motion, wave mechanics has a unique method and meaning, and its concept and connotation are significantly different from classical mechanics. For example, wave velocity (whether phase velocity or group velocity) is a scalar quantity. "Negative wave velocity" does not mean that the direction of motion is reversed, but a special phenomenon that does not accord with causality from the surface view. In any case, research in recent decades has shown that negative wave velocity is not only theoretically possible, it has also been repeatedly shown to exist experimentally. Moreover, negative wave velocity is a special form of superluminal speed; The wave with negative velocity is the advanced wave. It corresponds to the leading solution of the basic equation of electromagnetic field and electromagnetic wave. The past practice (discarding the advanced solution) is wrong! As for causality, scientists have provided a new definition and interpretation. In short, both wave mechanics and quantum optics take a different approach from classical mechanics. This paper points out that it is unusual for the 2022 Nobel Prize in Physics to be awarded to Alain Aspect and two others, since Aspect's experiments on the Bell inequality were completed in 1982, Which still stands today as a crucial experiment that proved Einstein's EPR paper wrong and quantum mechanics (QM) correct. Aspect's award in 2022 showed that mainstream physical community had been forced to accept that quantum entanglement existed and that the "light-speed limit" theory of SR was a mistake. This paper discusses the proposition "negative characteristic motion of electromagnetic wave" put forward by the author in 2013, pointing out that it is an inherent physical phenomenon reflecting symmetry in nature. In this paper, the faster-than-light motion of waves found in the near-field of antennas is discussed. In addition, the realizability of "time travel" is also discussed.Newton mechanics is great, Newton's contribution is indelible. But from Newton's classical mechanics, the definition of velocity (=) must mean that this physical parameter is a vector, so "negative velocity" only means the opposite direction of motion, no other meaning. It is difficult to understand negative velocity within the framework of classical mechanics. However, wave is a special form of material motion, wave mechanics has a unique method and meaning, and its concept and connotation are significantly different from classical mechanics. For example, wave velocity (whether phase velocity or group velocity) is a scalar quantity. "Negative wave velocity" does not mean that the direction of motion is reversed, but a special phenomenon that does not accord with causality from the surface view. In any case, research in recent decades has shown that negative wave velocity is not only theoretically possible, it has also been repeatedly shown to exist experimentally. Moreover, negative wave velocity is a special form of superluminal speed; The wave with negative velocity is the advanced wave. It corresponds to the leading solution of the basic equation of electromagnetic field and electromagnetic wave. The past practice (discarding the advanced solution) is wrong! As for causality, scientists have provided a new definition and interpretation. In short, both wave mechanics and quantum optics take a different approach from classical mechanics. This paper points out that it is unusual for the 2022 Nobel Prize in Physics to be awarded to Alain Aspect and two others, since Aspect's experiments on the Bell inequality were completed in 1982, Which still stands today as a crucial experiment that proved Einstein's EPR paper wrong and quantum mechanics (QM) correct. Aspect's award in 2022 showed that mainstream physical community had been forced to accept that quantum entanglement existed and that the "light-speed limit" theory of SR was a mistake. This paper discusses the proposition "negative characteristic motion of electromagnetic wave" put forward by the author in 2013, pointing out that it is an inherent physical phenomenon reflecting symmetry in nature. In this paper, the faster-than-light motion of waves found in the near-field of antennas is discussed. In addition, the realizability of "time travel" is also discussed.

Physical Science International JournalMay 13, 20230 citations

Wave Mechanics

Marianne Breinig

Compendium of Quantum PhysicsJan 1, 20091 citation

Phase and Group Momenta from the Prince Equation: Eliminating Superluminal Phase Velocity

Mustafa Khan

We show that the de Broglie (Prince) equation λ p = h naturally produces two conjugate momenta—a group momentum p_group = mv_n and a phase momentum p_phase = mc^2/v—whose product equals the square of the photon momentum:p_phase ×p_group = p^2(ϕ).This reformulation eliminates the superluminal phase velocity problem that has plagued wave mechanics since 1924. We first derive the constancy of Planck’s constant from the classical work equation and the Leibniz product rule, requiring no quantum postulates. The photon boundary at v = c fixes H = h and eliminates the integration constant, rendering the assumption v = 0 unnecessary. Velocity quantization v_n = c (n−1)/(n+1) emerges from first principles, producing discrete momentum levels where pn = mvn recovers classical mechanics at each level. The imaginary unit 'i' emerges algebraically when the Prince equation forbids rest. By expressing the wave’s phase structure as a momentum rather than a velocity, the physical ontological reality of matter waves becomes manifest without any quantity exceeding c.We show that the de Broglie (Prince) equation λ p = h naturally produces two conjugate momenta—a group momentum p_group = mv_n and a phase momentum p_phase = mc^2/v—whose product equals the square of the photon momentum:p_phase ×p_group = p^2(ϕ).This reformulation eliminates the superluminal phase velocity problem that has plagued wave mechanics since 1924. We first derive the constancy of Planck’s constant from the classical work equation and the Leibniz product rule, requiring no quantum postulates. The photon boundary at v = c fixes H = h and eliminates the integration constant, rendering the assumption v = 0 unnecessary. Velocity quantization v_n = c (n−1)/(n+1) emerges from first principles, producing discrete momentum levels where pn = mvn recovers classical mechanics at each level. The imaginary unit 'i' emerges algebraically when the Prince equation forbids rest. By expressing the wave’s phase structure as a momentum rather than a velocity, the physical ontological reality of matter waves becomes manifest without any quantity exceeding c.

Feb 8, 20260 citations

Wave Mechanics

Stephen J. Nettel

Advanced texts in physicsJan 1, 20030 citations

Reasons are given for believing that the boundary conditions at a discontinuity in the potential (

V

) in wave mechanics may be ${\ensuremath{\varphi}}_{1}={\ensuremath{\varphi}}_{2}{\ensuremath{\mu}}^{\frac{1}{2}}$ and $\frac{{\ensuremath{\varphi}}_{1}d{\ensuremath{\varphi}}_{1}}{\mathrm{dx}}=\frac{{\ensuremath{\varphi}}_{2}d{\ensuremath{\varphi}}_{2}}{\mathrm{dx}}$. Here $\ensuremath{\varphi}$ is the wave function and $\ensuremath{\mu}$ the ratio of the group velocities. With these boundary conditions the waves are either completely transmitted or completely reflected according as the energy

E

is greater or less than

V

. The particles therefore need not be supposed to enter the region in which

V>E

where their velocities would be imaginary and their kinetic energies negative. It is suggested that the group of waves associated with a particle should be identified with the particle and that the energy and momentum of the particle are the energy and momentum of the group of waves. When the group is stopped or absorbed it contracts to small dimensions and so becomes a particle.