Translations in quantum mechanics revisited. The point spectrum case

2016 ◽  
Vol 94 (12) ◽  
pp. 1365-1368 ◽  
Author(s):  
Armando Martínez-Pérez ◽  
Gabino Torres-Vega
Keyword(s):  
Quantum Mechanics ◽  
Point Spectrum ◽  
Time Operator

We study translations in quantum mechanics for the case of a point spectrum, including translations by non-allowed amounts. We find that we obtain a copy of the original interval if we want to move to the outside of it, or to a mixture of states when moving to non-spectrum values (i.e., an interpolation eigenfunction). These results will clarify the meaning of the Pauli statement about the existence of a time operator in quantum mechanics.

Conditional interpretation of time in quantum gravity and a time operator in relativistic quantum mechanics

2020 ◽  
Vol 35 (21) ◽  
pp. 2050114
Author(s):  
M. Bauer ◽  
C. A. Aguillón ◽  
G. E. García
Keyword(s):  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Relativistic Quantum ◽  
Canonical Quantization ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Problem Of Time ◽  
Quantization Of Gravity ◽  
Dynamical Time ◽  
Spatial Coordinates

The problem of time in the quantization of gravity arises from the fact that time in Schrödinger’s equation is a parameter. This sets time apart from the spatial coordinates, represented by operators in quantum mechanics (QM). Thus “time” in QM and “time” in general relativity (GR) are seen as mutually incompatible notions. The introduction of a dynamical time operator in relativistic quantum mechanics (RQM), that follows from the canonical quantization of special relativity and that in the Heisenberg picture is also a function of the parameter [Formula: see text] (identified as the laboratory time), prompts to examine whether it can help to solve the disfunction referred to above. In particular, its application to the conditional interpretation of time in the canonical quantization approach to quantum gravity is developed.

scholarly journals A dynamical time operator in Dirac's relativistic quantum mechanics

2014 ◽  
Vol 29 (06) ◽  
pp. 1450036 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Phase Velocity ◽  
Relativistic Quantum ◽  
Rate Of Change ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Instantaneous Rate ◽  
Expectation Value ◽  
Dynamical Time

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.

scholarly journals de Broglie clock, electron channeling, and time in quantum mechanics

2019 ◽  
Vol 97 (1) ◽  
pp. 37-41 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Resonance Energy ◽  
Intrinsic Property ◽  
Theoretical Description ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Electron Channeling ◽  
Quantum Mechanical Description ◽  
Dynamical Time

De Broglie’s association of a wave to particles is a fundamental concept in the quantum mechanical description of nature. The wave oscillation is referred to alternatively as the “de Broglie clock”, the “Compton clock”, or the “de Broglie periodic phenomenon”. In the present paper it is shown that Dirac’s relativistic quantum mechanics, complemented with the dynamical time operator recently introduced, provides a consistent theoretical description of: (i) the generation of the de Broglie wave through Lorentz boosts; and (ii) the characteristics of the resonance observed in electron channeling through thin crystals as responding to both the periodicity derived from the adjustment of the de Broglie period to the crystal interatomic distance (resonance energy) and the periodicity of the predicted trembling motion (Zitterbewegung). One can conclude that the channeling experiments provide the first direct evidence of the electron Zitterbewegung, and that the de Broglie period is an intrinsic property of matter arising from a self-adjoint dynamical time operator.

scholarly journals Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1583
Author(s):  
Sara Cruz y Cruz ◽  
Alejandro Romero-Osnaya ◽  
Oscar Rosas-Ortiz
Keyword(s):  
Refractive Index ◽  
Quantum Mechanics ◽  
Optical Waveguides ◽  
Point Spectrum ◽  
Optical Power ◽  
Refractive Indices ◽  
Exactly Solvable ◽  
Gain And Loss ◽  
Time Invariant ◽  
Complex Valued

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.

A time operator in the simulations of the Dirac equation

2019 ◽  
Vol 34 (22) ◽  
pp. 1950114 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Einstein Condensate ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Electron Motion ◽  
Klein Paradox ◽  
Bose Einstein Condensate ◽  
Bose Einstein

Simulations of the Dirac equation have allowed to mimic measurably the predicted unusual characteristics of the electron motion, e.g. Zitterbewegung and Klein paradox that are beyond current technical capabilities. In this paper, it is shown that a Bose–Einstein condensate experiment carried out corroborates these results, but in addition exhibits a particular feature of an observable represented by a Dirac self-adjoint time operator introduced in relativistic quantum mechanics.

scholarly journals TIME IN RELATIVISTIC AND NONRELATIVISTIC QUANTUM MECHANICS

2009 ◽  
Vol 07 (03) ◽  
pp. 595-602 ◽  
Author(s):  
HRVOJE NIKOLIĆ
Keyword(s):  
Quantum Mechanics ◽  
Time Operator ◽  
Position Operator ◽  
Nonrelativistic Quantum ◽  
Time Dynamics ◽  
Bell Theorem ◽  
Time Position ◽  
Nonrelativistic Quantum Mechanics ◽  
Bohmian Interpretation

The kinematic time operator can be naturally defined in relativistic and nonrelativistic quantum mechanics (QM) by treating time on an equal footing with space. The space–time position operator acts in the Hilbert space of functions of space and time. Dynamics, however, makes eigenstates of the time operator unphysical. This poses a problem for the standard interpretation of QM and reinforces the role of alternative interpretations such as the Bohmian one. The Bohmian interpretation, despite of being nonlocal in accordance with the Bell theorem, is shown to be relativistic covariant.

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