scholarly journals “Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–Nordström Black Hole

2020 ◽  
Vol 2 (3) ◽  
pp. 414-441
Author(s):  
Theodoros Pailas
Keyword(s):  
Black Hole ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Canonical Quantization ◽  
Hamiltonian Density ◽  
Reduced Equations ◽  
Spacetime Singularity ◽  
Classical Singularity ◽  
Pp Wave ◽  
Do So

A “time”-covariant Schrödinger equation is defined for the minisuperspace model of the Reissner–Nordström (RN) black hole, as a “hybrid” between the “intrinsic time” Schrödinger and Wheeler–DeWitt (WDW) equations. To do so, a reduced, regular, and “time(r)”-dependent Hamiltonian density was constructed, without “breaking” the re-parametrization covariance r→f(r˜). As a result, the evolution of states with respect to the parameter r and the probabilistic interpretation of the resulting quantum description is possible, while quantum schemes for different gauge choices are equivalent by construction. The solutions are found for Dirac’s delta and Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside this, a criterion is presented to adjudicate which, between two singular spacetimes, is “more” or “less” singular. Two ways to adjudicate the existence of singularities are compared (vanishing of the probability density at the classical singularity and semi-classical spacetime singularity). Finally, an equivalence of the reduced equations with those of a 3D electromagnetic pp-wave spacetime is revealed.

scholarly journals Time-covariant Schrödinger equation and invariant decay probability: the $$\Lambda $$-Kantowski–Sachs universe

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Theodoros Pailas ◽  
Nikolaos Dimakis ◽  
Petros A. Terzis ◽  
Theodosios Christodoulakis
Keyword(s):  
Schrödinger Equation ◽  
Wave Packet ◽  
Time Dependence ◽  
Schrodinger Equation ◽  
Canonical Quantization ◽  
Dynamical Equations ◽  
Quadratic Constraint ◽  
Decay Probability ◽  
Flrw Universe ◽  
Arbitrary Time Dependence

AbstractThe system under study is the $$\Lambda $$ Λ -Kantowski–Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minisuperspace Lagrangian describing the system, is reduced to a regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) possessing an explicit (though arbitrary) time dependence; thus a time-covariant Schrödinger equation arises. Additionally, an invariant (under transformations $$t=f({\tilde{t}})$$ t = f ( t ~ ) ) decay probability is defined and thus “observers” which correspond to different gauge choices obtain, by default, the same results. The time of decay for a Gaussian wave packet localized around the point $$a=0$$ a = 0 (where a the radial scale factor) is calculated to be of the order $$\sim 10^{-42}{-}10^{-41}~\text {s}$$ ∼ 10 - 42 - 10 - 41 s . The acquired value is near the end of the Planck era (when comparing to a FLRW universe), during which the quantum effects are most prominent. Some of the results are compared to those obtained by following the well known canonical quantization of cosmological systems, i.e. the solutions of the Wheeler–DeWitt equation.

Soliton solutions to the nonlocal non-isospectral nonlinear Schrödinger equation

2020 ◽  
Vol 34 (25) ◽  
pp. 2050219 ◽  
Author(s):  
Wei Feng ◽  
Song-Lin Zhao
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Reverse Time ◽  
Nonlinear Schrödinger ◽  
Reduced Equations ◽  
Double Wronskian ◽  
The One

In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.

Solitons to the derivative nonlinear Schrödinger equation: Double Wronskians and reductions

2021 ◽  
pp. 2150410
Author(s):  
Shu-Zhi Liu ◽  
Hua Wu
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Reduction Technique ◽  
Bilinear Method ◽  
Nonlinear Schrödinger ◽  
Reduced Equations ◽  
Double Wronskian ◽  
Derivative Nonlinear Schrödinger Equation

In this paper, we derive solutions to the derivative nonlinear Schrödinger equation, which are associated to real and complex discrete eigenvalues of the Kaup–Newell spectral problem. These solutions are obtained by investigating double Wronskian solutions of the coupled Kaup–Newell equations and their reductions by means of bilinear method and a reduction technique. The reduced equations include the derivative nonlinear Schrödinger equation and its nonlocal version. Some obtained solutions allow not only periodic behavior, but also solitons on periodic background. Dynamics are illustrated.

Superstatistics and canonical quantization of the damped harmonic oscillator

2019 ◽  
Vol 34 (14) ◽  
pp. 1950108 ◽  
Author(s):  
S. Sargolzaeipor ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Thermodynamic Properties ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Canonical Quantization ◽  
Quantization Method ◽  
Dirac Delta ◽  
Energy Eigenvalues ◽  
Damped Harmonic Oscillator ◽  
Gamma Distributions

In this paper, we investigate the behavior of the energy eigenvalues of the Schrödinger equation by using the canonical quantization method. We obtain the Hamiltonian of the Schrödinger equation by the Lagrangian in terms of the new coordinates. Then we calculate the partition function by the eigenvalues and the thermodynamic properties of the system in the superstatistics formalism for the modified Dirac delta and the Gamma distributions. All results in the limiting cases satisfy that of the harmonic oscillator. Furthermore, the effects of the all parameters in the problem of energy eigenvalues and thermodynamic properties are calculated and shown graphically.

ON THE WHEELER-DEWITT EQUATION FOR BLACK HOLES

1994 ◽  
Vol 03 (03) ◽  
pp. 579-591 ◽  
Author(s):  
M.D. POLLOCK
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Apparent Horizon ◽  
Hamiltonian Formalism ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Two Dimensional ◽  
Superstring Theory

Integration over the angular coordinates of the evaporating, four-dimensional Schwarzschild black hole leads to a two-dimensional action, for which the Wheeler-DeWitt equation has been found by Tomimatsu, on the apparent horizon, where the Vaidya metric is valid, using the Hamiltonian formalism of Hajicek. For the Einstein theory of gravity coupled to a massless scalar field ζ, the wave function Ψ obeys the Schrödinger equation [Formula: see text], where M is the mass of the hole. The solution is [Formula: see text], where k2 is the separation constant, and for k2>0 the hole evaporates at the rate Ṁ=−k2/4M2, in agreement with the result of Hawking. Here, this analysis is generalized to the two-dimensional theory [Formula: see text], which subsumes the spherical black holes formulated in D≥4 dimensions, when A = ½ (D - 2) (D - 3)ϕ2 (D - 4)/(D - 2), B=2(D−3)/(D−2), C=1, and also the twodimensional black hole identified by Witten and by Gautam et al., when A=4/α′, B=2, C=1/8π, c=+8/α′ being (minus) the central charge. In all cases an analogous Schrödinger equation is obtained. The evaporation rate is [Formula: see text] when D≥4 and [Formula: see text] when D=2. Since Ψ evolves without violation of unitarity, there is no loss of information during the evaporation process, in accord with the principle of black-hole complementarity introduced by Susskind et al. Finally, comparison with the four-dimensional, cosmological Schrödinger equation, obtained by reduction of the ten-dimensional heterotic superstring theory including terms [Formula: see text], shows in both cases that there is a positive semi-definite potential which evolves to zero, this corresponding to the ground state, which is Minkowski space.

scholarly journals Schrödinger equation in a general curved spacetime geometry

2022 ◽  
Author(s):  
Qasem Exirifard ◽  
Ebrahim Karimi
Keyword(s):  
Black Hole ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Quantum Field ◽  
Relativistic Quantum Field Theory ◽  
Schwarzschild Black Hole ◽  
Curved Spacetime ◽  
Spacetime Geometry ◽  
Energy Physics

In this paper, we consider relativistic quantum field theory in the presence of an external electric potential in a general curved spacetime geometry. We utilize Fermi coordinates adapted to the time-like geodesic to describe the low-energy physics in the laboratory and calculate the leading correction due to the curvature of the spacetime geometry to the Schrödinger equation. We then compute the nonvanishing probability of excitation for a hydrogen atom that falls in or is scattered by a general Schwarzschild black hole. The photon emitted from the excited state by spontaneous emission extracts energy from the black hole, increases the decay rate of the black hole and adds to the information paradox.

scholarly journals New Schrödinger wave mathematics changes experiments from saying there is, to denying there is quantum weirdness

2020 ◽  
Vol 18 ◽  
pp. 72-117
Author(s):  
Jeffrey Boyd
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
New Interpretation ◽  
Elementary Waves ◽  
Do So

With a clever new interpretation of the Schrödinger equation, those quantum experiments that allegedly prove that the quantum world is weird, no longer do so. When we approach the math from an unexpected angle, experiments that appeared to prove that time can go backwards in the quantum world, no longer say that. Experiments that appeared to demonstrate that a particle can be in two places at the same time, no longer say that. This requires that we take a counter-intuitive approach to the math, rather than a counter-intuitive approach to the quantum world. QM makes sensible assumptions and discovers that the quantum world is weird. Our math from the Theory of Elementary Waves (TEW) makes weird assumptions and discovers that the quantum world is sensible. We pay the weirdness tax up front. QM does not pay the weirdness tax and is penalized with a permanent misperception of the quantum world. This article is paired with a lively YouTube video that explains the same thing in 18 minutes: “New Schrödinger wave mathematics changes experiments from saying there is, to denying there is quantum weirdness.” That video can be found at the website ElementaryWave.com.

On the equations governing the perturbations of the Schwarzschild black hole

1975 ◽  
Vol 343 (1634) ◽  
pp. 289-298 ◽  
Keyword(s):  
Black Hole ◽  
Schrödinger Equation ◽  
Time Dependence ◽  
Spherical Harmonics ◽  
Schrodinger Equation ◽  
Schwarzschild Black Hole ◽  
Schwarzschild Metric ◽  
One Dimensional ◽  
The Subject

A coherent self-contained account of the equations governing the perturbations of the Schwarzschild black hole is given. In particular, the relations between the equations of Bardeen & Press, of Zerilli and of Regge & Wheeler are explicitly established. The equations governing the perturbations of the vacuum Schwarzschild metric - the Schwarzschild black hole-have been the subject of many investigations (Regge & Wheeler 1957; Vishveshwara 1970; Edelstein & Vishveshwara 1970; Zerilli 1970 a, b ; Fackerell 1971; Bardeen & Press 1972; Friedman 1973). Nevertheless, there continues to be some elements of mystery shrouding the subject. Thus, Zerilli (1970a) showed that the equations governing the perturbation, properly analysed into spherical harmonics (belonging to the different l values) and with a time dependence iot , can be reduced to a one dimensional Schrodinger equation of the form

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