scholarly journals Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation

2011 ◽  
Vol 79 (8) ◽  
pp. 882-885 ◽  
Author(s):  
A. Deriglazov ◽  
B. F. Rizzuti
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Classical Mechanics ◽  
Invariant Formulation

scholarly journals Position Dependent Planck’s Constant in a Frequency-Conserving Schrödinger Equation

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 490
Author(s):  
Rand Dannenberg
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Reference Frames ◽  
Classical Mechanics ◽  
Planck’S Constant ◽  
The Earth ◽  
Planck's Constant ◽  
Lagrangian Densities ◽  
The Impact ◽  
Potential Frequency

There is controversial evidence that Planck’s constant shows unexpected variations with altitude above the earth due to Kentosh and Mohageg, and yearly systematic changes with the orbit of the earth about the sun due to Hutchin. Many others have postulated that the fundamental constants of nature are not constant, either in locally flat reference frames, or on larger scales. This work is a mathematical study examining the impact of a position dependent Planck’s constant in the Schrödinger equation. With no modifications to the equation, the Hamiltonian becomes a non-Hermitian radial frequency operator. The frequency operator does not conserve normalization, time evolution is no longer unitary, and frequency eigenvalues can be complex. The wavefunction must continually be normalized at each time in order that operators commuting with the frequency operator produce constants of the motion. To eliminate these problems, the frequency operator is replaced with a symmetrizing anti-commutator so that it is once again Hermitian. It is found that particles statistically avoid regions of higher Planck’s constant in the absence of an external potential. Frequency is conserved, and the total frequency equals “kinetic frequency” plus “potential frequency”. No straightforward connection to classical mechanics is found, that is, the Ehrenfest’s theorems are more complicated, and the usual quantities related by them can be complex or imaginary. Energy is conserved only locally with small gradients in Planck’s constant. Two Lagrangian densities are investigated to determine whether they result in a classical field equation of motion resembling the frequency-conserving Schrödinger equation. The first Largrangian is the “energy squared” form, the second is a “frequency squared” form. Neither reproduces the target equation, and it is concluded that the frequency-conserving Schrödinger equation may defy deduction from field theory.

The Schrödinger Equation

2020 ◽  
pp. 265-288
Author(s):  
M. Suhail Zubairy
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Quantum Tunneling ◽  
Schrodinger Equation ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Particle Dynamics ◽  
Quantization Condition ◽  
De Broglie Waves ◽  
Macroscopic Objects

In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.

scholarly journals ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR A TWO-LEVEL SYSTEM WITH A HAMILTONIAN DEPENDING QUASI–PERIODICALLY ON TIME

2000 ◽  
Vol 12 (01) ◽  
pp. 25-64 ◽  
Author(s):  
JOÃO C. A. BARATA
Keyword(s):  
Power Series ◽  
External Field ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Classical Mechanics ◽  
Energy Difference ◽  
Level System ◽  
Large Coupling ◽  
Periodic External Field ◽  
Formal Power

We consider the Schrödinger equation for a class of two-level atoms in a quasi-periodic external field for large coupling, i.e. for which the energy difference 2∊ between the unperturbed levels is sufficiently small. We show that this equation has a solution in terms of a formal power series in ∊, with coefficients which are quasi-periodical functions of the time, in analogy to the Lindstedt–Poincaré series in classical mechanics.

scholarly journals Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

2019 ◽  
Vol 55 (3) ◽  
pp. 319-324
Author(s):  
Yu. A. Kurochkin
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Scattering Amplitude ◽  
Separation Of Variables ◽  
Free Particle ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Quantum Mechanical ◽  
Lobachevsky Space

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

scholarly journals From Schrödinger Equation to Quantum Conspiracy

2021 ◽  
Author(s):  
Francis T.S. Yu
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Superposition Principle ◽  
Classical Mechanics ◽  
Modern Science ◽  
Empty Space ◽  
Principle Of Superposition ◽  
Time Space ◽  
Double Slit

Schrödinger’s quantum mechanics is a legacy of Hamiltonian’s classical mechanics. But Hamiltonian mechanics was developed from an empty space paradigm, for which Schrödinger’s equation is a timeless (t = 0) or time-independent deterministic equation, which includes his fundamental principle of superposition. When one is dealing Schrödinger equation, it is unavoidable not to mention about Schrödinger ‘s cat. Which is one of the most elusive cats in modern science since disclosed the half-life cat hypothesis in 1935. The cat is alive or not had been debated by score of world renounced scientists it is still debating. Yet I will show Schrödinger ‘s hypothesis is not a physically realizable hypothesis, for which it has nothing for us to debate about. But quantum communication and computing rely on qubit information algorithm, I will show that qubit information logic is as elusive as Schrödinger’s cat. It exists only within an empty space, but not exists within our temporal (t > 0) universe. Since there is always a price to pay within our universe, I will show that every physical subspace needs a section of time ∆t and an amount of energy ∆E to create and it is not free. Although, double slit hypothesis had been fictitiously confirmed that superposition principle exists, but I will show that double-slit postulation is another non-physically realizable hypothesis that had let us to believing superposition principle is actually existed within our time–space. Yet one of the worst coverup must be particles behaved differently within a micro space to justify the spooky superposition principle, which is one of greatest quantum conspiracy in modern science. Nevertheless, the art of quantum mechanics is all about a physically realizable equation, we see that everything existed within our universe, no matter how small it is, it has to be temporal (t > 0) which includes all the laws, principles, and equations. Otherwise, it is virtual as mathematics is since Schrodinger equation is mathematics, but mathematics is not equaled to science. Finally, when science turns to virtual reality for solution it is not a reliable answer. But when science turns to physical reality for an answer it is a reliable solution.

Accidental Degeneracy of the Hydrogen Atom and its Non-accidental Solution in Parabolic Coordinates

2021 ◽  
Author(s):  
P.C. Deshmukh ◽  
Aarthi Ganesan ◽  
Sourav Banerjee ◽  
Ankur Mandal
Keyword(s):  
Quantum Mechanics ◽  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Dynamical Symmetry ◽  
Coordinate Systems ◽  
Casimir Operators

The degeneracy associated with dynamical symmetry of a potential can be identified in quantum mechanics, by solving the Schrödinger equation analytically, using the method of separation of variables in at least two different coordinate systems, and in classical mechanics by solving the Hamilton-Jacobi equation. In the present pedagogical article, the notion of separability and superintegrability of a potential, with profound implications is discussed. In an earlier tutorial paper, we had addressed the n<sup>2</sup>-fold degeneracy of the hydrogen atom using the Casimir operators corresponding to the SO(4) symmetry of the 1/r potential. The present paper is a sequel to it, in which we solve the Schrödinger equation for the hydrogen atom using separation of variables in the parabolic coordinate systems. In doing so, we take the opportunity to revisit some excellent classical works on symmetry and degeneracy in classical and quantum physics, if only to draw attention to these insightful studies which unfortunately miss even a mention in most undergraduate and even graduate level courses in quantum mechanics and atomic physics.

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