Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold

1981 ◽  
Vol 22 (10) ◽  
pp. 2144-2166 ◽  
Author(s):  
David Elworthy ◽  
Aubrey Truman
Keyword(s):  
Riemannian Manifold ◽  
Schrödinger Equation ◽  
Heat Equation ◽  
Schrodinger Equation ◽  
Classical Mechanics

scholarly journals Inverse problem for the heat equation and the Schrödinger equation on a tree

2011 ◽  
Vol 28 (1) ◽  
pp. 015011 ◽  
Author(s):  
Liviu I Ignat ◽  
Ademir F Pazoto ◽  
Lionel Rosier
Keyword(s):  
Inverse Problem ◽  
Schrödinger Equation ◽  
Heat Equation ◽  
Schrodinger Equation

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.

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