Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9911001

The Schrodinger Equation From a Quadratic Hamiltonian System

Wai Bong Yeung

Published Nov 2, 1999• 0 citations
arXivHTMLPDF

Abstract

We regard the real and imaginary parts of the Schrodinger wave function as canonical conjugate variables.With this pair of conjugate variables and some other 2n pairs, we construct a quadratic Hamiltonian density. We then show that the Schrodinger Equation follows from the Hamilton's Equation of motion when the Planck frequency is much larger than the characteristic frequencies of the Hamiltonian system. The Hamiltonian and the normal mode solutions coincide, respectively, with the energy expectation value and the energy eigenstates of the corresponding quantum mechanical system.

DOI
10.48550/arxiv.quant-ph/9911001

Related papers

Time-dependent general quantum quadratic Hamiltonian system

Kyu Hwang Yeon, Chung‐In Um, Thomas F. George

Solutions of the Schr\"odinger equation and propagators for the general quadratic and the linear Hamiltonian system of canonical variables whose coefficients are arbitrary and time dependent are obtained when the corresponding classical solution is oscillatory, linear, or monotonic. All of them are given by the coefficients of the Hamiltonian and the classical solution of the system under the conditions for which the classical solution exists.

Physical Review A•Nov 21, 2003•33 citations

The Schrödinger Equation

Uri Peskin

The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.

Cambridge University Press eBooks•May 11, 2023•0 citations

Schrödinger's Wave Quantum Mechanics

Caio Lima Firme

In the first part of this chapter, the Hamiltonian function and Hamilton-Jacobi equation are presented. The latter is the starting point of Schrödinger’s work to obtain his wave equation. Next, the derivation of the wave equation is given along with the foundations of Schrödinger’s wave mechanics, i.e., the Hamilton’s analogy between optics and mechanics. It is also introduced some applications of the Schrödinger’s new quantum theory to several problems (radial function of the hydrogen atom, Compton effect and quantum harmonic oscillator). Other applications are also indicated in this chapter. The Dirac notation is introduced followed by the derivation of the expectation value in quantum mechanics and its derivation with respect to time. Next, Born’s probability theory and the superposition principle are introduced followed by the derivation of the quantum linear momentum operator. Afterwards, the normalization of a wave function and the orthogonality of a wave function are given followed by the properties of the Hermitian operator. In the end of the chapter, several examples and exercises are presented (168 words).

CRC Press eBooks•Mar 21, 2022•0 citations

Coherent states, Schrödinger equation, and the Hamilton–Jacobi equation

Rachad Mounir Shoucri

The self-adjoint form of the classical equation of motion of the harmonic oscillator is used to derive a Hamiltonian-like equation and the Schrödinger equation in quantum mechanics. A phase variable ϕ(t) instead of time t is used as an independent variable. It is shown that the Hamilton–Jacobi solution in this case is identical with the solution obtained from the Schrödinger equation without the need to introduce the idea of hidden variables or quantum potential.

Canadian Journal of Physics•Jul 1, 1995•0 citations

The Relation between Classical Mechanics and Wave Mechanics

Yoshio Nishiyama

Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.

Progress of Theoretical Physics•Nov 1, 1963•1 citation

QUANTUM TREATMENT OF TIME DEPENDENT COUPLED OSCILLATORS

M. Sebawe Abdalla

In this paper we consider the most quadratic time dependent Hamiltonian. An exact solution of the wave function in both the Schrödinger picture and coherent states representation is given. Linear and quadratic invariants are discussed. The eigenvalues and the corresponding eigenfunctions are obtained. The expectation values for the energy are also given.

International Journal of Modern Physics B•Jul 30, 2002•21 citations
We regard the real and imaginary parts of the Schrodinger wave function as canonical conjugate variables.With this pair of conjugate variables and some other 2n pairs, we construct a quadratic Hamiltonian density. We then show that the Schrodinger Equation follows from the Hamilton's Equation of motion when the Planck frequency is much larger than the characteristic frequencies of the Hamiltonian system. The Hamiltonian and the normal mode solutions coincide, respectively, with the energy expectation value and the energy eigenstates of the corresponding quantum mechanical system.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
In the first part of this chapter, the Hamiltonian function and Hamilton-Jacobi equation are presented. The latter is the starting point of Schrödinger’s work to obtain his wave equation. Next, the derivation of the wave equation is given along with the foundations of Schrödinger’s wave mechanics, i.e., the Hamilton’s analogy between optics and mechanics. It is also introduced some applications of the Schrödinger’s new quantum theory to several problems (radial function of the hydrogen atom, Compton effect and quantum harmonic oscillator). Other applications are also indicated in this chapter. The Dirac notation is introduced followed by the derivation of the expectation value in quantum mechanics and its derivation with respect to time. Next, Born’s probability theory and the superposition principle are introduced followed by the derivation of the quantum linear momentum operator. Afterwards, the normalization of a wave function and the orthogonality of a wave function are given followed by the properties of the Hermitian operator. In the end of the chapter, several examples and exercises are presented (168 words).
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.