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

Progress of Theoretical Physics

Miscellanea in Elementary Quantum Mechanics, II

K. Husimi

Published Apr 1, 1953• 482 citations
Publisher

Abstract

It is a rather curious fact that the exact quantal treatment of forced oscillation has only recently been discovered to be feasible. There may be a lot of general arguments which make the possibility of such a treatment very convincing, but I myself find the ground for it in an elementary observation that the wave equation of forced harmonic oscillation, including the case of subharmonic resonance, admits a solution of Gaussian type, i.e. of the form: an exponential of a quadratic. On this basis the complete solution and the related problems are analyzed here.

DOI
10.1143/ptp.9.381

Related papers

The Harmonic Oscillator

Abstract The problem of a particle moving under the influence of a linear restoring force, F (x) = −Kx, or equivalently a quadratic potential, V (x) = Kx 2/2, is a problem which is studied at all levels of theoretical physics, from elementary classical mechanics through quantum field theory.É One of its most useful features is that, at every stage of development, it is exactly soluble and so can be easily used as a closed-form, analytic example. If it had no connection to real physical systems, however, that fact would be of only academic interest.

•Apr 13, 2006•0 citations

A concise quantum mechanical treatment of the forced damped harmonic oscillator

Ti Li

Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator $$ \hat X $$ in x representation. The evolution of $$ \left\langle {\hat x} \right\rangle $$ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.

Open Physics•Jul 31, 2008•0 citations

Intuitive Solution to Quantum Harmonic Oscillator at Infinity

Cory Christopher Pye

In the solution to the quantum harmonic oscillator problem, the approximate Gaussian solution at large displacements is typically presented without proof. While it is relatively easy to verify that this form is an approximate solution, it is not obvious how this form can be obtained. What follows is a simple intuitive argument that gives the exact Gaussian form required.

Journal of Chemical Education•Jun 1, 2004•1 citation

Harmonic oscillator with variable mass

P. G. L. Leach

A general treatment of the quantal harmonic oscillator with variable mass is given. Various examples additional to those obtained by Colegrave and Abdalla for which a closed form solution is possible are given.

Journal of Physics A Mathematical and General•Oct 1, 1983•63 citations

Linear Invariants and the Solution of a Time-Dependent Coupled Quantal Oscillator.

S. Bös, U. B. Dub

Summary. - We obtain here the solution of a system of two coupled oscillators under the influence of damping force and time-dependent external force. The solution is obtained by first constructing linear invariants for this system. The exact dynamics of time-dependent quantal harmonic oscillator with timedependent frequency has been extensively studied in recent years exploiting the explicitly time-dependent invariants (1-5). The fact that there exists a simple relationship between the eigenstates of the invariant and the solution of the corresponding SehrSclinger equation has prompted people to find general form of invariants for time-dependent oscillator-type Hamiltonian. In these approaches quadratic type of invariants have mostly been considered. Further, these invariants involve a simple function ~(t), related to the amplitude of the classical free oscillator, and satisfies a nonlinear equation which should be solved for practical application. However, there has been attempt so far to study the dynamics of a time-dependent coupled-oscillator system. We, therefore, consider in the present work a system of two coupled oscillators, both damped and under time-dependent external force. To get the solution of the SchrSdinger equation for this system, we first obtain explicitly timedependent invariants which are linear in position and momentum variables. The two advantages with the linear invariants is that the eigenvalues of the invariant can be made continuous with suitable initial condition on the motion of oscillators and, further, that it involves only the amplitude of the classical oscillator and thus can be easily diagonalized.

•Jan 1, 1985•0 citations

The Quantized Harmonic Oscillator: Vibrational Spectroscopy

Don Shillady

Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was first solved [1]. Then we proceed to application in the form of worked examples. We have tried to give sufficient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.

CRC Press eBooks•Jul 27, 2011•0 citations
It is a rather curious fact that the exact quantal treatment of forced oscillation has only recently been discovered to be feasible. There may be a lot of general arguments which make the possibility of such a treatment very convincing, but I myself find the ground for it in an elementary observation that the wave equation of forced harmonic oscillation, including the case of subharmonic resonance, admits a solution of Gaussian type, i.e. of the form: an exponential of a quadratic. On this basis the complete solution and the related problems are analyzed here.
Abstract The problem of a particle moving under the influence of a linear restoring force, F (x) = −Kx, or equivalently a quadratic potential, V (x) = Kx 2/2, is a problem which is studied at all levels of theoretical physics, from elementary classical mechanics through quantum field theory.É One of its most useful features is that, at every stage of development, it is exactly soluble and so can be easily used as a closed-form, analytic example. If it had no connection to real physical systems, however, that fact would be of only academic interest.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ \hat X X^ in x representation. The evolution of ⟨x^⟩ \left\langle {\hat x} \right\rangle ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.
Summary. - We obtain here the solution of a system of two coupled oscillators under the influence of damping force and time-dependent external force. The solution is obtained by first constructing linear invariants for this system. The exact dynamics of time-dependent quantal harmonic oscillator with timedependent frequency has been extensively studied in recent years exploiting the explicitly time-dependent invariants (1-5). The fact that there exists a simple relationship between the eigenstates of the invariant and the solution of the corresponding SehrSclinger equation has prompted people to find general form of invariants for time-dependent oscillator-type Hamiltonian. In these approaches quadratic type of invariants have mostly been considered. Further, these invariants involve a simple function ~(t), related to the amplitude of the classical free oscillator, and satisfies a nonlinear equation which should be solved for practical application. However, there has been attempt so far to study the dynamics of a time-dependent coupled-oscillator system. We, therefore, consider in the present work a system of two coupled oscillators, both damped and under time-dependent external force. To get the solution of the SchrSdinger equation for this system, we first obtain explicitly timedependent invariants which are linear in position and momentum variables. The two advantages with the linear invariants is that the eigenvalues of the invariant can be made continuous with suitable initial condition on the motion of oscillators and, further, that it involves only the amplitude of the classical oscillator and thus can be easily diagonalized.
Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was first solved [1]. Then we proceed to application in the form of worked examples. We have tried to give sufficient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.