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Springer eBooks

One-Dimensional Quantum Mechanics: Motion within a Potential, Stationary Bound States

S. Brandt, Hans Dieter Dahmen

Published Jan 1, 1995• 0 citations
Publisher
DOI
10.1007/978-1-4684-0233-9_6

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Exactly Solvable Systems I: Bound States

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This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.

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Quantum Hamilton equations of motion for bound states of one-dimensional quantum systems

Jeanette Köppe, Markus Patzold, Wilfried Grecksch +1 more

On the basis of Nelson’s stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton’s principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson’s diffusion paths in the ground state.

Journal of Mathematical Physics•Jun 1, 2018•6 citations
This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.
On the basis of Nelson’s stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton’s principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson’s diffusion paths in the ground state.