The solution of the time‐dependent Schrödinger equation by the (t,t’) method: Complex scaled multiphoton ionization/dissociation resonance wave functions are square integrable

For the two time-dependent coupled oscillators model we derive its time-dependent invariant in the context of Lewis–Riesenfeld invariant operator theory. It is based on the general solutions to the Schrödinger equation which is obtained and turns out to be the superposition of the generalized atomic coherent states in the Schwinger bosonic realization. The energy eigenvectors and eigenvalues of the corresponding time-independent Hamiltonian are also obtained as a by-product.

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Introduction

10.1093/oso/9780198805014.003.0001 ◽

2018 ◽

Author(s):

Niels Engholm Henriksen ◽

Flemming Yssing Hansen

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Degrees Of Freedom ◽

State Level ◽

Boltzmann Distribution ◽

Theoretical Description ◽

Time Dependent ◽

Nuclear Dynamics ◽

Molecular Reaction ◽

Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

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Quantum Boxes, Stringed Instruments

10.1093/oso/9780198808275.003.0008 ◽

2017 ◽

Author(s):

Frank S. Levin

Keyword(s):

Energy Level ◽

Schrödinger Equation ◽

Quantum System ◽

Schrodinger Equation ◽

Probability Distributions ◽

Wave Functions ◽

Energy Level Diagram ◽

One Dimensional ◽

Stringed Instruments ◽

Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

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