Energy-dependent point interaction: Self-adjointness

2006 ◽  
Vol 84 (11) ◽  
pp. 991-1005 ◽  
Author(s):  
F AB Coutinho ◽  
Y Nogami ◽  
L Tomio ◽  
F M Toyama
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Point Interaction ◽  
One Dimensional ◽  
Nonstationary State ◽  
Complete Set ◽  
Energy Dependent ◽  
Dependent Point ◽  
03.65 Ge

Recently, we constructed an energy-dependent point interaction (EDPI) in its most general form in one-dimensional quantum mechanics. In this paper, we show that stationary solutions of the Schrodinger equation with the EDPI form a complete set. Then any nonstationary solution of the time-dependent Schrodinger equation can be expressed as a linear combination of stationary solutions. This, however, does not necessarily mean that the EDPI is self-adjoint and the time-development of the nonstationary state is unitary. The EDPI is self-adjoint provided that the stationary solutions are all orthogonal to one another. We illustrate situations in which this orthogonality condition is not satisfied.PACS Nos.: 03.65.–w, 03.65.Nk, 03.65.Ge

Stationary solutions to a nonlinear Schrödinger equation with potential in one dimension

2003 ◽  
Vol 133 (2) ◽  
pp. 409-437 ◽  
Author(s):  
Mihai Mariş
Keyword(s):  
Schrödinger Equation ◽  
Sound Velocity ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Nonlinear Schrodinger Equation ◽  
One Dimension ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
The One

We study the one-dimensional Gross-Pitaevskii-Schrödinger equation with a potential U moving at velocity v. For a fixed v less than the sound velocity, it is proved that there exist two time-independent solutions if the potential is not too big.

Quantum Boxes, Stringed Instruments

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.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

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