Solution to One-Dimensional Schrödinger Equation for an Arbitrary Potential; Application to Radial Equation in Three Dimensions

1972 ◽  
Vol 6 (10) ◽  
pp. 4084-4086 ◽  
Author(s):  
James L. Sigel
Keyword(s):  
Schrödinger Equation ◽  
Potential Application ◽  
Schrodinger Equation ◽  
Three Dimensions ◽  
Radial Equation ◽  
One Dimensional ◽  
Arbitrary Potential

scholarly journals On the asymptotic distribution of eigenvalues

1950 ◽  
Vol 200 (1063) ◽  
pp. 572-580 ◽  
Keyword(s):  
Approximate Solution ◽  
Schrödinger Equation ◽  
Asymptotic Distribution ◽  
Schrodinger Equation ◽  
Eigenvalue Distribution ◽  
Three Dimensions ◽  
One Dimension ◽  
One Dimensional ◽  
Distribution Of Eigenvalues ◽  
The One

It is well known that the asymptotic distribution of the eigenvalues of the one-dimensional Schrödinger equation is provided by the so-called W. K. B. formula. Most proofs of this depend on the approximate solution of the equation in two regions and the joining up of these solutions at the boundaries of the regions in a certain way. These methods are not easily generalized to the Schrödinger equation for dimensions greater than one. In the present paper the methods of Courant & Hilbert are applied to this problem and they lead very simply to a proof of the known result in one dimension and to analogous formulae for the eigenvalue distribution of the Schrödinger equation in two and three dimensions.

scholarly journals On the one dimensional cubic NLS in a critical space

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marco Bravin ◽  
Luis Vega
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Three Dimensions ◽  
Critical Space ◽  
Initial Value ◽  
One Dimensional ◽  
Geometrical Problem ◽  
The One

<p style='text-indent:20px;'>In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.</p>

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.

scholarly journals Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Li Chen ◽  
Jinyeop Lee ◽  
Matthew Liew
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Mean Field ◽  
Weak Limit ◽  
Weak Compactness ◽  
Three Dimensions ◽  
Uniform Estimates ◽  
Fermionic Systems ◽  
Semiclassical Limits ◽  
Semiclassical Regime

AbstractWe study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

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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