Schrodinger equation

The celebrated work of Yau and Yau solved the nonlinear filtering problem in theory in the following manner. They reduced the problem of solving the Duncan-Mortensen-Zakai equation in real-time to the off-time solution of a Kolmogorov type equation. For the Yau filtering system, this Kolmogorov equation can be transformed as the Schrodinger equation. In this paper, we shall describe the fundamental solution of this Schrodinger equation with quartic potential.

QM Demystified A Self Teaching Guide by David McMahon. I purchased this
book in 2007. Read chapter 1 and 2 in that year. Did not get past chapter 2, but did complete chapter 2. May had gotten busy with other things, most probably found the subject matter incomprehensible, difficult, tough, .... This book made it possible for me to understand the basic mathematics required for the topic. I revisited it again in 2019 and 2020, as you can see.

Had it not been for David McMahon's book, Schrodinger Equation may not been a subject matter I could understand. I admit I may still be incomplete on its understanding. So there maybe mistakes here that you can easily spot. Apologies in advance.

Example were needed to supplement the theory, this is why this book was suitable for me. There are a few examples from other textbook(s) in the study notes.

Its mainly the mathematics technique required and applied to understand some of the undergraduate Schrodinger Equation subject matter.

Approximately 85 pages. Tried to make it as step by step as possible.

21 examples in all.

You may need to zoom in at a higher magnification for the subscripts.

Any errors and omissions apologies in advance.

KSB Dhaliwal.

All'interno del lavoro viene mostrato come la rotazione di Wick e il formalismo del tempo immaginario permettano in maniera semplice di identificare lo stato fondamentale di un sistema quantistico. Vengono poi sviluppate due tecniche di risoluzione numerica basate su questo formalismo: lo splitting degli operatori e la diffusione montecarlo.
Si descrivere poi un metodo di integrazione numerica dell'equazione di Schrödinger  indipendente dal tempo, l'algoritmo di Numerov, associato alla procedura di shooting.
I tre metodi vengono infine utilizzati per studiare la buca di potenziale infinita, l'oscillatore armonico e l'oscillatore quartico.

“The Schrödinger equation and the Dirac equation are wrong because length can not be smaller than Planck length and time can not be smaller than Planck time.”
Adrian Ferent

“To quantize the energy operator, the wave function must be multiplied by ħ:”
Adrian Ferent

“The Schrödinger equation is wrong because time can not be smaller than Planck time”
Adrian Ferent

“The Schrödinger equation is wrong because length can not be smaller than Planck length”
Adrian Ferent

“The Dirac equation is wrong because length can not be smaller than Planck length and time can not be smaller than Planck time.”
Adrian Ferent

Tinjau sebuah partikel dengan massa m, yang dibatasi bergerak sepanjang sumbu-x, yang mengalami gaya tertentu F(x,t) (Gambar 1.1). Dalam mekanika klasik programnya adalah menentukan kedudukan partikel itu pada suatu saat: x(t). Setelah mengetahuinya, kita akan dapat mengira kecepatan (v = dx/dt), momentum (p=mv), energi kinetik (T=(1/2)mv 2), atau variabel dinamika lain yang menjadi perhatian. Dan bagaimanakah kita menentukan x(t)? Kita mempergunakan Hukum Newton kedua: F = ma. (Untuk sistem yang konservatif-satu-satunya yang akan kita tinjau, dan, kebetulan, merupakan satu-satunya macam yang terdapat pada level mikroskopis-gaya dapat dinyatakan sebagai turunan dari sebuah fungsi energi potensial,1F = V/x, dan hukum Newton menjadi md 2 x/dt 2 = V/x). Hal ini, beserta kondisi awal yang tepat (biasanya kedudukan dan kecepatan pada t = 0), menentukan x(t). Mekanika kuantum mendekati permasalahan yang sama secara amat berbeda. Dalam hal ini kita mencari fungsi gelombangpartikel, (x,t), dan kita memperolehnya dengan mencari solusi persamaan Schrödinger:. [1.1] Gambar 1.1: Sebuah "partikel' dibatasi bergerak dalam satu dimensi di bawah pengaruh sebuah gaya tertentu. Di sini i adalah akar dari 1, dan ħ adalah konstanta Planck-atau lebih tepat konstanta Planck asli (h) dibagi 2:. [1.2] 1 Bab1 FUNGSI GELOMBANG 1 Gaya magnetik merupakan pengecualian, namun marilah kita tidak merisaukan hal itu dulu. Satu hal lagi, dalam buku ini kita akan menganggap bahwa geraknya non-relativistik (v<<c)

The nature of the hydrogen bond is studied in the context of the Spacetime Wave Theory. The usual description of a covalent bond is: a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs, and the stable balance of attractive and repulsive forces between atoms, when they share electrons is known as covalent bonding.

Why does time reversal involve two operations, a temporal reflection and the operation of complex conjugation in quantum mechanics? Why is it that time reversal preserves position and reverses momentum and spin? This puzzle of time reversal in quantum mechanics has been with us since Wigner's first presentation. In this paper, I show that the standard account of time reversal in quantum mechanics can be derived from the natural requirement that time reversal reverses velocities by analyzing the continuity equation.

We illustrate a simple derivation of the Schrodinger equation, which requires only knowledge of the electromagnetic wave equation and the basics of Einstein&#39;s special theory of relativity. We do this by extending the wave equation for classical fields to photons, generalize to non-zero rest mass particles, and simplify using approximations consistent with non-relativistic particles.

I propose a model wherein a system is represented by a finite sequence of natural numbers. These numbers are thought of as population numbers in statistical ensemble formed as a sample with replacement of entities (microstates) from some abstract set. I derive the concepts of energy and of temperature. I show an analogy between energy spectra computed from the model and energy spectra of some known constructs, such as particle in a box and quantum harmonic oscillator. The presented model replaces the concept of wave function with knowledge vector. I derive Schrödinger-type equation for knowledge vector and discuss principal differences with Schrödinger equation. The model retains major QM hallmarks such as wave-particle duality, violation of Bell’s inequalities, quantum Zeno effect, uncertainty relations, while avoiding controversial concept of wave function collapse. Unlike standard QM and Newtonian mechanics, the presented model has the Second Law of Thermodynamics built-in; in particular, it is not invariant with respect to time reversal.

Quantum Mechanics is often used by philosophers of mind to develop a non mechanist approach to the cognitive domain. In this paper we propose the inverse: we show that the mechanist or computationalist hypothesis in philosophy of mind can justify some quantum logic of the &#39;observable&#39;. We suggest that the quantum nature of reality might be explained by the nature of the digital when &quot;seen from inside arithmetic&quot; in some precise technical sense.

El problema de la medición en la mecánica cuántica es que la partícula cuántica en el curso de la evolución, como la describe la ecuación lineal de Schrodinger, existe en todos sus estados posibles, pero en la medición, la partícula siempre se detecta en solo uno de sus estados. Esta propiedad se denomina "colapso de la función de onda" y fue formulada por Von Neumann como uno de los postulados de la mecánica cuántica.

Sin embargo, no está claro en qué momento y bajo qué leyes ocurre esta transición. Este artículo demuestra que el colapso de la función de onda puede deberse a la creación o aniquilación de partículas (cuasi-partículas). Los procesos de creación o aniquilación de partículas juegan un papel clave en las mediciones y se describen sobre la base de la teoría cuántica de campos. El sistema de ecuaciones de la teoría cuántica de campos de partículas y campos no es lineal; como resultado, el principio de superposición no es válido para la teoría.

El colapso de la función de onda es consecuencia de esta no linealidad y ocurre en el momento de la creación (aniquilación) de una partícula. Este resultado demuestra que el colapso de la función de onda puede ocurrir tanto en sistemas microscópicos como macroscópicos. Comprender los mecanismos del colapso de la función de onda puede conducir a la creación de dispositivos microscópicos involucrados en los cálculos basados ​​en la computación cuántica.

The solutions of the Schrödinger with more general exponential screened coulomb (MGESC), Yukawa potential (YP) and the sum of the mixed potential (MGESCY) have been presented using the Parametric Nikiforov-Uvarov Method (pNUM). The bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. Some derived equations were used to calculate numerical values for MGESC, YP, and MGESCY potentials for diatomic molecules with different screening parameters (α) for l = 0 and l = 1 state with V 0 = 2.75 MeV and V 1 = 2.075 MeV. We observed an increase in l value; the particles behave more repulsive than attractive. The numerical values for different l-states at different screening parameters for CO molecules (r = 1.21282) and NO molecule (r = 1.1508) were obtained using the bound state energy eigenvalue of the Schrodinger equation for MGESC, YP and MGESCY potentials. Potential variation with intermolecular distance (r) for some of the particles moving under the influence of MGESC, Yukawa and the mixed potential (MGESCY) were also studied. We also observed the variation of the MGESC potential How to cite this paper: Ita,

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in (14) in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrodinger equation and the central charge-free Virasoro algebra Vect(S1). We call sv the Schrodinger-Virasoro Lie algebra. We choose to present sv from a Newtonian

“Maxwell’s equations are wrong because length can not be smaller than Planck length and time can not be smaller than Planck time”
Adrian Ferent

“Maxwell’s equations are wrong, because the Electric field E and the Magnetic field B are not continuous functions”
Adrian Ferent

“The Electric field E and the Magnetic field B are discrete functions”
Adrian Ferent

L'oggetto principale della prova finale è la teoria degli operatori m-dissipativi su spazi di Banach.
Nella tesi si analizzano alcune proprietà di tale classe di operatori. Il caso più studiato in letteratura è quello degli operatori m-dissipativi definiti su un dominio denso su spazi di Hilbert. Alcuni esempi di operatori m-dissipativi sono forniti dall'operatore Laplaciano con diversi domini di definizione.
Consideriamo ora il problema di evoluzione generale, soffermandoci nei casi particolari in cui A è un operatore autoaggiunto o antisimmetrico in uno spazio di Hilbert.
Nella nostra trattazione introdurremo la teoria dei semigruppi di contrazione, con particolare attenzione al semigruppo di contrazione  generato da un operatore m-dissipativo con dominio denso, analizzando un importante risultato, il Teorema di Hille-Yosida-Phillips, utile nello studio delle equazioni di evoluzione.
In Natura, molti fenomeni di propagazione sono descritti da equazioni di evoluzione, per esempio l'equazione del calore che modellizza il trasporto dell'energia termica in un mezzo omogeneo: essa è il più semplice esempio di equazione di diffuusione che analizziamo nella prova finale come applicazione della teoria dei semigruppi.
La trattazione è basata sui Capitoli 2-3 del libro "An Introduction to Semilinear Evolution Equations" di Thierry Cazenave e Alain Haraux [4].

Lagrargian formalism was utilized in this work to obtain Schrödinger, Klein-Gordon and Dirac equation for frictional medium. Fortunately these equations are typical to that obtained by others. These equations were solved for lasing and absorption conditions. These conditions show that lasing and absorption take place due to the presence of frictional relaxation time term. This conforms to common sense and physical reality, since friction effect enables the medium to absorb energy from the incident beam. This absorbed energy can cause population inversion and lasing

In this paper, wavelets and fuzzy support vector machines are used to automated detect and classify power quality (PQ) disturbances. Electric power quality is an aspect of power engineering that has been with us since the inception of power systems. The types of concerned disturbances include voltage sags, swells, interruptions, switching transients, impulses, flickers, harmonics, and notches. Fourier transform and wavelet analysis are utilized to denoise the digital signals, to decompose the signals and then to obtain eight common features for the sampling PQ disturbance signals. A fuzzy support vector machines is designed and trained by 8-dimension feature space points for making a decision regarding the type of the disturbance. Simulation cases illustrate the effectiveness.

we have obtained the analytical solution of Schrödinger wave equation with Mie – type potential using factorization method. We have also obtained energy eigenvalues of our potential and the corresponding wave function using an ansatz and then compare the result to standard Laguerre's differential equation. Under special cases our potential model reduces two well known potentials such as Coulomb and the Kratzer Feus potentials.

The effective mass one-dimensional Schr\&quot;odinger equation for the generalized Morse potential is solved by using Nikiforov-Uvarov method. Energy eigenvalues and corresponding eigenfunctions are computed analytically. The results are also reduced to the case of constant mass. Energy eigenvalues are computed numerically for some diatomic molecules. The results are in agreement with the ones obtained before.

Abstract-We use the method of multiple scales borrowed from perturbation theory to derive a new time-domain transfer function of the nonlinear fiber-optic wave-division multiplexing (WDM) communications channel. The obtained channel response, derived from the nonlinear Schrodinger equation is shown to be equivalent to the multi-path fading frequency selective channel encountered in wireless links. In the linear regime, the channel response is shown to be equivalent to a standard intersymbol interference (lSI) channel and is used to derive new bounds on the capacity of the dispersive optical fiber channel. I.