The Schrödinger Equation, Path Integration and Applications

2021 ◽  
Vol 15 (01) ◽  
pp. 61-75
Author(s):  
Everaldo M. Bonotto ◽  
Felipe Federson ◽  
Márcia Federson
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Path Integration ◽  
Schrodinger Equation ◽  
Feynman Integral ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Henstock Integral ◽  
The Mean

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.

Solusi Persamaan Schrödinger Berbasis Panjang Minimal untuk Potensial Coulomb Termodifikasi

2018 ◽  
Vol 2 (2) ◽  
pp. 43-47
Author(s):  
A. Suparmi, C. Cari, Ina Nurhidayati
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Minimal Length ◽  
Energy Spectra ◽  
Atomic Particle ◽  
Approximate Analytical Solutions ◽  
Radial Schrödinger Equation

Abstrak – Persamaan Schrödinger adalah salah satu topik penelitian yang yang paling sering diteliti dalam mekanika kuantum. Pada jurnal ini persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Fungsi gelombang dan spektrum energi yang dihasilkan menunjukkan kharakteristik atau tingkah laku dari partikel sub atom. Dengan menggunakan metode pendekatan hipergeometri, diperoleh solusi analitis untuk bagian radial persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Hasil yang diperoleh menunjukkan terjadi peningkatan energi yang sebanding dengan meningkatnya parameter panjang minimal dan parameter potensial Coulomb Termodifikasi. Kata kunci: persamaan Schrödinger, panjang minimal, fungsi gelombang, energi, potensial Coulomb Termodifikasi Abstract – The Schrödinger equation is the most popular topic research at quantum mechanics. The  Schrödinger equation based on the concept of minimal length formalism has been obtained for modified Coulomb potential. The wave function and energy spectra were used to describe the characteristic of sub-atomic particle. By using hypergeometry method, we obtained the approximate analytical solutions of the radial Schrödinger equation based on the concept of minimal length formalism for the modified Coulomb potential. The wave function and energy spectra was solved. The result showed that the value of energy increased by the increasing both of minimal length parameter and the potential parameter. Key words: Schrödinger equation, minimal length formalism (MLF), wave function, energy spectra, Modified Coulomb potential

scholarly journals Non-Markovian wave-function collapse models are Bohmian-like theories in disguise

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 594
Author(s):  
Antoine Tilloy ◽  
Howard M. Wiseman
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Initial Conditions ◽  
Hidden Variables ◽  
Bohmian Mechanics ◽  
Primitive Ontology ◽  
Collapse Model ◽  
Collapse Models

Spontaneous collapse models and Bohmian mechanics are two different solutions to the measurement problem plaguing orthodox quantum mechanics. They have, a priori nothing in common. At a formal level, collapse models add a non-linear noise term to the Schrödinger equation, and extract definite measurement outcomes either from the wave function (e.g. mass density ontology) or the noise itself (flash ontology). Bohmian mechanics keeps the Schrödinger equation intact but uses the wave function to guide particles (or fields), which comprise the primitive ontology. Collapse models modify the predictions of orthodox quantum mechanics, whilst Bohmian mechanics can be argued to reproduce them. However, it turns out that collapse models and their primitive ontology can be exactly recast as Bohmian theories. More precisely, considering (i) a system described by a non-Markovian collapse model, and (ii) an extended system where a carefully tailored bath is added and described by Bohmian mechanics, the stochastic wave-function of the collapse model is exactly the wave-function of the original system conditioned on the Bohmian hidden variables of the bath. Further, the noise driving the collapse model is a linear functional of the Bohmian variables. The randomness that seems progressively revealed in the collapse models lies entirely in the initial conditions in the Bohmian-like theory. Our construction of the appropriate bath is not trivial and exploits an old result from the theory of open quantum systems. This reformulation of collapse models as Bohmian theories brings to the fore the question of whether there exists `unromantic' realist interpretations of quantum theory that cannot ultimately be rewritten this way, with some guiding law. It also points to important foundational differences between `true' (Markovian) collapse models and non-Markovian models.

scholarly journals A Refined Propensity Account for GRW Theory

2021 ◽  
Vol 51 (2) ◽  
Author(s):  
Lorenzo Lorenzetti
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Precise Nature ◽  
Dispositional Account ◽  
Collapse Theories ◽  
Dispositional Properties

AbstractSpontaneous collapse theories of quantum mechanics turn the usual Schrödinger equation into a stochastic dynamical law. In particular, in this paper I will focus on the GRW theory. Two philosophical issues that can be raised about GRW concern (a) the ontology of the theory, in particular the nature of the wave function and its role within the theory, and (b) the interpretation of the objective probabilities involved in the dynamics of the theory. During the last years, it has been claimed that we can take advantage of dispositional properties in order to develop an ontology for GRW theory, and also in order to ground the objective probabilities which are postulated by it. However, in this paper I will argue that the dispositional interpretations which have been discussed in the literature so far are either flawed or—at best—incomplete. If we want to endorse a dispositional interpretation of GRW theory we thus need an extended account which specifies the precise nature of those properties and which makes also clear how they can correctly ground all the probabilities postulated by the theory. Thus, after having introduced several different kinds of probabilistic dispositions, I will try to fill the gap in the literature by proposing a novel and complete dispositional account of GRW, based on what I call spontaneous weighted multi-track propensities. I claim that such an account can satisfy both of our desiderata.

Continuous Quantum Measurement: Local and Global Approaches

1997 ◽  
Vol 09 (08) ◽  
pp. 907-920 ◽  
Author(s):  
S. Albeverio ◽  
V. N. Kolokol'tsov ◽  
O. G. Smolyanov
Keyword(s):  
Schrödinger Equation ◽  
Quantum System ◽  
Quantum Measurement ◽  
Path Integral ◽  
Schrodinger Equation ◽  
Stochastic Equation ◽  
Point Of View ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Stochastic Schrödinger Equation

In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schrödinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menski's formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schrödinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator.

scholarly journals Schrodinger Equation in Cosmological Inertial Frame

2021 ◽  
Author(s):  
Sangwha Yi
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Inertial Frame ◽  
Special Theory ◽  
Theory Of Relativity ◽  
Special Theory Of Relativity ◽  
Klein Gordon

Schrodinger equation is a wave equation. Wave function uses as a probability amplitude in quantum mechanics. We make Schrodinger equation from Klein-Gordon free particle’s wave function in cosmological special theory of relativity.

The Schrödinger Equation and Negative Energies

2018 ◽  
Vol 73 (12) ◽  
pp. 1129-1135
Author(s):  
S.A. Bruce
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Schrödinger Equation ◽  
Discrete Symmetries ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Negative Energy ◽  
Space Time ◽  
Energy Eigenstates

AbstractIt is known that there is no room for anti-particles within the Schrödinger regime in quantum mechanics. In this article, we derive a (non-relativistic) Schrödinger-like wave equation for a spin-$1/2$ free particle in 3 + 1 space-time dimensions, which includes both positive- and negative-energy eigenstates. We show that, under minimal interactions, this equation is invariant under $\mathcal{P}\mathcal{T}$ and 𝒞 discrete symmetries. An immediate consequence of this is that the particle exhibits Zitterbewegung (‘trembling motion’), which arises from the interference of positive- and negative-energy wave function components.

The formulation of quantum mechanics in terms of phase space functions

1964 ◽  
Vol 60 (3) ◽  
pp. 581-586 ◽  
Author(s):  
D. B. Fairlie
Keyword(s):  
Wave Function ◽  
Distribution Function ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Time Dependence ◽  
Schrodinger Equation ◽  
Average Energy ◽  
Weighted Mean

AbstractA relationship between the Hamiltonian of a system and its distribution function in phase space is sought which will guarantee that the average energy is the weighted mean of the Hamiltonian over phase space. This relationship is shown to imply the existence of a wave function satisfying the Schrödinger equation, and dictates the possible forms of time-dependence of the distribution function.

The investigation of Carnot engine in the presence of deformed formalism

2021 ◽  
Vol 36 (35) ◽  
Author(s):  
H. Naseri Karimvand ◽  
B. Lari ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Thermodynamic Properties ◽  
Schrödinger Equation ◽  
Single Particle ◽  
Schrodinger Equation ◽  
Carnot Cycle ◽  
Energy Eigenvalues ◽  
Work Done

In this paper, after introducing a kind of [Formula: see text]-deformation in quantum mechanics, first [Formula: see text]-deformed form of Schrödinger equation for a single particle in a box is derived. Then, the energy eigenvalues and wave function in Schrödinger equation are studied. Also, we discuss the Carnot cycle by using of the energy eigenvalues. We obtain the thermodynamic properties such as force, heat transferred, work done and efficiency in the cycle. Finally, all results have satisfied what we had expected before.

PATH INTEGRAL APPROACH TO A SINGLE POLYMER CHAIN WITH RANDOM MEDIA

2004 ◽  
Vol 18 (10n11) ◽  
pp. 1465-1478 ◽  
Author(s):  
CH. KUNSOMBAT ◽  
V. SA-YAKANIT
Keyword(s):  
Polymer Chain ◽  
Path Integral ◽  
Random Media ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Short Range Correlation ◽  
Range Correlation ◽  
End Distance ◽  
Non Local ◽  
The Mean

In this paper we consider the problem of a polymer chain in random media with finite correlation. We show that the mean square end-to-end distance of a polymer chain can be obtained using the Feynman path integral developed by Feynman for treating the polaron problem and successfuly applied to the theory of heavily doped semiconductor. We show that for short-range correlation or the white Gaussian model we derive the results obtained by Edwards and Muthukumar using the replica method and for long-range correlation we obtain the result of Yohannes Shiferaw and Yadin Y. Goldschimidt. The main idea of this paper is to generalize the model proposed by Edwards and Muthukumar for short-range correlation to finite correlation. Instead of using a replica method, we employ the Feynman path integral by modeling the polymer Hamiltonian as a model of non-local quadratic trial Hamiltonian. This non-local trial Hamiltonian is essential as it will reflect the translation invariant of the original Hamiltonian. The calculation is proceeded by considering the differences between the polymer propagator and the trial propagator as the first cumulant approximation. The variational principle is used to find the optimal values of the variational parameters and the mean square end-to-end distance is obtained. Several asymptotic limits are considered and a comparison between this approaches and replica approach will be discussed.

Sign in / Sign up

Export Citation Format

Share Document