On the theory of variational principles in quantum mechanics

1986 ◽  
Vol 94 (2) ◽  
pp. 149-167 ◽  
Author(s):  
G. Fonte
Keyword(s):  
Quantum Mechanics ◽  
Variational Principles

The Physical World

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Particle Physics ◽  
Variational Principles ◽  
Black Body ◽  
Black Body Radiation ◽  
Physical World ◽  
Atomic Scale ◽  
Solid State Physics ◽  
Least Action ◽  
Dimensional Spacetime

The book is an inspirational survey of fundamental physics, emphasizing the use of variational principles. Chapter 1 presents introductory ideas, including the principle of least action, vectors and partial differentiation. Chapter 2 covers Newtonian dynamics and the motion of mutually gravitating bodies. Chapter 3 is about electromagnetic fields as described by Maxwell’s equations. Chapter 4 is about special relativity, which unifies space and time into 4-dimensional spacetime. Chapter 5 introduces the mathematics of curved space, leading to Chapter 6 covering general relativity and its remarkable consequences, such as the existence of black holes. Chapters 7 and 8 present quantum mechanics, essential for understanding atomic-scale phenomena. Chapter 9 uses quantum mechanics to explain the fundamental principles of chemistry and solid state physics. Chapter 10 is about thermodynamics, which is built around the concepts of temperature and entropy. Various applications are discussed, including the analysis of black body radiation that led to the quantum revolution. Chapter 11 surveys the atomic nucleus, its properties and applications. Chapter 12 explores particle physics, the Standard Model and the Higgs mechanism, with a short introduction to quantum field theory. Chapter 13 is about the structure and evolution of stars and brings together material from many of the earlier chapters. Chapter 14 on cosmology describes the structure and evolution of the universe as a whole. Finally, Chapter 15 discusses remaining problems at the frontiers of physics, such as the interpretation of quantum mechanics, and the ultimate nature of particles. Some speculative ideas are explored, such as supersymmetry, solitons and string theory.

Quantum Mechanics

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Variational Principles ◽  
Free Particle ◽  
Uncertainty Relations ◽  
Probabilistic Interpretation ◽  
Historical Account ◽  
Potential Wells ◽  
Harmonic Oscillator Potential ◽  
Ground State Energies

In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.

NONLINEAR QUANTUM MECHANICS WITH NONCLASSICAL GRAVITATIONAL SELF-INTERACTION II: NONSTATIONARY SITUATION

1993 ◽  
Vol 08 (16) ◽  
pp. 2683-2707 ◽  
Author(s):  
A. D. POPOVA ◽  
A. N. PETROV
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Variational Principles ◽  
Gravitational Interaction ◽  
Classical Field Theory ◽  
Classical Field ◽  
Stationary Case ◽  
Self Consistent ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

Quantum mechanics (first quantization) with self-consistent gravitational interaction, previously constructed for the stationary case, is extended to the general case. The two requirements for such a theory are realized: to obtain the theory maximally resembling a classical field theory and to achieve the invariance of the theory under the rescaling transformations of a wave function. The construction is not trivial, because it rejects the variational principles of extremality of any action and involves some principles of smoothed extremality which give relevant equations.

NONLINEAR QUANTUM MECHANICS WITH NONCLASSICAL GRAVITATIONAL SELF-INTERACTION III: RELATED TOPICS

1993 ◽  
Vol 08 (16) ◽  
pp. 2709-2734 ◽  
Author(s):  
A. D. POPOVA ◽  
A. N. PETROV
Keyword(s):  
Quantum Mechanics ◽  
Gauge Invariance ◽  
Ricci Tensor ◽  
Variational Principles ◽  
Uncertainty Relations ◽  
Quantum Problem ◽  
General Quantum ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics ◽  
The One

Some problems are considered in the framework of general quantum mechanics with gravitational self-interaction constructed earlier. A number of them were analyzed for the stationary situation. Here, the problem of gauge invariance generated by translations which do not violate the 3 + 1 splitting is studied. The notions of position and momentum operators are extended to the general case. The uncertainty relations are obtained for the uncertainty of the Ricci tensor and for uncertainties of the position and momentum of a particle. The correspondence between the stationary and nonstationary cases is examined at the level of variational principles. At least, the one-particle and two-particle problems in the Newtonian–Schrödingerian limit are considered; the latter problem is compared with the standard two-particle quantum problem to demonstrate the advantage of our approach.

Complementary variational principles for ∇2φ = f(φ) with applications to the Thomas–Fermi and Liouville equations

1969 ◽  
Vol 65 (2) ◽  
pp. 535-541 ◽  
Author(s):  
A. M. Arthurs ◽  
P. D. Robinson
Keyword(s):  
Quantum Mechanics ◽  
Statistical Mechanics ◽  
Boundary Value Problem ◽  
Liouville Equation ◽  
Variational Principles ◽  
Boundary Value ◽  
Equilibrium Statistical Mechanics ◽  
Thomas Fermi

AbstractComplementary variational principles associated with the boundary-value problem ∇2φ = f(φ) in ∇,φ = φ0 on the boundary of ∇, are presented. The theory is applied to the Thomas–Fermi equation in quantum mechanics and the Liouville equation in equilibrium statistical mechanics.

scholarly journals A Stochastic Fractional Calculus with Applications to Variational Principles

2020 ◽  
Vol 4 (3) ◽  
pp. 38
Author(s):  
Houssine Zine ◽  
Delfim F. M. Torres
Keyword(s):  
Numerical Simulation ◽  
Quantum Mechanics ◽  
Fractional Calculus ◽  
Theoretical Result ◽  
Calculus Of Variations ◽  
Variational Principles ◽  
Lagrange Equation ◽  
Stochastic Calculus ◽  
Fractional Calculus Of Variations ◽  
Stochastic Calculus Of Variations

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

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