From Classical to Quantum Mechanics

2021 ◽  
pp. 207-219
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Harmonic Oscillator ◽  
Path Integral ◽  
Classical Theory ◽  
Constrained Systems ◽  
Experimental Results ◽  
Basic Assumptions ◽  
Original Approach

In Chapter 2 we presented the method of canonical quantisation which yields a quantum theory if we know the corresponding classical theory. In this chapter we argue that this method is not unique and, furthermore, it has several drawbacks. In particular, its application to constrained systems is often problematic. We present Feynman’s path integral quantisation method and derive from it Schroödinger’s equation. We follow Feynman’s original approach and we present, in addition, more recent experimental results which support the basic assumptions. We establish the equivalence between canonical and path integral quantisation of the harmonic oscillator.

scholarly journals Relativistic quantum mechanics

1932 ◽  
Vol 136 (829) ◽  
pp. 453-464 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Correspondence Principle ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Relativistic Quantum Mechanics ◽  
Classical Electrodynamics ◽  
Quantum Phenomena

The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approxi­mation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.

scholarly journals Relativity quantum mechanics with an application to compton scattering

1926 ◽  
Vol 111 (758) ◽  
pp. 405-423 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Poisson Bracket ◽  
Classical Theory ◽  
Analytic Functions ◽  
Algebraic Theory ◽  
Dynamical Theory ◽  
Canonical Variables ◽  
Algebraic Laws ◽  
Points Of View

The new quantum mechanics, introduced by Heisenberg and since developed from different points of view by various authors, takes its simplest form if one assumes merely that the dynamical variables are numbers of a special type (called q-numbers to distinguish them from ordinary or c-numbers) that obey all the ordinary algebraic laws except the commutative law of multiplication, and satisfy instead of this the relations q r q s – q s q r =0, p r p s – p s p r = 0 } q r q s – p s q r = 0 ( r ≠ s ) or ih ( r = s ) where the p' s and q' s are a set of canonical variables and h is a c-number euqal to (2π) -1 times the usual Planck’s constant. Equations (1) may be regarded as replacing the commutative law of the classical theory, as one can, with their help, build up a complete algebraic theory of quantities that are analytic functions of a set of canonical variables. Further, it may easily be seen that the quantity [ x, y ] defined by xy – yz = ih [ x, y ] is completely analogous to the Poisson bracket of the classical theory. By means of this analogy the whole of the classical dynamical theory, in so far as it can be expressed in terms of P. B.’s instead of differential coefficients, may be taken over immediately into the quantum theory.

QUANTUM MECHANICS AND PATH INTEGRALS IN SPACES WITH CURVATURE AND TORSION

1989 ◽  
Vol 04 (24) ◽  
pp. 2329-2337 ◽  
Author(s):  
H. KLEINERT
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Path Integral ◽  
Affine Space ◽  
Correspondence Principle ◽  
Integral Formula ◽  
Path Integrals ◽  
Physical Systems ◽  
Curvature And Torsion ◽  
Quantization Rules

We point out that there is a natural geometric procedure for constructing the quantum theory of a particle in a general metric-affine space with curvature and torsion. Quantization rules are presented and expressed in the form of a simple path integral formula which specifies compactly a new combined equivalence and correspondence principle. The associated Schrödinger equation has no extra curvature nor torsion terms that have plagued earlier attempts. Several well-known physical systems are invoked to suggest the correctness of the proposed theory.

Quantum Theory

2018 ◽  
Author(s):  
Kate Atkinson
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum System ◽  
20Th Century ◽  
Classical Theory ◽  
Theoretical Physics ◽  
Early 20Th Century ◽  
Max Planck ◽  
Physical Phenomena

Developed in the early 20th century, quantum theory is a branch of theoretical physics that concerns the unpredictable quality of particles at the quantum, or subatomic, level. In 1900 Max Planck (1859–1947) inaugurated inquiry into quantum mechanics when he challenged the classical theory that light behaves as a wave, proposing instead that it is emitted in quanta, or discrete units. By 1927 this groundbreaking theory had been more clearly articulated by Niels Bohr’s (1885–1962) "Principle of Complementarity" and Werner Heisenberg’s (1901–1976) "Uncertainty Principle." Heisenberg proposed that all physical phenomena that can be observed are subject to a degree of indeterminacy and suggested that the act of scientific observation of a quantum system would change that system. These new proposals of quantum theory unseated the authority of classical deterministic physics and challenged the perceived objectivity of science. Attracted by quantum theory’s revolutionary ideas, various modernist critics adapted its principles of uncertainty and indeterminacy to studies in the humanities. For instance, I. A. Richards (1893–1970) and William Empson (1906–1984) employed Bohr’s concepts in their work on irony, ambiguity, and paradox. Heisenberg suggested, however, that both modern artistic innovations and quantum theory were the products of "profound transformations in the fundamentals of our existence" (1958: 95).

scholarly journals Emergent quantum mechanics as a thermal ensemble

2014 ◽  
Vol 11 (08) ◽  
pp. 1450068 ◽  
Author(s):  
P. Fernández De Córdoba ◽  
J. M. Isidro ◽  
Milton H. Perea
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Thermal Fluctuations ◽  
Irreversible Processes ◽  
Quantum Mechanical ◽  
Transformation Theory ◽  
Time Irreversibility ◽  
Quantum Mechanical Systems ◽  
Emergent Quantum Mechanics

It has been argued that gravity acts dissipatively on quantum-mechanical systems, inducing thermal fluctuations that become indistinguishable from quantum fluctuations. This has led some authors to demand that some form of time irreversibility be incorporated into the formalism of quantum mechanics. As a tool toward this goal, we propose a thermodynamical approach to quantum mechanics, based on Onsager's classical theory of irreversible processes and Prigogine's nonunitary transformation theory. An entropy operator replaces the Hamiltonian as the generator of evolution. The canonically conjugate variable corresponding to the entropy is a dimensionless evolution parameter. Contrary to the Hamiltonian, the entropy operator is not a conserved Noether charge. Our construction succeeds in implementing gravitationally-induced irreversibility in the quantum theory.

scholarly journals The quantum theory of dispersion

1927 ◽  
Vol 114 (769) ◽  
pp. 710-728 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Matrix Theory ◽  
Hamiltonian Function ◽  
Spectral Lines ◽  
Light Quantum ◽  
Original Matrix ◽  
The Matrix ◽  
Points Of View

The new quantum mechanics could at first be used to answer questions concerning radiation only through analogies with the classical theory. In Heisenberg’s original matrix theory, for instance, it is assumed that the matrix elements of the polarisation of an atom determine the emission and absorption of radiation analogously to the Fourier components in the classical theory. In more recent theories a certain expression for the electric density obtained from the quantum mechanics is used to determine the emitted radiation by the same formulæ as in the classical theory. These methods give satisfactory results in many cases, but cannot even be applied to problems where the classical analogies are obscure or non-existent, such as resonance radiation and the breadths of spectral lines. A theory of radiation has been given by the author which rests on a more definite basis. It appears that one can treat a field of radiation as a dynamical system, whose interaction with an ordinary atomic system may be described by a Hamiltonian function. The dynamical variables specifying the field are the energies and phases of its various harmonic conrponents, each of which is effectively a simple harmonic oscillator. One must, of course, in the quantum theory take these variables to be q-numbers satisfying the proper quantum conditions. One finds then that the Hamiltonian for the interaction of the field with an atom is of the same form as that for the interaction of an assembly of light-quanta with the atom. There is thus a complete formal reconciliation between the wave and light-quantum points of view.

The build of nonlinear quantum mechancs and variations of feature of microscopic particles as well as their experimental affirmation

2014 ◽  
Vol 5 (3) ◽  
pp. 871-981 ◽  
Author(s):  
Pang Xiao Feng
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Lagrange Equation ◽  
Minimum Principle ◽  
Superposition Principle ◽  
Type Equation ◽  
Experimental Results ◽  
Traditional Concept ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics

We establish the nonlinear quantum mechanics due to difficulties and problems of original quantum mechanics, in which microscopic particles have only a wave feature, not corpuscle feature, which are completely not consistent with experimental results and traditional concept of particle. In this theory the microscopic particles are no longer a wave, but localized and have a wave-corpuscle duality, which are represented by the following facts, the solutions of dynamic equation describing the particles have a wave-corpuscle duality, namely it consists of a mass center with constant size and carrier wave, is localized and stable and has a determinant mass, momentum and energy, which obey also generally conservation laws of motion, their motions meet both the Hamilton equation, Euler-Lagrange equation and Newton-type equation, their collision satisfies also the classical rule of collision of macroscopic particles, the uncertainty of their position and momentum is denoted by the minimum principle of uncertainty. Meanwhile the microscopic particles in this theory can both propagate in solitary wave with certain frequency and amplitude and generate reflection and transmission at the interfaces, thus they have also a wave feature, which but are different from linear and KdV solitary wave’s. Therefore the nonlinear quantum mechanics changes thoroughly the natures of microscopic particles due to the nonlinear interactions. In this investigation we gave systematically and completely the distinctions and variations between linear and nonlinear quantum mechanics, including the significances and representations of wave function and mechanical quantities, superposition principle of wave function, property of microscopic particle, eigenvalue problem, uncertainty relation and the methods solving the dynamic equations, from which we found nonlinear quantum mechanics is fully new and different from linear quantum mechanics. Finally, we verify further the correctness of properties of microscopic particles described by nonlinear quantum mechanics using the experimental results of light soliton in fiber and water soliton, which are described by same nonlinear Schrödinger equation. Thus we affirm that nonlinear quantum mechanics is correct and useful, it can be used to study the real properties of microscopic particles in physical systems.

Quantum Becoming?

2017 ◽  
Author(s):  
Craig Callender
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Time ◽  
Quantum Non Locality ◽  
Time Quantum ◽  
Temporal Becoming ◽  
Non Locality ◽  
Quantum Wavefunction

Two of quantum mechanics’ more famed and spooky features have been invoked in defending the idea that quantum time is congenial to manifest time. Quantum non-locality is said by some to make a preferred foliation of spacetime necessary, and the collapse of the quantum wavefunction is held to vindicate temporal becoming. Although many philosophers and physicists seek relief from relativity’s assault on time in quantum theory, assistance is not so easily found.

Sign in / Sign up

Export Citation Format

Share Document