Path Integral Formulation of Quantum Mechanics Notes (original) (raw)
Related papers
On the Fundamental Nature of the Quantum Mechanical Probability Function
2012
The probability of occurrence of an event or that of the existence of a physical state has no relative existence in the sense that motion is strongly believed to only exist in the relative sense. If the probability of occurrence of an event or that of the existence of a physical state is known by one observer, this probability must be measured to have the same numerical value by any other observer anywhere in the Universe. If we accept this bare fact, then, the probability function can only be a scalar. Consequently, from this fact alone, we argue that the quantum mechanical wavefunction can not be a scalar function as is assumed for the Schrödinger and the Klein-Gordon wavefunctions. This has fundamental implications on the nature of the wavefunction insofar as translations from one reference system to the other is concerned.
Classical" propagator and path integral in the probability representation of quantum mechanics
Journal of Russian Laser Research, 1999
In the probability representation of the standard quantum mechanics, the explicit expression (and its quasiclassical van-Fleck approximation) for the "classical" propagator (transition probability distribution), which completely describes the quantum system's evolution, is found in terms of the quantum propagator. An expression for the "classical" propagator in terms of path integral is derived. Examples of free motion and harmonic oscillator are considered. The evolution equation in the Bargmann representation of the optical tomography approach is obtained.
MEASUREMENT IN QUANTUM PHYSICS
International Journal of Modern Physics E, 1999
The conceptual problems in quantum mechanics -related to the collapse of the wave function, the particle-wave duality, the meaning of measurement -arise from the need to ascribe particle character to the wave function. As will be shown, all these problems dissolve when working instead with quantum fields, which have both wave and particle character. Otherwise the predictions of quantum physics, including Bell's inequalities, coincide with those of the conventional treatments. The transfer of the results of the quantum measurement to the classical realm is also discussed.
A detailed interpretation of probability, and its link with quantum mechanics
Eprint Arxiv 1011 6331, 2010
In the following we revisit the frequency interpretation of probability of Richard von Mises, in order to bring the essential implicit notions in focus. Following von Mises, we argue that probability can only be defined for events that can be repeated in similar conditions, and that exhibit 'frequency stabilization'. The central idea of the present article is that the mentioned 'conditions' should be well-defined and 'partitioned'. More precisely, we will divide probabilistic systems into object, environment, and probing subsystem, and show that such partitioning allows to solve a wide variety of classic paradoxes of probability theory. As a corollary, we arrive at the surprising conclusion that at least one central idea of the orthodox interpretation of quantum mechanics is a direct consequence of the meaning of probability. More precisely, the idea that the "observer influences the quantum system" is obvious if one realizes that quantum systems are probabilistic systems; it holds for all probabilistic systems, whether quantum or classical.
The interpretation of quantum mechanics and of probability: Identical role of the 'observer
Eprint Arxiv 1106 3584, 2011
The aim of the article is to argue that the interpretations of quantum mechanics and of probability are much closer than usually thought. Indeed, a detailed analysis of the concept of probability (within the standard frequency theory of R. von Mises) reveals that the latter concept always refers to an observing system. The enigmatic role of the observer in the Copenhagen interpretation therefore derives from a precise understanding of probability. Besides explaining several elements of the Copenhagen interpretation, our model also allows to reinterpret recent results from 'relational quantum mechanics', and to question the premises of the 'subjective approach to quantum probabilities'.
Rules of probability in quantum mechanics
Foundations of Physics, 1988
We show that the quantum mechanical rules for manipulating probabilities follow naturally from standard probability theory. We do this by generalizing a result of Khinchin regarding characteristic functions. From standard probability theory we obtain the methods" usually assoeiated with quantum theory, that is', the operator method, eigenvalues, the Born rule, and the fact that only the eigenvalues of the operator have nonzero probability. We discuss the general question as to why quantum mechanics seemingly necessitates different methods than standard probability theory and argue that the quantum mechanical method is much richer in its ability to generate a wide variety of probability distributions which are inaccessibe by way of standard probability theory.
Quantum probability: New perspectives for the laws of chance
The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.
Advanced Topics in Measurements, 2012
Measurement Systems 1.3 Advanced single photon experiments 1.3.1 Applying the bohr complementarity principle 1. general introduction (above) 2. key historical debates on the foundations of QM 340 Advanced Topics in Measurements www.intechopen.com From Conditional Probability Measurements to Global Matrix Representations on Variant Construction 3 3. analysis of key issues of QM 4. conditional construction proposed 5. exemplar results 6. analysis of visual distributions 7. using the variant solution to resolve longstanding puzzles 8. main results 9. final conclusions 2. Wave and particle debates in QM developments 2.1 Heisenberg uncertain principle The Heisenberg Uncertainty Principle HUP was established in 1927 [Heisenberg (1930)]. The HUP represented a milestone in the early development of quantum theory [Jammer (1974)]. It implies that it is impossible to simultaneously measure the present position of a particle while also determining the future motion of a particle or any system small enough to require a Quantum mechanical treatment. From a mathematical viewpoint, the HUP arises from an equation following the methodology of Fourier analysis for the motion [Q, P]=QP − PQ = ih. The later form of HUP is expressed as △p •△q ≈ h. This equation shows that the non-commutativity implies that the HUP provides a physical interpretation for the non-commutativity. 2.2 Bohr complementarity principle The HUP provided Bohr with a new insight into quantum behaviors [Bohr (1958)]. Bohr established the BCP to extend the idea of complementary variables for the HUP to energy and time, and also to particle and wave behaviors. One must choose between a particle model, with localized positions, trajectories and quanta or a wave model, with spreading wave functions, delocalization and interferences [Jammer (1974)]. Under the BCP, complementary descriptions e.g. wave or particle are mutually exclusive within the same mathematical framework because each model excludes the other. However, a conceptual construction allowed the HUP, the BCP and wave functions together with observed results to be integrated to form the Copenhagen Interpretation of QM. In the context of double slit experiments, the BCP dictates that the observation of an interference pattern for waves and the acquisition of directional information for particles are mutually exclusive. 2.3 Bohr-Einstein debates on wave and particle issues Bohr and Einstein remained lifelong friends despite their differences in opinion regarding QM [Bohr (1949; 1958)]. In 1926 Born proposed a probability theory for QM without any causal explanation. Einstein's reaction is well known from his letter to Born [Born (1971)] in which he said "I, at any rate, am convinced that HE [God] does not throw dice." Then in 1927 at the Solvay Conference, Heisenberg and Bohr announced that the QM revolution was over with nothing further being required. Einstein was dismayed [Bohr (1949); Bolles (2004)] for he believed that the underlying effects were not yet properly understood.
Quantum Probability: An Historical Survey
What is quantum probability Quantum Probability (QP) is a new branch of mathematics interconnecting classical probability, functional analysis, pure algebra, quantum physics and information and communication engineering. The mid seventies is the period that marks the beginning of QP as an autonomous discipline. Since then this cross disciplinary nature has accompanied the development of QP and still now it is one of its points of strength, making it an original new trend in contemporary mathematics as well as one of the earliest pioneers of non-commutative mathematics: a field now flourishing with the more recent development of quantum groups, non-commutative geometry, quantum computer, . . .
Interpretations of Quantum Mechanics and the measurement problem
2010
We present a panoramic view on various attempts to "solve" the problems of quantum measurement and macro-objectivation, i.e. of the transition from a probabilistic quantum mechanic microscopic world to a deterministic classical macroscopic world.