A reading of the book of Dirac's life entitled The Strangest Man is a most stirring experience, bringing one back to the beginnings of quantum mechanics where every attempt was made "to establish a basis for theoretical quantum mechanics founded exclusively on relationships between quantities which are in principle observable." The prime movers in this quest were Heisenberg and Dirac. One of Dirac's most important contributions in the passage from classical to quantum mechanics, a passage that consumed much of his early efforts, was unfortunately published in an obscure Russian journal where it remained largely unread until it was found by Feynman while a graduate student at Princeton. The paper posed the question, "what corresponds in the quantum theory to the Lagrangian method of classical mechanics?", a method that, as Dirac pointed out, is clearly superior in the simplicity of its structure to that of the classical Hamiltonian approach. Dirac's partial answer to this question provided the key to solving the problem of introducing the action integral into quantum mechanics that occupied Feynman's mind, leading to his formulation of the path integral technique. His contribution was followed two years later by Schwinger's independently derived statement of the quantum action principle, each contribution providing a complete formulation of quantum mechanics stated in terms of single principle. The present paper points out that the successful introduction of the action principle into quantum mechanics made possible by Dirac, enables one to proceed still further by extending Schwinger's quantum action principle to an open system, to an atom in a molecule. Thus the quantum theory of an atom in a molecule has its roots in the question posed by Dirac in 1933. The paper proposes a return to a greater use of the theorems of quantum mechanics in interpretive chemistry from that begun by Slater in 1933, a staunch advocate of theory following in the footsteps of observation.
A reading of the book of Dirac's life entitled The Strangest Man is a most stirring experience, bringing one back to the beginnings of quantum mechanics where every attempt was made "to establish a basis for theoretical quantum mechanics founded exclusively on relationships between quantities which are in principle observable." The prime movers in this quest were Heisenberg and Dirac. One of Dirac's most important contributions in the passage from classical to quantum mechanics, a passage that consumed much of his early efforts, was unfortunately published in an obscure Russian journal where it remained largely unread until it was found by Feynman while a graduate student at Princeton. The paper posed the question, "what corresponds in the quantum theory to the Lagrangian method of classical mechanics?", a method that, as Dirac pointed out, is clearly superior in the simplicity of its structure to that of the classical Hamiltonian approach. Dirac's partial answer to this question provided the key to solving the problem of introducing the action integral into quantum mechanics that occupied Feynman's mind, leading to his formulation of the path integral technique. His contribution was followed two years later by Schwinger's independently derived statement of the quantum action principle, each contribution providing a complete formulation of quantum mechanics stated in terms of single principle. The present paper points out that the successful introduction of the action principle into quantum mechanics made possible by Dirac, enables one to proceed still further by extending Schwinger's quantum action principle to an open system, to an atom in a molecule. Thus the quantum theory of an atom in a molecule has its roots in the question posed by Dirac in 1933. The paper proposes a return to a greater use of the theorems of quantum mechanics in interpretive chemistry from that begun by Slater in 1933, a staunch advocate of theory following in the footsteps of observation.
Starting from the earlier notions of stationary action principles, we show how Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. The connection between the two is brought out, and applications are discussed. The Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.
Through a very careful analysis of Dirac's 1932 paper on the Lagrangian in Quantum Mechanics as well as the second and third editions of his classic book {\it The Principles of Quantum Mechanics}, I show that Dirac's contributions to the birth of the path-integral approach to quantum mechanics is not restricted to just his seminal demonstration of how Lagrangians appear naturally in quantum mechanics, but that Dirac should be credited for creating a path-integral which I call {\it Dirac path-integral} which is far more general than Feynman's while possessing all its desirable features. On top of it, the Dirac path-integral is fully compatible with the inevitable quantisation ambiguities, while the Feynman path-integral can never have that full consistency. In particular, I show that the claim by Feynman that for infinitesimal time intervals, what Dirac thought were analogues were actually proportional can not be correct always. I have also shown the conection between Dirac path-integrals and the Schrödinger equation. In particular, it is shown that each choice of Dirac path-integral yields a {\it quantum Hamiltonian} that is generically different from what the Feynman path-integral gives, and that all of them have the same {\it classical analogue}. Dirac's method of demonstrating the least action principle for classical mechanics generalizes in a most straightforward way to all the generalized path-integrals.
Feynman's path integral formulation arose from his attempt to incorporate the Lagrangian framework into quantum mechanics, offering what he regarded as a more fundamental perspective than the Hamiltonian approach, particularly in the context of quantum electrodynamics. Inspired by an analogy proposed by P. A. M. Dirac, Feynman showed that Schrödinger's equation could be recovered by promoting this analogy to an equality, up to a constant factor. This insight laid the foundation for the path integral formalism. In this article, we first outline the historical development of dynamical frameworks in physics and then trace Feynman's reasoning as he constructed his formulation, beginning with the double-slit experiment and its interpretation. We subsequently derive the Schrödinger equation from the path integral and demonstrate the conservation of probability. These final sections aim to provide students with a clear link between this elegant yet less commonly presented approach and the more standard methods typically taught in undergraduate courses.
WHAT IS QUANTUM MECHANICS? A remarkable feature of the 1968 conference of Nobel prize winners in physics at Lindau is that it was possible for me to ask such a question in the presence of two of the founders of quantum mechanics, Werner Heisenberg and P.A.M. Dirac, more than 30 years after the discovery, in a lecture attended by 400 students who had recently begun their study of the subject. Several answers to the question are possible. The only easy one is that quantum mechanics is a discipline that provides a wonderful set of rules for calculating physical properties of matter. For such simple systems as hydrogen and helium atoms the calculated energy levels agree with experiment to fantastic accuracy. In more complicated cases the computations are difficult and the accuracy is lower, but it is reasonable to believe, in principle at least, that the theory would be adequate if only the calculational problems could be overcome.
In an attempt to give an exact mathematical formulation of Bohr*s Correspondence Principle, Heisenberg (June 1925) discovered the rules governing the behaviour of quantum-theoretical magnitudes.In fall 1925 Born, Heisenberg and Jordan and, independently, Dirac, formulated consistent algebraic schemes of quantum mechanics.Early in 1926 Schrodinger developed wave mechanics.In quick succession were discovered : Bora's probability interpretation of the wave function, the transformation theory of Dirac, Jordan and E. London, Heisenberg's uncertainty relations and Bohr's principle of complementarity.By September 1927 the basis of a complete theory of atomic phenomena had been established.Aspects of this development, in which Heisenberg played a central role, will be presented as a tribute to his memory.
This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.