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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

The evaluation of matrix elements in functional integral form

J. L. Martin

Published Dec 9, 1958• 1 citation
Publisher

Abstract

Abstract The source-function technique of Schwinger is used to express any transformation matrix element in a form involving a functional derivative operator. An elementary algebra of such expressions is developed, and in the case of quantum mechanics is found to lead to a natural evaluation procedure; the result expresses the matrix element as an indefinitely repeated integral, with the integrations arranged in order of importance. Discrete as well as continuous eigenvalues can be dealt with by this method. In so far as the procedure is a kind of ‘sum-over-histories’, the result is related to the Feynman principle. The field-theoretic situation requires further developments in technique.

DOI
10.1098/rspa.1958.0261

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Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.