Symbol calculus by symbolic computation and semi-classical expansions

Abstract

We present a survey of our work about applications of symbol calculus to quantum mechanics. Symbol calculus embodies the rules of algebra for the linear partial differential operators with variable coefficients, when such operators are expressed as functions of the non-commuting quantities xj and ~/~xj (j=l ..... n). In quantum mechanics the symbol calculus generates S~-C~sica£ expansions, i.e. perturbative series in powers of Planck's constant ~, that are both numerically useful and mathematically interesting. As the complexity of such expansions increases very rapidly with the order, algebraic computer systems have proven most useful in their study [l] . We present here a few of our results, obtained with the extensive use of the formula manipulation language REDUCE (except when specified) : in particular we have found new algorithms to solve the stationary SchrBdinger equation formally in powers of ~. Basic facts about quantum mechanics and semi-classical methods are found in [3] any textbook such as . More precise references, about symbol calculus for instance, are listed in our earlier articles [4j.~ ~