Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus". h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula
where is the reduced Planck constant.
In the q-calculus and h-calculus, differentials of functions are defined as
In the limit, as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.
A function F(x) is a q-antiderivative of f(x) if DqF(x)=f(x). The q-antiderivative (or q-integral) is denoted by and an expression for F(x) can be found from the formula which is called the Jackson integral of f(x). For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if |f(x)x^α| is bounded on the interval (0,A] for some 0 <= α < 1.
The q-integral is a Riemann-Stieltjes integral with respect to a step function having infinitely many points of increase at the points qj, with the jump at the point qj being qj. If we call this step function gq(t) then dgq(t) = dqt.
A function F(x) is an h-antiderivative of f(x) if DhF(x)=f(x). The h-antiderivative (or h-integral) is denoted by . If a and b differ by an integer multiple of h then the definite integral is given by a Riemann sum of f(x) on the interval [a,b] partitioned into subintervals of width h.
The derivative of the function (for some positive integer ) in the classical calculus is . The corresponding expressions in q-calculus and h-calculus are
with the q-bracket
respectively. The expression is then the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of is instead the falling factorial, One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine.
The h-calculus is just the calculus of finite differences, which had been studied by George Boole and others, and has proven useful in a number of fields, among them combinatorics and fluid mechanics. The q-calculus, while dating in a sense back to Leonhard Euler and Carl Gustav Jacobi, is only recently beginning to see more usefulness in quantum mechanics, having an intimate connection with commutativity relations and Lie algebra.
- FUNCTIONS q-ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS, LUIS DANIEL ABREU, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 04–32
- F. H. Jackson (1908), "On q-functions and a certain difference operator", Trans. Roy. Soc. Edin., 46 253-281.
- Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538