# Quantum calculus

**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", where 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.

## Contents

## Differentiation[edit]

In the q-calculus and h-calculus, differentials of functions are defined as

and

respectively. Derivatives of functions are then defined as fractions by the q-derivative

and by

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.

## Integration[edit]

### q-integral[edit]

A function *F*(*x*) is a q-antiderivative of *f*(*x*) if *D*_{q}*F*(*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 *q*^{j}, with the jump at the point *q*^{j} being *q*^{j}. If we call this step function *g*_{q}(*t*) then *dg*_{q}(*t*) = *d*_{q}*t*.^{[1]}

### h-integral[edit]

A function *F*(*x*) is an h-antiderivative of *f*(*x*) if *D*_{h}*F*(*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*.

## Example[edit]

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

and

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.

## History[edit]

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.

## See also[edit]

## References[edit]

**^**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 - Victor Kac, Pokman Cheung, Quantum calculus
*, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8*