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.

Differentiation

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

q-integral

A function F is a q-antiderivative of f if DqF = f. The q-antiderivative is denoted by and an expression for F can be found from the formula
which is called the Jackson integral of f. For, the series converges to a function F on an interval xα| is bounded on the interval then dgq = dqt.

h-integral

A function F is an h-antiderivative of f if DhF = f. The h-antiderivative 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 on the interval partitioned into subintervals of width h.

Example

The derivative of the function 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

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.