Jump to content

Quantum calculus

From Wikipedia, the free encyclopedia

Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. The two types of calculus in quantum calculus are q-calculus and h-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit, the original object is returned. In q-calculus, the limit as q tends to 1 is taken of the q-analog. Likewise, in h-calculus, the limit as h tends to 0 is taken of the h-analog. The parameters and can be related by the formula .


The q-differential and h-differential are defined as:



respectively. The q-derivative and h-derivative are then defined as


respectively. By taking the limit as of the q-derivative or as of the h-derivative, one can obtain the derivative:



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:, 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)xα| 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..The jump at the point qj is qj. Calling this step function gq(t) gives dgq(t) = dqt.[1]


A function F(x) is an h-antiderivative of f(x) if DhF(x) = f(x). The 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 sub-intervals of equal width h. The motivation of h-integral comes from the Riemann sum of f(x). Following the idea of the motivation of classical integrals, some of the properties of classical integrals hold in h-integral. This notion has broad applications in numerical analysis, and especially finite difference calculus.


In infinitesimal calculus, the derivative of the function is (for some positive integer ). The corresponding expressions in q-calculus and h-calculus are:

where is the q-bracket


respectively. The expression is then the q-analog and is the h-analog of the power rule for positive integral powers. The q-Taylor expansion allows for the definition of q-analogs of all of the usual functions, such as the sine function, whose q-derivative is the q-analog of cosine.


The h-calculus is the calculus of finite differences, which was studied by George Boole and others, and has proven useful in combinatorics and fluid mechanics. In a sense, q-calculus dates back to Leonhard Euler and Carl Gustav Jacobi, but has only recently begun to find usefulness in quantum mechanics, given its intimate connection with commutativity relations and Lie algebras, specifically quantum groups.

See also[edit]


  1. ^ Abreu, Luis Daniel (2006). "Functions q-Orthogonal with Respect to Their Own Zeros" (PDF). Proceedings of the American Mathematical Society. 134 (9): 2695–2702. doi:10.1090/S0002-9939-06-08285-2. JSTOR 4098119.

Further reading[edit]