Quantum calculus

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.

q-integral

A function F(x) is a q-antiderivative of f(x) if D_qF(x) = f(x). The q-antiderivative (or q-integral) is denoted by ∫ f ( x ) d q x and an expression for F(x) can be found from the formula ∫ f ( x ) d q x = ( 1 − q ) ∑ j = 0 ∞ x q j f ( x q j ) 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_qt.

h-integral

A function F(x) is an h-antiderivative of f(x) if D_hF(x) = f(x). The h-antiderivative (or h-integral) is denoted by ∫ f ( x ) d h x . If a and b differ by an integer multiple of h then the definite integral ∫ a b f ( x ) d h x is given by a Riemann sum of f(x) on the interval [a,b] partitioned into subintervals of width h.

Example

The derivative of the function x n (for some positive integer n ) in the classical calculus is n x n − 1 . The corresponding expressions in q-calculus and h-calculus are

with the q-bracket

and

respectively. The expression [ n ] q x n − 1 is then the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function x n is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of x n is instead the falling factorial, ( x ) n := x ( x − 1 ) ⋯ ( x − n + 1 ) . 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.