# 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". h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula

$q = e^{i h} = e^{2 \pi i \hbar} \,$

where $\scriptstyle \hbar = \frac{h}{2 \pi} \,$ is the reduced Planck constant.

## Differentiation

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

$d_q(f(x)) = f(qx) - f(x) \,$

and

$d_h(f(x)) = f(x + h) - f(x) \,$

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

$D_q(f(x)) = \frac{d_q(f(x))}{d_q(x)} = \frac{f(qx) - f(x)}{(q - 1)x}$

and by

$D_h(f(x)) = \frac{d_h(f(x))}{d_h(x)} = \frac{f(x + h) - f(x)}{h}$

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(x) is a q-antiderivative of f(x) if DqF(x)=f(x). The q-antiderivative (or q-integral) is denoted by $\int f(x)d_qx$ and an expression for F(x) can be found from the formula $\int f(x)d_qx = (1-q)\sum_{j=0}^\infty xq^jf(xq^j)$ 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 qj, with the jump at the point qj being qj. If we call this step function gq(t) then dgq(t) = dqt.[1]

### h-integral

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 $\int f(x)d_hx$. If a and b differ by an integer multiple of h then the definite integral$\int_a^b f(x)d_hx$ 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 $nx^{n-1}$. The corresponding expressions in q-calculus and h-calculus are

$D_q(x^n) = \frac{q^n - 1}{q - 1} x^{n - 1} = [n]_q\ x^{n - 1}$

with the q-bracket

$[n]_q = \frac{q^n - 1}{q - 1}$

and

$D_h(x^n) = x^{n - 1} + h x^{n - 2} + \cdots + h^{n - 1}$

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, $(x)_n := x(x-1)\cdots(x-n+1).$ 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.