# 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

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

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

## Differentiation

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

${\displaystyle d_{q}(f(x))=f(qx)-f(x)\,}$

and

${\displaystyle d_{h}(f(x))=f(x+h)-f(x)\,}$

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

${\displaystyle D_{q}(f(x))={\frac {d_{q}(f(x))}{d_{q}(x)}}={\frac {f(qx)-f(x)}{(q-1)x}}}$

and by

${\displaystyle D_{h}(f(x))={\frac {d_{h}(f(x))}{d_{h}(x)}}={\frac {f(x+h)-f(x)}{h}}}$

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 ${\displaystyle \int f(x)\,d_{q}x}$ and an expression for F(x) can be found from the formula ${\displaystyle \int f(x)\,d_{q}x=(1-q)\sum _{j=0}^{\infty }xq^{j}f(xq^{j})}$ 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)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, 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 ${\displaystyle \int f(x)\,d_{h}x}$. If a and b differ by an integer multiple of h then the definite integral${\displaystyle \int _{a}^{b}f(x)\,d_{h}x}$ 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 ${\displaystyle x^{n}}$ (for some positive integer ${\displaystyle n}$) in the classical calculus is ${\displaystyle nx^{n-1}}$. The corresponding expressions in q-calculus and h-calculus are

${\displaystyle D_{q}(x^{n})={\frac {q^{n}-1}{q-1}}x^{n-1}=[n]_{q}\ x^{n-1}}$

with the q-bracket

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

and

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

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