Quantum calculus

Quantum calculus 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

:$q\; =\; exp(ih)$

We can define differentials of functions in the q-calculus and h-calculus by

:$d\_q(f(x))\; =\; f(qx)\; -\; f(x)$:$d\_h(f(x))\; =\; f(x\; +\; h)\; -\; f(x)$

We may then further define derivatives of functions as fractions by

:$D\_q(f(x))\; =\; frac\{d\_q(f(x))\}\{d\_q(x)\}\; =\; frac\{f(qx)\; -\; f(x)\}\{(q\; -\; 1)x\}$:$D\_h(f(x))\; =\; frac\{d\_h(f(x))\}\{d\_h(x)\}\; =\; frac\{f(x\; +\; h)\; -\; f(x)\}\{h\}$

In the limit, as h goes to 0, or equivalently as q goes to 1, we may reconstitute the derivative of the classical calculus. Now consider the function $x^n$ for some positive integer $n$. Its derivative in the classical calculus is simply $nx^\{n\; -\; 1\}$. We can calculate

:$D\_q(x^n)\; =\; frac\{q^n\; -\; 1\}\{q\; -\; 1\}\; x^\{n\; -\; 1\}$:$D\_h(x^n)\; =\; x^\{n\; -\; 1\}\; +\; h\; x^\{n\; -\; 2\}\; +\; cdots\; +\; h^\{n\; -\; 1\}$

By setting

:$\{n\}\_q\; =\; frac\{q^n\; -\; 1\}\{q\; -\; 1\}$

We can see that $D\_q\{x^n\}\; =\; \{n\}\_q\; x^\{n\; -\; 1\}$. This is the q-calculus analogue of the simple power rule forpositive integral powers. In this sense, the function $x^n$ is still "nice" in the q-calculus, but ratherugly in the h-calculus. 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.

Of course, 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, on the other hand, while dating in a sense back to Euler and Jacobi, is only recently beginning to see more usefulness in quantum mechanics, having an intimate connection with commutativity relations and Lie algebra.

See also

* Noncommutative geometry
* Time scale calculus
* Q-derivative

Notice that

References

*Victor Kac, Pokman Cheung, Quantum calculus", Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8