Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. Unlike traditional calculus, which applying the concept of limits to analyze function. in fact the property of traditional calculus it is also holds in Quantum Calculus. In Quantum Calculus there are two distinct type of calculus i.e 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 ${\textstyle \hbar ={\frac {h}{2\pi }}}$ 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

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

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

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.

See also

• Noncommutative geometry
• Quantum differential calculus
• Time scale calculus
• q-analog
• Basic hypergeometric series
• Quantum dilogarithm

Further reading

• George Gasper, Mizan Rahman, Basic Hypergeometric Series, 2nd ed, Cambridge University Press (2004), ISBN 978-0-511-52625-1, doi:10.1017/CBO9780511526251

References

• Jackson, F. H. (1908). "On q-functions and a certain difference operator". Transactions of the Royal Society of Edinburgh. 46 (2): 253–281. doi:10.1017/S0080456800002751. S2CID 123927312.
• Exton, H. (1983). q-Hypergeometric Functions and Applications. New York: Halstead Press. ISBN 0-85312-491-4.
• Kac, Victor; Cheung, Pokman (2002). Quantum calculus. Universitext. Springer-Verlag. ISBN 0-387-95341-8.