Quantum calculus

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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

dq(f(x)) = f(qx) − f(x)
dh(f(x)) = f(x + h) − f(x)

Then 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 xn for some positive integer n. Its derivative in the classical calculus is simply nxn − 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 Dqxn = {n}qxn − 1. This is the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function xn is still nice in the q-calculus, but rather ugly 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.

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[edit] References

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