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(i h)

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

d q (f (x)) = f (q x) - f (x)

d h (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 $x n$ for some positive integer $n$ . Its derivative in the classical calculus is simply $n x n - 1$ . 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 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. 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.