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". h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula

q = e^{i h} = e^{2 \pi i \hbar} \,

where \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

d_q(f(x)) = f(qx) - f(x) \,

and

d_h(f(x)) = f(x + h) - f(x) \,

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

D_q(f(x)) = \frac{d_q(f(x))}{d_q(x)} = \frac{f(qx) - f(x)}{(q - 1)x}

and by

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

D_q(x^n) = \frac{q^n - 1}{q - 1} x^{n - 1} = [n]_q\ x^{n - 1}

with the q-bracket

[n]_q = \frac{q^n - 1}{q - 1}

and

D_h(x^n) = x^{n - 1} + h x^{n - 2} + \cdots + h^{n - 1}

respectively. The expression [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 x^n is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of x^n is instead the falling factorial, (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

References

  1. ↑ FUNCTIONS q-ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS, LUIS DANIEL ABREU, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 04–32