Quantum calculus, typically referred to as calculus with out limits, is equal to conventional infinitesimal calculus with out the notion of limits. It defines “q-calculus” and “h-calculus”, the place h ostensibly stands for Planck’s fixed whereas q stands for quantum. The 2 parameters are associated by the formulation
the place
ℏ
=
h
2
π
{displaystyle scriptstyle hbar ={frac {h}{2pi }},}
is the decreased Planck fixed.
Contents
Differentiation[edit]
Within the q-calculus and h-calculus, differentials of features are outlined as
and
respectively. Derivatives of features are then outlined as fractions by the q-derivative
and by
Within the restrict, as h goes to 0, or equivalently as q goes to 1, these expressions tackle the type of the spinoff of classical calculus.
Integration[edit]
q-integral[edit]
A operate F(x) is a q-antiderivative of f(x) if DqF(x) = f(x). The q-antiderivative (or q-integral) is denoted by
∫
f
(
x
)
d
q
x
{displaystyle int f(x),d_{q}x}
and an expression for F(x) could be discovered from the formulation
∫
f
(
x
)
d
q
x
=
(
1
−
q
)
∑
j
=
∞
x
q
j
f
(
x
q
j
)
{displaystyle int f(x),d_{q}x=(1-q)sum _{j=0}^{infty }xq^{j}f(xq^{j})}
which is named the Jackson integral of f(x). For 0 < q < 1, the sequence converges to a operate F(x) on an interval (0,A] if |f(x)xα| 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 operate having infinitely many factors of improve on the factors qj, with the bounce on the level qj being qj. If we name this step operate gq(t) then dgq(t) = dqt.[1] h-integral[edit] A operate F(x) is an h-antiderivative of f(x) if DhF(x) = f(x). The h-antiderivative (or h-integral) is denoted by ∫ f ( x ) d h x {displaystyle int f(x),d_{h}x} . If a and b differ by an integer a number of of h then the particular integral ∫ a b f ( x ) d h x {displaystyle int _{a}^{b}f(x),d_{h}x} is given by a Riemann sum of f(x) on the interval [a,b] partitioned into subintervals of width h.
Instance[edit]
The spinoff of the operate
x
n
{displaystyle x^{n}}
(for some constructive integer
n
{displaystyle n}
) within the classical calculus is
n
x
n
−
1
{displaystyle nx^{n-1}}
. The corresponding expressions in q-calculus and h-calculus are
with the q-bracket
and
respectively. The expression
[
n
]
q
x
n
−
1
{displaystyle [n]_{q}x^{n-1}}
is then the q-calculus analogue of the easy energy rule for
constructive integral powers. On this sense, the operate
x
n
{displaystyle x^{n}}
continues to be good within the q-calculus, however somewhat
ugly within the h-calculus – the h-calculus analog of
x
n
{displaystyle x^{n}}
is as an alternative the falling factorial,
(
x
)
n
:=
x
(
x
−
1
)
⋯
(
x
−
n
+
1
)
.
{displaystyle (x)_{n}:=x(x-1)cdots (x-n+1).}
One could proceed additional and develop, for instance, equal notions of Taylor enlargement, et cetera, and even arrive at q-calculus analogues for the entire standard features one would need to have, akin to an analogue for the sine operate whose q-derivative is the suitable analogue for the cosine.
Historical past[edit] – “q calculus”
The h-calculus is simply the calculus of finite variations, which had been studied by George Boole and others, and has confirmed helpful in a variety of fields, amongst them combinatorics and fluid mechanics. The q-calculus, whereas courting in a way again to Leonhard Euler and Carl Gustav Jacobi, is barely lately starting to see extra usefulness in quantum mechanics, having an intimate reference to commutativity relations and Lie algebra.
See additionally[edit]
References[edit]
“q calculus”