Quantum Groups, q-Oscillators and q-Deformed Quantum Mechanics
99
For q-exponential functions,ex p q X · ex p q Y = ex p q (X + Y ) . The q-analogues of
trigonometric functions are defined as
sin q x =
1
2i
ex p q (i x) − ex p q (−i x)
(36)
cos q x =
1
2
ex p q (i x) + ex p q (−i x)
(37)
The q-difference operator D x is defined by [15]
D x f (x) =
f (qx) − f (x)
x(q − 1)
(38)
This operator is defined on a q-lattice in which the lattice points are in a geometric
sequence. This q-difference operator does not possess q ↔ q
−1 symmetry. As q →
1, D x →
d
dx
, if it exists. It is evidently a linear operator. Also, it is easy to show that
D x (x
n
) = [n] q x
n−1
(39)
Hence we have the following results:
D x ex p q (x) = ex p q (x)
(40)
D x sin q (x) = cos q (x)
(41)
D x cos q (x) = − sin q (x)
(42)
Hence sin q (x) and cos q (x) are solutions of the q-difference equation
D
2
x + k
2
f (x) = 0
(43)
The q-analogues of the Leibinitz product rule and the quotient rule are found as
D x (u(x)v(x)) = u(qx)D x v(x) + v(x)D x u(x)
= u(x)D x v(x) + v(qx)D x u(x)
(44)
D x (u(x)/v(x)) =
v(x)D x u(x) − u(x)D x v(x)
v(qx)v(x)
(45)
q-integration is defined by
b
a
f (x)d(qx) = (1 − q)
b
∞
r =0
q
r f (q
r b) − a
∞
r =0
q
r f (q
r a)
(46)
Product rule for this q-integral is