Quantum Groups, q-Oscillators and q-Deformed Quantum Mechanics
103
where θ is the Heaviside step function defined by
θ(x) = 1 f or x > 0
= 0 f or x ≤ 0
(75)
4 q-Deformed Quantum Mechanics
The q-deformed harmonic oscillator inspire the search for a q-deformed quantum mechanics which produce the results of standard quantum mechanics when
the deformation parameter approaches a particular value. The q-deformed calculus, when applied to quantum mechanics with its fundamental postulates preserved,
gives rise to q-quantum mechanics. Quantum mechanics is usually deformed in two
different ways: either one may replace the canonical commutation relations by a
q-commutation relation or may replace the momentum operator in the Schrödinger
equation by a q-deformed one. The form of the q-momentum operator depends on
the definition of the q-difference operator that we use. Wess and Zumino [21] have
constructed a q-deformed momentum operator which is Hermitian when q is a root
of unity. Vinod et al. has developed a q-deformed Schrödinger equation with a nonHermitian momentum operator [22].
4.1 General Formulation of q-Quantum Mechanics
In the linear vector space E, we define a norm mapping E → R
+
: v → →v such
that
αv = |α| v; ;v + u ≤ ≤v + +u; ;v = 0 ⇒ v = 0
u, v ∈ E, α ∈ C
For square integrable function in the position space, the norm can be written as
v =
v
∗
(x)v(x)d(qx)
1/2
(76)
The scalar product of two vectors is defined as
E × E → C : (u, v) → u|v
such that