1. Introduction

In 1950 Wigner [¹] proposed a deformed Heisenberg algebra including reflection operator in the form,

where P denotes the reflection operator Pf(x) = f(-x). The realization of this algebra [²,³] was given by

where we set ℏ = 1 and the Dunkl derivative [³] is defined as

In fact, Wigner was the first who discussed the question on supreme level of the Heisenberg-Lie equations on the commutation relation between the momentum and position operator. In 1951 Yang simply developed the problem treated by Wigner and obtained the well-known non-canonical description of the non-relativistic momentum operator [²]. In 1980 N. Mukunda et al investigated Energy position and momentum eigenstates of para-Bose oscillator operators. They found that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent [³]. Also in next year coherent states and the minimum uncertainty states of para-Bose oscillator operators investigate by J. K. Sharma et al. [⁴]. In 1989 Dunkl constructed a commutative set of first-order differential difference operators associated to the second-order operator [⁵]. The canonical approach to the non-relativistic quantum mechanics with v = 0 is a special case of the non-canonical approach for arbitrary ν that was proposed by Wigner. Reference [⁶] shows the Wigner function of the ground state for the para-Bose oscillator (it is an oscillator that was discussed in Refs. [³,⁴]) that generalizes Gaussian distribution almost overlaps with the Wigner function of the canonical non-relativistic bosonic harmonic oscillator. If there is overlap of the Wigner function of non-canonical ground state with the any excited canonical state, then it means that not only Ramsauer-Townsend effect, but also a lot of other physical effects can be obtained theoretically if to replace canonical momentum operator with non-canonical one. Some studies related to the Wigner-Dunkl quantum mechanics have been accomplished in [⁷-¹²].

In one-dimensional Wigner-Dunkl quantum mechanics, the inner product is given by [⁷,¹²]

where |x| ^2v is a weight function. The expectation value of a physical operator 𝒪 with respect to the state ψ(x, t) is defined by

and 𝒪 is a Hermitian operator if it obeys

For the weight function (6) the momentum operator p^=1/iDx is a Hermitian operator.

In this paper we study the continuity Equation for Wigner-Dunkl-Schrödinger equation. We discuss some properties of v-deformed functions related to Dunkl derivative. Using these we discuss the step potential and Ramsauer-Townsend effect in Wigner-Dunkl quantum mechanics. This paper is organized as follows: In 2 we discuss continuity equation for Wigner-Dunkl-Schrödinger equation. In 3 we discuss the v-deformed functions. In 4 we discuss Step potential. In 5 we discuss Ramsauer-Townsend effect.

2. Continuity equation for Wigner-Dunkl-Schrödinger equation

Now let us consider the time-dependent Wigner-Dunkl-Schrödinger equation,

Now let us derive the continuity equation for the Wigner-Dunkl-Schrödinger equation. From Eq.(7) we have

The wave function can be split into the even part and odd part,

Now let us multiply the weight function K(x)=|x|2ν by Eq.(8),

Let us set

where we set

Then we have

where

Thus we have the continuity equation of the form,

where the flux is

and the source is

The derivation of Eq.(17) is given in Appendix A. Then we have

where we have

because f is odd. Thus, |x|2ν|ψ|2 can be interpreted as the probability density function.

Now let us consider the time-independent Wigner-Dunkl-Schrödinger equation,

or

This means that Dx2ψ(x) is not continuous in general because the potential can be discontinuous.

To find the continuity condition in the Wigner-Dunkl-Schrödinger equation, we should find the inverse of Dunkl derivative. The Dunkl derivative can be written as

The inverse of Dunkl derivative is

where ∂x-1 denotes the ordinary integration. Thus we have

where Dunkl number is

Because the inverse of Dunkl derivative is expressed in terms of multiple of the ordinary integration, from Eq. (23) we know that both Dxψ and ψ are continuous.

3. The v-deformed functions

Now let us find the v-exponential function obeying

We consider the ν-deformed differential equation

Let us set

where y _e (x) is the even function obeying Pye(x)=ye(x) while y _o (x) is the odd function obeying Pyo(x)=-yo(x). Inserting Eq. (30) into Eq. (28) and splitting into the even part and odd part we get

Let us set

Inserting Eqs. (33,34) into Eq. (31,32), we get

From the above equations we have

Thus, we have

Thus, we have

where

and

and

One can also express the ν-deformed hyperbolic functions as

where I _α (x) denotes the modified Bessel function. These deformed hyperbolic functions reduce to cosh(ax) and sinh(ax) in the limit v → 0. The v-deformed hyperbolic functions obey

Action of the v-derivative gives

If we replace x → ix, , we have the v-deformed Euler relation

where

One can also express the v-deformed trigonometric functions as

where Jα(x) denotes the Bessel function. The v-deformed trigonometric functions obey the following relations

Figure 1 shows the plot of y=cosν(x) for v = 0 (Pink), v = 0.1 (Brown) and for v = -0.1 (Gray). Figure 2 shows the plot of y=sinν(x) for v = 0 (Pink), v = 0.1 (Brown) and for v = -0.1 (Gray).

4. Step potential

Now let us consider the step potential problem whose potential is given by

Now let us consider the case of 0 < V ₀ < E. Then, Wigner-Dunkl-Schrödinger equation reads

for x < 0, while it reads

for x > 0. The solution is given by

and

where

From the continuity of ψ and Dxψ we have

Thus we have

which is the same as the case of v = 0. In this case the transmission and reflection flux are

We see that transmission and reflection flux are independent of the v-deformed parameter. Figure 3 shows the plot of R and T versus E with V ₀ = 1.

5. Ramsauer-Townsend effect

Let us consider the quantum well whose potential is given by

where V ₀ is a positive constant and we assume E > 0.

Now let us consider the Wigner-Dunkl-Schrödinger equation for three cases:

Solving three equations, we get

where

Now the boundary conditions are the continuity of the wave functions and their first Dunkl derivatives at the boundaries, which are

Solving Eqs. (76-79) for A we get

The reflection probability density is given by

where

In Fig. 4 shows the behavior of the reflection probability density versus energy. Now let us investigate the Ramsauer-Townsend effect. This effect is a physical phenomenon involving the scattering of low-energy electrons by atoms of a noble gas. Ramsauer-Townsend effect is no reflection condition. From |A|² = 0, we have

Thus, we have

where αν+1/2,p denotes p-th zero of Jν+1/2(x).

6. Conclusion

In this paper we derived the continuity equation for Wigner-Dunkl-Schrödinger equation. We found the flux and probability density. We found that the probability conserved in time. From the Dunkl integral ( inverse of Dunkl derivative ), we found that the wave function and first order Dunkl derivative are continuous although the potential is not continuous. We introduced the v-deformed functions related to Dunkl derivative and investigated their mathematical properties. Using the continuity condition in the Wigner-Dunkl-Schrödinger equation we discussed two examples; the step potential problem and Ramsauer-Townsend effect.