
Application of Radi al Wave Equation for HCB Mode
Abstract:
The Phase shifts needed for the calculation of the transport properties of hard convex body (HCB) fluid in quantum mechanics is the solution of the radial wave equation described by the HCB coordinate systems. The radial wave equation described by HCB coordinate system has been deduced and expressed for the pair intermolecular potential specified in terms of the function h(x) and surfacetosurface coordinate representation. The radial wave equation has been expressed in the reduced form.
Key Words:
Radial wave equation, HCB Coordinate System.
Introduction:
In quantum mechani cal calculation of the transport phenomena, the major problem is the evaluation of the Radial wave equation. The phase shifts is the solution of the radial wave equation. The expression for the radial wave equation of a HCB Model coordinate system has been described first and expressed for the pair intermolecular potential specified in terms of the function h(x) and surfacetosurface coordinate representation. The properties of hard convex bodies (HCB’s) necessary for our analysis are due to Kihara [1].
Expression for in t erms of HCB coordinate
Let us first assume that the convex body has a smooth surface and that each suppor ting plane has a contact of first order with the convex body. Let r ( ? , Ø) be the radius vector from the origin to the contact point of the body with the supporting plane in the direction ( ? , Ø). Then by use of the unit vector k ( ? , Ø) in the direction ( ? , Ø). So, the identity In terms of HCB’S coordinates The expression for in te rms of HCB’S coordinate system is
Radial Wave Equation
The radial wave equation described by the HCB co ordinate system is obtained from the expression (2) by the method of separation of variables. This method results the expression in one variable. The method most commonly used work by removing one or more partial derivative terms so that an equation with fewer variables is obtained. This may be repeated until an ordinary differential equation in one variable result. The Schrodinger equation for two particles interacting according to a potential function Ø(k), may be written as 

Thus for, the expression for in terms of the above identities, it is desirable to use the procedure for making the transformation from Cartesian coordinates to the required coordinate system. This is done by using the concept of orthogonal curvilinear co ordinates. The expression for (Laplacian) in orthogonal curvilinear coordinates is given by
Where u 1 , u 2 and u 3 are called orthogonal curvilinear coordinates and h 1 , h2 and h 3 are called scale factors. The essential task is determing the explicit form of is that of determing the scale factors. The condition for this transformation is that the Jacobin
is non zero.
in which E is the total energy of the system.
g being the relative speed of the colliding pair before the collision take place and is the reduced mass. If we define J by h J = µ g, the Schrodinger equation assumes the form [2]
This equation will be solved by the method of separation of variables by putting
where Y (q, f ) are the spherical harmonics and Y (K) satisfy the radial wave Equation.

The left side of equation (8) depends only on K, and the right side depends only on q and f, both sides must be equal to a constant that is  l ( l +1). Thus equation (8) gives us a radial equation
Where u (k) satisfies the equation (10) and th is equation corresponds to the angular momentum quantum The expression for Intermolecular pair potential [3] for HCB Model, f (K) = 4E [h(x) The radial wave equation (10) may be written in the reduced form
m is the mass of the particle, m is the reduced mass and is
is a reduced quantum parameter. The reduced quantum mechanical parameter
When x = cosq = 1, the orientation is along semi  major axis ‘a’ and
When x = cosq = 0, the orientation is along semiminor axis ‘b
Result and Discussion
The asymptotic solution of the radial wave equation for real(interacting) and ideal (non interacting) pairs of mol ecules are sinusoidal and differ only in the phase of the sine functions, the difference being the phase shifts,h l (J*). The phase shift depends upon the angular momentum quantum number l and the wave number of relative motion. 1. It is in general not possible to give an exact solution of the radial wave equation for the phases. The expression for the phase shifts has been given by N. F. mott [4] and applied by Hulthen to calculate h l for different potentials which gives very satisfactory results.


References:
1. T. Kihara, Adv. Chem. Phys. 5, 147 (1963).
2. S Chapman and T G Cowling The Mathematical Theory of Nonuniform Gases (Cambridge: University Press) (1970)
3. V S Giri and B P Akhouri Indian J. Phys. 77B 233 (2003). 4. N F Mott and H S W Massey The Theory of Atomic Collisions (Oxford: Clarendon) (1949). 
