Application of Radi al Wave Equation for HCB Mode


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 surface-to-surface coordinate representation. The radial wave equation has been expressed in the reduced form.
Key Words:

Radial wave equation, HCB Coordinate System.
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 co-ordinate system has been described first and expressed for the pair intermolecular potential specified in terms of the function h(x) and surface-to-surface co-ordinate 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 co-ordinate

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 7o ui In terms of HCB’S co-ordinates ui The expression for in te rms of HCB’S co-ordinate system is uio
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 co-ordinate system. This is done by using the concept of orthogonal curvilinear co- ordinates. The expression for (Laplacian) in orthogonal curvilinear co-ordinates is given by

Where u 1 , u 2 and u 3 are called orthogonal curvilinear co-ordinates 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. uio
in which E is the total energy of the system. uio

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] uio

This equation will be solved by the method of separation of variables by putting uio

where Y (q, f ) are the spherical harmonics and Y (K) satisfy the radial wave Equation. uioi



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 semi-minor 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.

1. T. Kihara, Adv. Chem. Phys. 5, 147 (1963).

2. S Chapman and T G Cowling The Mathematical Theory of Non-uniform 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).

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