^{1}

^{*}

^{2}

^{*}

In this paper, the Schr?dinger equation is solved by Modified separation of variables (MSV) method suggested by Pishkoo and Darus. Using this method, Meijer’s G-function solutions are derived in cylindrical coordinate system for quantum particle in cylindrical can. All elementary functions and most of the special functions which are the solution of extensive problems in physics and engineering are special cases of Meijer’s G-functions.

Perhaps, Cornelis Simon Meijer (1936) was the first to introduce the G-function in mathematics. This very general function intended to include most of the known special functions as particular cases; and for a long time, many studies have been done involving this type of functions. These functions have a lot of practical applications in the fields of mathematical physics, theoretical physics, mathematical analysis, etc.

Meijer’s G-functions are defined as Mellin-Barnes contour integrals which have been in existence for over 60 years [1-5]. Meijer’s G-function satisfies the linear ordinary differential equation (LODE) of the generalised hypergeometric type whose order is equal to

Our previous works had focused on the introduction of the Modified separation of variables method (MSV), and applying it to solve partial differential equation related to the Reaction-Diffusion process [

We begin with the definition of Meijer’s G-function as the following:

Definition 1 A definition of the Meijer’s G-function is given by the following path integral in the complex plane, called Mellin-Barnes type integral:

Here, the integers

Exercise 1 Using (1.1), wo obtain the follows

The Meijer’s G-function

whose order is equal to

Choosing appropriate values for

We start with using Modified separation of variables method (MSV) in cylindrical coordinates system as follows:

When the geometry of the boundaries is cylindrical, the appropriate coordinate system is the cylindrical one. Separation of variables leads to ODEs in which certain constants (eigenvalues) appear. Different choices of signs for these constants can lead to different functional forms of the general solution. Thus general form of the solution is indeterminate. However, once the boundary conditions are imposed, the unique solutions will emerge regardless of the initial functional form of the solutions. Writing

where in anticipation of the correct BCs, we have written the constants as

is a combination of two independent solutions deduced by “separation of variables method” or

is a combination of two independent solutions deduced by “modified separation of variables method”. Similarly

is a combination of two independent solutions deduced by “separation of variables method” or

is a combination of two independent solutions deduced by “modified separation of variables method”.

1) For

By changing

By multiplying both sides of the equation by −4, we have

On the other hand, let Bessel equation

The conditions for equivalence or these two differential equations are

and its solutions is

Exercise 2 Considering a quantum particle in a cylindrical can, for an atomic particle of mass μ confined in a cylindrical can of length L and radius a, the relevant Schrödinger equation is

Let us solve this equation subject to the BCs that

Here is the steps: A separation of variables,

The second DE, the

Since the extra condition of periodicity is usually imposed on the potential for variable

If we let

Then, the energy eigenvalues are

On the other hand,

The condition for the equivalence of these two differential equations is given by the solution of t-component. Thus, the general solution can be written as

&