Module 1 : Atomic Structure
Lecture 4 : The Schrodinger Equation
  4.1

Introduction

In the lecture 2, we have seen that the quantum mechanical description of a particle moving in one, two and three dimensions brings out the essential aspects of operators, wave functions and boundary conditions. The hydrogen atom represents the simplest atomic system in which an electron moves in three-dimensions under the influence of the nuclear charge of + e = 4.8 x 10 -10 e.s.u. We shall examine the quantum mechanical structure of the hydrogen atom in considerable detail as it provides important insights regarding the structure of multi-electron atoms, molecules and solids.

 

The Schrodinger equation for hydrogen atom can be written as,

 

(4.1a)

or

(4.1b)
 

* (Cartesian coordinates)
 

*(1 / r) ( 2 / r 2) r + (1/r 2 ) (1/sin2 ) ( 2 / 2 ) + (1/sin ) ( / ) sin ( / ) (Spherical polar coordinates)

(4.1c)
 

The potential energy term arises form the Coulombic interaction between the electron (charge of – e) and the proton (charge of + e) and separated by a distance ‘r'.( 0 is the permittivity of vacuum. We have written the columbic interaction in MKS units here, so that your will be familiar with both the MKS and the CGS units used later.) The Schrodinger equation for the hydrogen atom has been solved exactly i.e; the solutions can be expressed in terms of standard mathematical / analytical functions. The energies predicted by the equation match with the experimentally observed energy levels.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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