Step wise derivation of Schrodinger equation – Part 1 of 3

Introduction

The Schrodinger equation is one of the key concept in the quantum mechanics framework. It is equivalent of Newton’s second law in the quantum world. The Newton’s second law of motion predicts the evolution of a body based on the initial condition. The Schrodinger equation determines the evolution of a quantum system over the time.


Classical Mechanics
The Newton’s laws of motion are being used in the non relativistic system or bodies (cars, rocket, etc). Relativisitic systems are the one which are moving close to the speed of light. The classical mechanics does not work when it comes to the quantum world.

In the classical mechanics, the total energy of a system is sum of kinetic energy and potential energy.





In the above equation, multiply ‘m’ on both denominator and numerator on right hand side of the the equation, we get,



The momentum of a moving body is given by product of mass and velocity of the body.



Substituting the above in equation (1) we get,



Quantization
In quantum world, all the energy released or absorbed are quantized. The quantized unit of the energy is



Schrodinger Equation #1/3

Photo Electric Effect
Let us take Einstein’s photo electric effect. When electromagnetic radiation falls on a metal surface, if the frequency of the electromagnetic radiation is above a threshold frequency, the radiation has enough energy to break the electro static force holding the electrons in the metal surface. Once it breaks the electro static force, the electron flows from the metal surface. The flow of electron is current. The energy of the electron released from the metal surface depends on the frequency of the radiation falls on the metal surface. The energy of electron is multiple of . Where h is the Plank’s constant. The value of the Plank’s constant is very small. It is

Energy equation in angular terms
As stated in the photoelectric effort, the energy released in the quantum mechanics framework depends on the frequency. I prefer to call frequency as instead of . The Plank’s constant can be expressed in angular terms and denoted by . The angular frequency is given by . The energy can be expressed in terms of and .