Schrodinger Equation – Part 1 of 3

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.

$Total Energy = K.E + P.E$

$K.E = \frac{mv^2}{2}$

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

$K.E = \frac{m^2v^2}{2m}$

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

$p = mv$

Substituting the above in equation (1) we get,

$K.E = \frac{p^2}{2m}$

Quantization
In quantum world, all the energy released or absorbed are quantized. The quantized unit of the energy is  $E = nh\nu$
 $n = 0,1,2,3,4.....$

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 $h\nu$. Where h is the Plank’s constant. The value of the Plank’s constant is very small. It is $6.62 *10^{-34} J S$

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  $f$ instead of  $\nu$. The Plank’s constant can be expressed in angular terms and denoted by  $\hbar = \frac{h}{2\pi}$. The angular frequency is given by  $\omega=2\pi f$ . The energy can be expressed in terms of  $\hbar$ and $\omega$.

 $E = \hbar \omega$

Cricket Match #2 (Season 2008) against A2C2

Falcons Vs A2C2 (Game #2, Season 2008 )

Falcon’s Batting

Detroit Falcons played their second match of 2008 season yesterday – Jun-07-2008. We came up with great win against A2C2 (Ann Arbor Cricket Club) team. A2C2 won the toss and elected to field first. I and Babu went as the openers. We had a great start (around 22 in 3 overs) and was slowly picking up the aggressive strokes. Made critical singles , doubled easy singles and occasional boundaries. When I was at 8, Babu signaled me for a double which was meant to be an easy single and I was rushing towards the danger end. As usual, I run fast when required and slow down when required. I was running really fast to reach my end. Babu was running at the same straight line as I was. He made me to slow down completely and I tried to gain the speed after a sudden stop and not good enough to reach my end before the wicket keeper removed the bails. It was an easy decision for the leg umpire and my good start ended in vain.

Falcon batsmen belted A2C2 all over the ground and scored Falcons highest score ever. 150/7 in 25 overs. Matta scored 50+, Hari scored 26+, Sudhir P,Babu,Sai scored 14+ each, and rest scored around 8-10 each.

Falcon’s Bowling

I love wicket keeping. I have been doing wicket keeping since I was in 8th grade. During my school days, I thought I would at least play for my state as a wicket keeper. That dream evaporated long ago when I was in the college. That is a long story and deserve a separate blog posting.

My team mates and others tells me that sometime I take spectacular catches and unfortunately I can’t see those catches by myself. In this match, there were 5 catching opportunities for me. First one was a spectacular catch, I was told. Second was could have been another spectacular catch but I reacted slow. I missed it. Third one was an easy catch but umpire did not hear it. He did a very good job except that call. Fourth and five th catching opportunities were easy catches and were given out. I took three catches in this match.

Run outs

There were three run out opportunities for me. First run out was an easy and was given out. It was a good throw by Hari. Second was very close but the leg umpire was not convinced. It was an excellent throw by Sudhir from cover. Third was a direct throw but I screwed it up. It was a good direct hit by Hari and I was running towards the stump to collect it and collected it before the stump and hit the stumps. I interrupted the direct hit. It is wicket keeping 101. DON’T COLLECT THE BALL IN FRONT OF THE STUMPS. Even it was a direct hit, it would have been a very close call and it was very unlikely the leg umpire (who is from the batting side) would have given him out.

Excellent display by Falcons over all and A2C2 was bundled for 44 runs. Falcons won by 106 runs. Biggest win margin by Falcons ever.