Category: Physics

Step wise of Schrodinger Equation – Part 2 of 3

de Broglie’s Hypothesis

Light is a wave or particle debate was going on for centuries and eventually we all came to know it behaves both. Sometime light has wave properties and sometime light has particle properties. It is like humans. Behaves differently with out using a well known rationale.

de Broglie brought an insight that the both wave & particle properties applies to matter and also the wave length of the matter inversely proportional to momentum.

The above relationship between the momentum and wavelength is given below.

\lambda  \alpha \frac{1}{p}

The wave length is equal to the ratio of Plank’s constant and momentum.

\lambda =  \frac{h}{p}

Wave Number:
It is equivalent of frequency in the spatial domain. It is a measure of unit of repeating waves per unit of space.

I was not specific in the video. The angular wave number or circular wave number is denoted by k and is given by
k =  \frac{2\pi}{\lambda}

The presenting the Plank’s constant in angular form is given by
\hbar =  \frac{h}{2\pi}

Using the above equations in de Brogile’s equation representing the relationship between wavelength and momentum we get
p =  {\hbar}{k}

Wave Function:
The wave function is the crux of the Schrodinger equation. The plane wave is taken as the wave function.


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