Question

What is the Schrödinger equation and its applications and how it is deduced and what is boundary conditions I have studied a lot and did not understand from the professor at the

University

please help me

Answer 1

Schrödinger equation is a fundamental law of Quantum Mechanics as you know in classical mechanics Newton's Second law is a fundamental law of classical mechanics.

In classical mechanics If you know applied force on object at given time to( t not) ,you can tell the position and velocity of object at any time .

But classical mechanics does not work on micro particle.

so schrödinger developed another mechanics named Quantum Mechanics for micro particle.

In schrödinger mechanics If you know potential energy and total energy of particle , you can find wave function of the particle using time independent schrödinger equation.

Now you have time independent wave functiofunction. With the help of this wave function using time dependent schrödinger equations we can find wave function at time t.

Now question is ,what is wave function?

I want to explain with the help of classical mechanics.

In classical mechanics If you know displacement S(x,t).

You can find everything about particle as velocity , differentiat with respect to t, Acceleration etc.

same in wave function if you know wave function .

you can find everything about particle at any time t.

Fundamental laws can't be derived. You can't derived Newton's law, schrödinger law.

We can only do verification of schrödinger equation with the help wave equation.

Schrödinger told some boundary conditions for acceptable wave function.

(1)It must be single valued.

(2) it must finite everywhere.

(3)it must be continous everywhere.

(4)it must be square integrable.

​​​​​​Applications-

(1) for finding energy level in Linear Harmonic Oscillator.

(2) to explain properties of Hydrogen atom.

(3) particle in potential well.

I suggest you to read the book Quantum Physics By H. C Verma (Indian Writer).

So If you will satisfy with my answer.

Please upvote me.