Schrodinger Wave Equation explained in simple words

Newtonian Mechanics don't work in the quantum world.
Quantum objects are very microscopic that they can't be located but they can be probably located. And this probability can be determined by using wave functions of the quantum particles




Wave function gives us the probabilities of where the electron is likely to be. The act of not knowing where the electron is, allows its probability distribution to be spread out over a large space kind of wave. Therefore wave function is the function which describes the wave shape of the probability distribution of the electron.


The Heisenberg Uncertainty principle says that we can't predict the exact position and momentum of the quantum objects. But we can know about things like Energy levels and wave functions.
THE SCHRODINGERS WAVE EQUATION :

                                                                                                          = Wave function (psi)                    E= Energies of the electron is allowed to have 

E[(psi)(x)] = The energy levels of the wave function of the electron 

If we consider an electron inside a box where it is confined to only a few frequencies the wavefunction always needs to be 0. In other words, the electron has 0 probability to be found outside the box.

 

And we know the famous Einsteins equation  E=hf where h is Planks constant. Since only a few frequencies are allowed only certain energy levels are allowed.
That is what is meant by QUANTISATION.

Derivation :

For more detailed information :
http://physics.mq.edu.au/~jcresser/Phys201/LectureNotes/SchrodingerEqn.pdf
http://lejpt.academicdirect.org/A26/031_048.pdf

"No one undertakes research in physics with the intention of winning a prize. It is the joy of discovering something no one knew before."

Casimir Effect

Scientists are often astonished observing the moment of things around, on their own with no visible force acting on them. 
When two mirrors which are later called the Casimir plates, placed in a strong box maintaining one-atmosphere pressure at 100 nanometers apart move towards each other with no well-known forces between. Here comes the Casimir effect into the picture.
This was discovered by Hendrick Casimir in 1948 and named after him.

To understand the Casimir effect we have to first know these bullet points:

• Heisenberg's Uncertainty Principle proposes that both the position and momentum of a subatomic particle cannot be determined simultaneously.                                                                                                                                                   And also energy and duration cannot be predicted simultaneously. Thus particles don't have both nill energy and duration. Therefore particles that don't exist can even have a high chance of having huge energy causing duration and forcing them to exist for a very short period. These are called virtual particles which further turn up into matter and antimatter to annihilate rapidly.
• Wave-Particle Duality says us that particles can turn into waves and again into their particle nature.
• Stationary wave theory tells us that only a finite number of waves can exist in closed space which is generally termed as Harmonics.
• Infinities are distinct which means decimal infinities are greater than real number infinities. Thus a number of waves exist outside the mirror
• Force exerted is generally defined to be the change in momentum times the inverse of time. We know that there are a number of virtual particles outside the Casimir plates than in between and thus they exert more force than the particles in between them.