Schrodinger's equation. How to Understand It??

What does the Schrodinger wave equation tell us?

At the beginning of the twentieth century, experimental evidence suggested that atomic particles were also wave-like in nature. For example, electrons were found to give diffraction patterns when passed through a double slit in a similar way to light waves. Therefore, it was reasonable to assume that a wave equation could explain the behaviour of atomic particles.

Schrodinger was the first person to write down such a wave equation. Much discussion then centred on what the equation meant. The eigenvalues of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen atom, and the energy levels were found to be in accord with Rydberg's Law. 

It was initially much less obvious what the wavefunction of the equation was. After much debate, the wavefunction is now accepted to be a probability distribution. The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wavefunction gives the probability of finding the particle at a certain position.

The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behaviour of a particle in a field of force. Schrodinger’s Equation tells you how to calculate the properties of any particle at the atomic level, for example, an electron or a photon. Particles at this level of size have a limited number of properties. They include position, momentum, a quantum property called “spin,” and a few others.

Schrodinger’s Equation is the key equation of quantum physics. It’s a parallel to Newton’s Laws of Motion in classical physics but it is not deterministic in the same way Newton’s Laws are. In classical physics, if you know the position and momentum of an object, you can use Newton’s Laws of Motion to calculate the future position and momentum of the object. Given knowledge of the exact initial position and momentum and the measurements of all the forces acting on the object, Newton’s laws are deterministic – they tell you how the forces will interact and so, where the object is going to be at a next point in time. 

Schrodinger’s Equation will not tell you the position (or other properties) of an individual subatomic particle at a future point in time. It will tell you only its possible positions and the probabilities of its being in each of those possible positions. For example, if you used a laser to shoot a lot of photons towards a photographic plate, Schrodinger’s Equation could be used to calculate the overall pattern of pixels that would form on the plate, but not the position of which pixel any particular photon will light up. So, Schrodinger’s Equation is deterministic but at the statistical level rather at the individual particle level.

What is the real difference between the time dependent Schrodinger's equation and time independent equation?

The time dependent form of the Schrödinger equation depends on the physical situation . The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time.

The time-independent Schrödinger equation is the equation describing stationary states. The time    independent Schrödinger equation predicts that wave functions can form standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals). These states are important in their own right, and if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state

Limitation:

. It is not relativistic, and it doesn't take spin into account. It only applies to spin 1/2 particles, which means it doesn't apply to photons. It works to model simple atoms, but anything more complicated requires a lot of corrections. 

Summary:-

1) 

It is like "Newton's law" in "Classical Mechanics"!!!

First of all, Newton's law predicts the future behavior of a dynamic system. In the same way Schrodinger's wave equation predicts the future behavior in "Quantum Mechanics"!!!

The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems. The associated wave-function gives the probability of finding the particle at a certain position.

2) 

The solution of the equation will give the allowed energy levels of quantum mechanical systems. If it's time-dependent we can describe the system as it changes over time. If it's time-independent then wen get a stationary value. 

3) 

Schrodinger equation just tells the wave nature of an electron when it moves in one particular orbit bounded to certain potential and nothing else.

This equation is non-relativistic and free from spin of an electron.