The Schrodinger Wave Equation (SWE) was a landmark in the history of mankind, for it lay the basic framework of one of the most successful theory a.k.a quantum mechanics. Before understanding the real meaning of this equation, let us see what it looks like.
Understanding of SWE is very amusing. You already know what this equation actually means. In one line, this equations means, "
The total energy of a particle is the sum of its kinetic and potential energy."
What???? Are you kidding me! Is that what this equation tells?
Yes! If you look closely, this is what the Schrodinger wave equation means. The left part of the equation adds kinetic energy(h^2/2m ∇^2) and potential energy (V) and sums it equal to the total energy E. SWE is the most fundamental equation of quantum mechanics. All the information about a quantum system is already embedded in this equation just like everything about a mass m is contained in Newton’s second law. The difference is that in classical mechanics, the fundamental quantity is position and in quantum mechanics, it is wavefunction.
Consider throwing a particle in a box, you can only tell where the particle will probably be. You can visualize it using an analogy. Suppose your friend is in his bedroom. Knowing the nature of your friend, you can tell that there is 80% probability that he is playing video games, 15% probability that he is sleeping and 5% probability that he is studying. So there is a spread of probability (probability distribution) but when you open the door, he is, say, sleeping. The distribution collapses into one value.
This is what this ψ(x) tells us. ψ(x) is the wave function of the particle. Since by being probabilistic, a particle exhibits a wave like nature in the box, ψ(x) is hence known as the wave function. Itself it doesn’t represent anything physical but its square represents the probability of finding a particle in the particular state.
The image above shows a typical wavefunction of a particle in a box. One feature of the wavefunction is that it must be zero at the boundary of the box. This means the probability of finding the particle at the boundary is 0. Look carefully, this is exactly what every graph shows.
Another important implication of SWE, after energy conservation is quantization of energy. Consider the same problem of particle in a box as taken above. When you solve SWE for this particle, the energy levels that we get for a particle are quantized. This means the particle can only take certain energy values, unlike in classical mechanics. See Particle in a box.
So the Shrodinger wave equation tells the evolution of the wavefunction of a particle. If you know the wavefunction, you can derive all other quantities of the system.