The quantum theory that was established by Max Planck, Albert Einstein and Niels Bohr enjoyed experimental success for nearly two decades. But by the year 1920, this theory was soon referred as the 'old quantum theory'. From here on, a 'new quantum theory' would be developed by a radical group of scientists comprising Schrodinger, de-Broglie, Dirac, Heisenberg, Ehrenfest, Born and Jordan.
The old quantum theory showed that the radiation or the light has particle like nature associated with it. Scientists then questioned the converse of this: Can we associate a wave with all the material objects? The answer was Yes! The de-Broglie hypothesis, that every particle has a wave nature associated with it was confirmed by the famous Davisson-Germer experiment.
Quantum mechanics now needed a sound mathematical platform. With time, two mathematical formulations of quantum theory emerged: Wave mechanics by Schrodinger and Matrix mechanics by Heisenberg. Both the formulations had different approach towards the same problems and yet both provided the correct results. Today, let us talk about the flagship equation of quantum mechanics, the analogue to Newton's second law in classical physics: The Schrodinger Wave Equation.
This equation tells something you all know. You have studied the meaning in your high school. The meaning of the equation is:
The total energy of a particle is the sum of its kinetic and potential energy.“
Is that all it means? 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.
The reason why this equation is the analogue of Newton's second law from classical mechanics is that in the latter, we solve for the position of the mass. Once we know the position, we get everything: velocity, acceleration, momentum, energy etc. Similarly in quantum mechanics, the SWE solves for the wavefunction of the particle, from which we can learn about the entire quantum system. 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.