In the previous lectures, we have been studying the kinematics of quantum mechanics. Starting from the wave particle duality, we conclude that we had better to use a wave function to describe this quantum state, the quantum particle and then what is the nature, the interpretation of the wave function as a probability density amplitude. Then how do we get information from this wave function, namely the theory of measurement and what is the limitation of measurement, namely uncertainty principle. This is the kinematics of quantum mechanics and you can guess what is in the next step. The next step we are going to study the dynamics of quantum mechanics. What I mean, is how this wave function evolves as a function of time and also how a quantum particle described by a wave function interact with other things. And the simplest thing we can imagine is how a quantum particle moves in a potential. So time dependence and how the particle moves in a potential, these are what we are going to study and this is known as the Schrodinger equation. The Schrodinger equation is a fundamental postulate quantum mechanics that I'm unable, nobody is able to really derive it for you. However, we can make an educated guess that, from what we knew in the past, we can actually have an idea why Schrodinger proposes his Schrodinger equation and why it should work in more general cases. We can have an educated guess, have some intuition, have some ideas. Where do the ideas come from? The ideas come from that we know if we have a constant potential, then how a quantum particle should evolve. The particle should evolve like a plane wave or superpositions of plane waves. For example, if we have psi 1, which psi 1 is e to the I P1 x minus E1 t over h bar and psi 2 correspondingly I change the P2 and E2. We can have for example, superposition, psi equals to psi 1 plus psi 2. For sure we can have more general superpositions as integrations with some constants c_p as continuously weighting functions. But here, just think about some very simple cases, some plane wave and superposition of the plane wave. Now, what should be the time evolution equation for these plane waves? What do I mean by time evolution equation? I mean that I would like to find an equation which contains some time derivative partial t and by partial t by the way, I mean partial t acting acting on something. We would like to find time derivative partial t and where can we find a time derivative in here? We just bruteforcely, we act this time derivative on the wave function and let's check what we get. We act this time derivative on psi 1, for example, psi 1, what do we get? We get a minus I, E over h bar psi 1.Okay? Now we are making progress. We're making progress that at least we find something, we find a time derivative, the evolution of this plane waves state, what it looks like. Lets make it slightly better looking, make it better looking by multiplying a I h bar here. Then this I cancels this minus I and this h bar cancels this h bar, so it will become just the E1 multiplying psi 1. This equation looks pretty nice, however, is that the equation that we are looking for? Is it the time evolution equation that we're looking for? Not yet. Why not yet? Because there is a E1 in here. That means this equation makes reference to some nature of this state. This equation is not very universal and even worse, what if we study not psi 1, but psi? In general, what if we study psi? If we study that I h bar partial t plus psi, what we get will be E1 ps1 plus E2 psi 2 since we have assumed that psi equals to psi 1 plus psi 2. We know the linearity of quantum mechanics, so the time evolution equation has to be linear, that now we end up with E1 psi 1 plus E2 psi 2. This cannot be a fundamental equation. That depends on what is the component of the state, how this state is made up by superpositions, is more and more complicated. As you consider superpositions, this cannot be a fundamental equation. We try to find out the equation which doesn't have reference to this state, namely it doesn't have reference to this E1.Okay? How do we find that? Then we have an idea, what is E1, What is E2? From the non-relativistic quantum mechanics, we are saying we started nine relativistic quantum mechanics. We don't mix in relativity because then it will become more complicated. For non-relativistic quantum mechanics, what is energy? Energy in Newtonian mechanics, what's that? that will be p squared divided by 2m plus a potential energy. Kinetic energy plus potential energy. This is energy. Then we have the experience that we would like to make it universal, we would like to make it applying to all forms of plane waves. In that case, how do I modify this p? We have the experience, where we are talking about observables. We would like to modify this p, into the operator p. Once it is an operator p, which is minus ih partial x. Once we have the operator p, no matter we apply on E1, which has momentum p1 or E2 which has momentum p2. We have the same form of operator p. Since we have modified this equation by changing this p number into an operator, we had better to give this a new name. Let me give it a new name H as an operator. This H is known as the Hamiltonian of the system. Hamiltonian operator. The name Hamiltonian has very deep root in classical mechanics, Hamiltonian mechanics, but we're not getting into that part. That just you know, this is the Hamiltonian operator and that has relation with energy that you have the Hamiltonian operator acting on a state with definite energy, you will get the energy of the state. Then I can replace this Hamiltonian operator to here. This E1 acting Psi1 equals to the Hamiltonian operator acting on Psi1. This E2 acting Psi2 equals to the same Hamiltonian operator acting on Psi2. Because now we have p-hat and p-hat is universal. If you don't believe, try it for yourself. Plus the same Hamiltonian multiplying Psi2. Now, it's great, because now we can write this equation into a universal form that we can write it as Hamiltonian multiplying Psi1 plus Psi2. And Psi1 plus Psi2 is Psi. In other words, we have written this equation into a universal form that ih bar partial tPsi equals to the Hamiltonian operator multiplying Psi. This is a universal equation. This universal equation is known as the Schrodinger equation. The Schrodinger equation is not only a universal equation for all the plain waves, but there is a further possible generalization. Recall what is our assumption for the plane waves? The assumption is we have a potential, which is a constant. If the potential is not a constant, the solution will be no longer plane wave. Now this Schrodinger equation here, I have been a little bit cheating, I have been secretly writing a potential as a function of x as a function of position, which is not necessarily a constant. The postulate, the real magic is the Schrodinger equation not only applies for this constant potential, but applies for all possible potentials. We've removed the assumption of a constant potential. For all kinds of potentials, this Schrodinger equation applies. This is the key equation in quantum mechanics and a large part of quantum mechanic when you are learning a full course of quantum mechanics, will be the techniques, how to solve the Schrodinger equation, and then how to read off the information from the solution of the Schrodinger equation? This Schrodinger equation is of key importance in quantum mechanics. It tells you how a state evolve with time and how the state interacts with a nontrivial potential.
