 The previous four postulates of quantum mechanics have described features of the wave function and importantly how to calculate properties given the wave function. The fifth and last postulate of quantum mechanics has a slightly different flavor and that tells us about the time dependence of the wave function. So far, whenever we've talked about a wave function, we've described it as a function of position. Maybe if it's a one-dimensional particle, we just say psi of x. If it's three-dimensional, we might say psi of x, y, and z. But notice that there's no time in that formulation of the wave function. So we call that a time-independent wave function. It's independent of the time, the value of the time, whether it's today or yesterday doesn't change the value of that wave function. And we can contrast that with, more generally, in fact the wave function of any particle might not be the same today as it was yesterday or as it will be tomorrow. So in general, there might be some time dependence of the wave function. The wave function might be changing as time goes on. So we would call that a time-dependent wave function. So if we want to understand how it is that the wave function changes as time goes on, then we need the fifth postulate of quantum mechanics, which tells us, again, for a system with a particular wave function, if it's in a wave function described by psi, then the time evolution or the way that the wave function changes with time, the way that the wave function changes with time is telling us something about how the wave function changes with time, the rate at which it changes with time. And in particular, if I take i times h over 2 pi, that collection of constants, that happens to equal the Hamiltonian acting on the wave function. So not only does h psi equal e psi, but h psi also equals i h over 2 pi d psi d t. So this tells us something about the time-dependent behavior of the wave function. I'll say that treating the time-dependent behavior of the wave function is something that we won't spend a lot of time on in undergraduate physical chemistry. That's a somewhat more advanced topic. For the most part, we will consider wave functions to be in what's called a stationary state. We can understand most of the p-chem we want to understand without explicitly treating the time-dependence of the wave function with a few exceptions that will come later on in the course. But for the most part, we will consider time-independent wave functions or wave functions that we call wave functions in a stationary state. But for completeness, we do have this fifth postulate. And just to understand what it means, it's essentially not that surprising that there's some statement like this. For example, if we forget about quantum mechanics and think only about classical mechanics, think back to physics one or physics two or whatever it is you learned about cannonballs and predicting the motions of objects. If I describe completely the state of some system, a cannonball flying through the air, it has some position, it has some momentum, and it has some mass. If I describe the state of that system completely, you can predict not only the current properties of that system. It's kinetic energy, it's potential energy, it's momentum, and so on. But you can also predict where that system is going to be in the future. You can predict how that state of the system is changing as time goes on. So this is the equivalent statement in quantum mechanics as Newton's laws of motion, for example, would be in classical mechanics. It tells us how the state of the system described by the wave function is changing as time goes on, and it gives a prescription for how to calculate that. What's a little bit remarkable, if you think about it, is the same equation that predicts the time evolution of the system depends on the same operator as the operator that tells us something about the energy of the system. So that's a pretty remarkable fact. So we have now completed our tour of the postulates of quantum mechanics. This is the fifth and last postulate of quantum mechanics. Remember that what these postulates overall tell us how to do is to, once we know the wave function, how to calculate the properties, those properties include the energy. Remember also that we've seen that Boltzmann's distribution and statistical mechanics tell us what to do with those energies. They tell us how to calculate the probability that particles will occupy any one of those individual energy levels, and also we can use those properties to go on and then calculate more thermodynamic properties. So that's the next thing we'll do is use quantum mechanics to calculate some energy, energies, and then use those to calculate additional properties.