 So, the first postulate of quantum mechanics told us that everything is possible to know about a system. A quantum mechanical system can be obtained from its wave function, so all the different properties of that system can be obtained in some way from the wave function. So, before we can understand how to do that, we should talk a little bit more about what these properties are that we can obtain from the wave function. And to do that, it's important to distinguish between properties that are what we call physical observables and those that aren't. So a physical observable is any property that can be measured and given a concrete value to without anyone disagreeing about what that value is. So that may seem like a silly definition. Of course, if there's a property, it has a value and we can measure it. But some properties, especially for us as chemists, we like to talk about a lot of properties that are somewhat ambiguous or subjective. For example, in organic chemistry, we might talk about the quality of a leaving group or we might talk about the amount of electron withdrawing character that a functional group has or something. So those are somewhat poorly defined. Different people may disagree about how to define them or about whether one leaving group has more or less leaving group character than another. So those will be properties that are not physical observables. There's no instrument we can put them into and measure the leaving group quality. But properties like mass or energy or momentum or things like that that describe the physical properties of a system that are not ambiguous or subjective. Those are the ones that we talk about as physical observables. And in particular, for those carefully defined properties or physical observables, each one of those physical observables has a corresponding quantum mechanical operator. So we've seen operators before and I'll remind you of what a few of those are. But what this postulate says is whatever property we want to measure, whether it's the energy or the momentum or whatever it is, there is some operator that corresponds to that property. There's an energy operator, there's a momentum operator, and so on. So we have in fact seen several examples of these operators that correspond to individual properties before. So let me give you a few examples of those. So let's start with a list of properties. And for each one of those, I'll write down the operator symbol that we use to refer to it. So for example, the kinetic energy we have seen and briefly talked about the kinetic energy operator that sometimes we call t-hat. And we've seen that as it shows up in the Schrodinger equation as this second derivative operator. If I take a wave function and I do this to it, minus h squared over 8 pi squared mass times the second derivative of the wave function. So it's an operator because it acts on the thing that comes after it. That gives us the kinetic energy in some way that we haven't talked about a great deal yet, but this is the kinetic energy operator. So this is in one dimension, that's our kinetic energy operator. If we're solving a three-dimensional problem, it's actually a little bit easier to write, perhaps harder to think about. We use del squared for the d squared dx squared plus d squared dy squared plus d squared dz squared in three dimensions. So there's an example of an operator that we've seen. Another operator that shows up in the Schrodinger equation is the potential energy operator. And that shows up just as potential energy times the wave function that follows it. Same thing in 3D, except here we would have a potential energy as a function of three-dimensional coordinates multiplying whatever wave function comes after it. And of course the whole purpose of the Schrodinger equation is to use this Hamiltonian operator, the sum of these two things, the kinetic energy and potential energy operators added together. So minus constants times second derivative plus potential energy, that operator, again, if it acts on some wave function that comes after it, we take the second derivative times constants and we add that to the potential energy multiplying those constants. That's the Hamiltonian operator. That's the sum of the kinetic and potential energy, so that's the total energy operator. Remember what Schrodinger's equation says is that operator acting on the wave function gives us back the total energy multiplying the wave function. So that's what it means to be the total energy operator. And we can write that down in three dimensions as well. To get to a few properties that we haven't already seen, there's a momentum operator. If I want to find out the momentum of a particle, the operator that I need to use is this momentum operator, p hat. And that looks a little bit like the kinetic energy operator in a way, but it's a single derivative instead of a second derivative. It's got h and the 2 pi to the first power instead of higher powers. And it's complex. Notice the i here, the square root of negative one that shows up in the momentum operator. Likewise in 3D, it looks similar. It's minus i h over 2 pi times del, which includes the d dx, d dy, d dz derivatives in that vector. Along with momentum, we often want to know just what is the position of a particle? The operator that will tell us the position of a particle for that property is just x times, or in three dimensions, the r vector times. So if you want the operator corresponding to position, it's just the position multiplied. So some operators like momentum and energy involve derivatives. Some operators like the position of potential energy only involve multiplication. And then the last operator that we'll discuss for now that will really only be relevant once we get to talk about three dimensional problems is the angular momentum. So if I have a particle that's orbiting, an electron that's orbiting an atom, and I want to talk about its angular momentum, that operator we'll call l hat. It's also complex. It has this imaginary i in the numerator. And in fact, if you remember enough physics to know that angular momentum should be r cross p, that's exactly what we have here. This is a cross product between the r vector, i.e. the operator corresponding with the position, and minus i h over 2 pi del, which is the operator corresponding with momentum. So in quantum mechanics as well, this operator is the cross of the position operator with the momentum operator. So again, second postulate tells us each property. Any physical observable property that we're interested in has some operator associated with it. It's a little bit unclear for now how I use that operator to actually obtain a value for the property. So we need one more postulate to be able to tell us how to do that, and that's the next step.