 Okay, so we've seen when we do quantum mechanics that when we solve a quantum mechanical problem very often the energies are quantized or discretized only certain energies are allowed. The same thing is going to be true for other properties that we tend to measure and that's because the third postulate of quantum mechanics tells us something about which of those values of certain properties are allowed. So the postulate says a property or a physical observable maybe the energy maybe the momentum maybe whatever property I'm interested in if there's some property let's just call it a remember the second postulate told us each one of those properties has its own operator so whatever property that is it has some operator associated with it so if I call the operator a hat corresponding to that property the only values that that property can take on are the eigenvalues of the operator so remember what an eigenvalue is I'll give an example in just a second but an eigenvalue is the if I operate with operator on some function I get back a constant times the same function those constants are the eigenvalues so the discretized the specific values of the property that are allowed are only the eigenvalues of the operator so we've seen that in one example already for particle in a box the one of the operators for a one dimensional problem for the particle in a box with potential energy zero that Hamiltonian operator was just the kinetic energy portion of the operator so second derivative multiplied by these constants with a negative sign so that's the Hamiltonian operator that Hamiltonian operator is the operator that corresponds to some property the property is the energy so we've talked about the fact that the energy corresponds to this Hamiltonian operator so if I'm looking for some function that I can act with the Hamiltonian on and get back some eigenvalue times the same function or the way we usually write that expression so that's writing it as an eigenvalue problem the way we usually write that problem is like this the Schrodinger equation so if I'm looking for eigenfunctions I'm looking for functions that I can act with the Hamiltonian operator on and get back some eigenvalues or constants times the original function then there's certain solutions when we solve the Schrodinger equation the goal of that is to find the various different solutions for the eigenfunctions and the eigenvalues and when we did that what we found was that's not right for the particle in a box the energies were these particular set of values h squared over 8ma squared times n squared so with the energy ladders that we've drawn a few times I can have this energy e1 or this energy e2 or this energy e3 and so on but I can only have these specific discrete quantized energy levels so this is an example of the third postulate in action the only allowed values of the energy are these specific values e1 e2 and e3 and so on and that's because those specifically are the eigenvalues of the Hamiltonian operator if I wanted to ask what values of the momentum what values of the position what values of some other property are allowed for the one-dimensional particle in a box I would have to find out which eigen for each one of those operators the momentum operator the position operator and so on what are the eigenvalues of that operator and we'll have occasion to do that but by far the most useful property in a chemistry context is often the Hamiltonian because we're interested in the energies of the molecules so this tells us how to find the allowed values of the energy what it hasn't told us yet is which values are the ones that we're actually going to see if I have a quantum mechanical system it's allowed to have energy e1 or e2 or 3 which one is it actually going to have what's the energy I can expect to see for that particle that requires one additional postulate and the fourth postulate and that's what we'll talk about in the next video lecture.