 So the wave function for a quantum mechanical particle describes the wave properties of that particle. In fact, it describes all the properties that we can calculate of that particle in ways that we haven't seen just yet. But also it's important to know that the wave function itself is a function that obeys an equation called the Schrodinger equation. So we need to understand the Schrodinger equation and how it can be solved to generate wave functions. So the Schrodinger's equation can be written in this way. So this collection of constants, Planck's constant squared over 8 pi squared mass with a negative sign times the second derivative of the wave function. If I add to that the potential energy times the wave function, what I'll get is the energy, the total energy, times the wave function. So that's an equation. We'll describe that in a little more detail. But that's an equation that this function, the wave function, obeys or satisfies. So the sum of the terms in that equation worth pointing out. So we've seen psi, of course, is the wave function. E, this capital E, on the right side of the equation is the total energy of the particle. That energy has two parts. V of x is the potential energy of a particle at some position x. So since total energy is the sum of the kinetic and the potential energy in some way that shouldn't be entirely clear yet, this term gives the kinetic energy. So the kinetic energy of the particle is given by the second derivative in some way. The potential energy of the particle is given by this term, and then the last term gives the total energy. Let's see what else we have to point out. This equation, the way I've written it down, is for a one-dimensional wave function, a wave function that depends only on the position x. For a three-dimensional particle that can move around in x and y and z, the function has a slightly different form. So in three dimensions, Schrodinger's equation looks like the same collection of constants. Now instead of second derivative with respect to x, I need to take partial derivatives because the wave function is going to be a function of three dimensions, x, y, and z. And the derivatives I need to take are the sum of the second derivatives with respects to x and y and z altogether. Their terms remain the same, so I still have a potential energy, except now that potential energy depends on the x, y, z position of the particle. Wave function is still three-dimensional. Those things summed together have to equal the total energy times the wave function. So that's the three-dimensional version of the wave function. This term, d squared dx squared plus d squared dy squared plus d squared dz squared, that may seem familiar from Calculus III. That quantity is called the Laplacian, and we have a shorthand form that we can use to write the Laplacian. We often, as a shorthand, just write del squared to represent the sum of the three second derivatives and the three Cartesian coordinates. And if I continue using shorthand without bothering you write down what the wave function depends on, some constants times the Laplacian acting on the wave function. If I add that to the potential energy times the wave function, I have to get back the energy times the wave function. So that's a more compact way of running Schrodinger's equation in three dimensions where I just haven't written out the full Laplacian and I haven't written out the x, y, z, but it's exactly the same as the equation above. So we have one-dimensional Schrodinger equation, two different ways of writing the three-dimensional Schrodinger equation. Let me point out here that the constants in front of del squared in this kinetic energy term, the h and the 8 and the pi are always the same constants. The mass is going to depend on what problem we're solving. The mass is the mass of the particle we're talking about. Maybe it's the mass of an electron or a hydrogen atom or a carbon atom or a larger molecule, whatever our quantum mechanical object is. But because the h and the 8 and the pi always show up together in this form, it's pretty common to do the following. Hes frequently show up with some pi's and some even numbers in the denominator. So if we define this new quantity called h bar to be Planck's constant, the regular Planck's constant that you can look up the value of on a table of constant divided by 2 pi, then that new constant, so I've defined that, I've chosen to define h bar is equal to h over 2 pi, that new constant h bar squared is going to be the square of h over 2 pi or h squared over 4 pi squared. So that looks almost like this h squared over 8 pi squared. And I can say that negative h squared over 8 pi squared is negative one-half h bar squared. I still have a 1 over m. So if I say, let me write that instead as h bar squared in a numerator, 2m in the denominator times the del squared. That term is the same as the kinetic energy term I've written down here. If I act on psi with that, add it to v times psi, I'm going to get e times psi. So that's yet another way of writing down Schrodinger's equation. Exactly the same as this one I've just written in terms of h bar instead of h. So I'm putting that out now so that you see if you see in one textbook or another some textbook authors prefer to use h bar, some prefer to use h. These two forms are exactly the same. So I've written down for you, let's see, 1, 2, 3, 4 different ways of writing Schrodinger's equation, either with or without h bar, with or without shorthand notation, with or without all Laplacian, when we're in 3D versus 1D. So I've got lots of ways of writing down Schrodinger's equation. But so what? What have we learned? Schrodinger's equation is only useful if we can proceed with solving that equation in order to figure out what the wave functions are so we can learn something about the particle that has that wave function. So the goal is to understand enough about Schrodinger's equation in order to solve it and obtain wave functions. But to do that, we're gonna have to step back and look at some properties of this equation to understand how to solve it. And that's what we'll do next.