 Okay, so now we understand what it means when we say a wave function is normalized. A wave function is normalized, let's say a wave function size of n is normalized. When the integral, if I integrate that wave function times itself, remembering to take the complex conjugate, if I need to of one of them, that integral comes out to be exactly one. And that's, remember, because this particular product is the probability of finding the wave function at a particular place. Probability of finding it anywhere has to be 100%. So a normalized wave function is one where this integral is equal to one. That's when I multiply a wave function by itself. We can also ask the question what happens when I take a wave function and I multiply it by a different one. So maybe I have wave function size of one or two for n equals one or two and a different wave function, let's say psi with m equals five. So product of two different wave functions, again, including a complex conjugation and integrate that product. And I can ask, what do I get when I take the product of two different wave functions times each other? So this thing that we're calculating, let's call, let's give that a name. So this integrated product between wave function m and wave function n, I'll call that O, capital O sub mn. And we can see what that's equal to for a specific example. The only problem we know how to write down wave functions for so far is the one-dimensional particle in a box. If I say the wave functions for that 1D particle in a box have this form, they look like sine waves, sine of some integer n pi x divided by the box length with a normalization constant out front. Let's say I want to know the value of this O, just for the simplest two wave functions, the first two, one of them for n equals one, one for n equals two. So with our definition, that's the integral of the product root two over a sine pi x over a when n equals one. And then root two over a sine two pi x over a when n equals two. And I don't have to worry about the complex conjugate because there's no complex or imaginary numbers in this wave function. I integrate this, this is the particle in a box, so I integrate from zero to a because the particles live in a box. The range is only between zero and a, and the wave function is zero outside that box. It's only non-zero inside the box, so size of one looks like this. Size of two looks like this. So we have this integral to evaluate. We could evaluate it formally. The way we'd go about it is, is probably use the double angle formula to do something with this sine of two pi x over a and turn it into some trig functions of pi x over a. And that would work fine. We could work a little while and get the answer. Or we could use symmetry. We could, we could use a shortcut and say, since we know what these wave functions look like, especially if I had drawn them as symmetrically as they're supposed to be, the area on the left hand side of the box where both psi one and psi two are positive is symmetrically equivalent to the area on the right hand side of the box where psi one is positive but psi two is negative. So every point like this point here where psi one is positive and psi two is negative is mirrored by a point over there here where the two values are the same but psi two has an opposite sign. So the area under the product on the left side of the box is positive and exactly equal but opposite sign to the area on the right side of the box. So whether we do it with actual trig and integration or whether we say we observe the symmetry of the problem and use that to just observe that this integral has to come out equal to zero. This overlap integral, this O sub one two is going to have the value of zero. So not one like for the normalized wave function multiplied by itself. When I take these two particular different wave functions multiply them together and integrate what I get is zero. So that's a different property. Two wave functions we say that they're orthogonal to one another. It should be an O orthogonal when that product of the two wave functions one of them complex conjugated integrated when that gives us zero. So that's the definition of this property orthogonal and so what we've just seen is that the psi sub one and the psi sub two wave functions for the one-dimensional particle in a box we can say those two wave functions are orthogonal to one another. And orthogonal means the same thing here as it does in the sense that you may have heard that word before when you're talking about vectors. I would say two vectors in 3D space or orthogonal to one another as a way of saying they're perpendicular to one another. The way you know if two vectors are orthogonal to one another as you multiply them in a particular way you take their dot product and if their dot product is zero then you can say the wave function that I'm sorry the vectors are orthogonal. So just like a dot product is a particular way of multiplying vectors together to see whether they're orthogonal. This integrating complex conjugating and then taking the integral that's a particular way of multiplying two wave functions together to find out whether they're orthogonal or not. And again wave function can be normalized you just have to multiply it by itself and integrate can be orthogonal to another vector if that product complex conjugated integrated works out to be zero. So we could continue with more examples for wave functions of the particle in a box one-dimensional particle in a box but it turns out that whichever pair we choose whether it's one and two or if I chose one and three or one and four or two and three or two and four and so on any pair of different vectors or I'm sorry wave functions that I choose are always going to turn out to be orthogonal to one another. So we can say a set of vectors so let's say all of the vectors and go a little further and say the one-dimensional particle in a box wave functions size of one, size of two, size of three and so on that complete set of vectors those are all mutually orthogonal to one another what I mean is that any pair of different wave functions I choose out of that set are going to be orthogonal to one another it's not just one and two it's every pair within that set each one is orthogonal to all the other so we say they're mutually orthogonal. There's one other important feature of these wave functions that we can point out since they're both normalized every one of these particle in a box wave functions is normalized once we've chosen this particular coefficient down in front and the set is mutually orthogonal we can say that a set of vectors is orthonormal another vocabulary word if this quantity o sub mn if I do that for two wave functions that are the same as one another n and n if these two subscripts are the same then if they're normalized when m is equal to n when those indices are the same then the integrated product is equal to one but when they're not the same when m is not equal to n for every pair if it comes out to be zero because they're mutually orthogonal then we say that that set of vectors is orthonormal in other words an orthonormal set of vectors is one where every vector is normalized and also the full set is mutually orthogonal and it turns out that having orthonormal vectors is particularly nice for reasons that we'll see soon coming up it's also convenient that not just for the one-dimensional particle in a box but for any quantum mechanical problem whichever quantum mechanical problem we choose to solve we'll get a set of wave functions and the wave functions will be we can normalize them and we can also make sure to make them orthogonal to one another if they don't come out automatically orthogonalized like the one-dimensional particle in a box wave functions are then there's a procedure we can take to make sure that they're mutually orthogonal to one another so it's very convenient that quantum mechanical problems can always be chosen so that the wave functions turn out to be orthonormal the solutions the wave functions all are orthonormal and that's convenient because then we can use those orthonormal wave functions to construct other functions and that's what we'll talk about next all right I'm back with a quick addendum to point out one more terminology piece of terminology that I forgot and that's to give a name to this quantity OMN that we've been talking about this OMN that's equal to either a 1 or a 0 if the wave functions are orthonormal that quantity is called an overlap integral the reason we call it an overlap integral first of all it's an integral two wave functions multiplied together and integrated so it's an integral and what we're doing when we calculate that overlap integral when we calculate the product of one wave function times another wave function that overlap integral the inside of that overlap integral is nonzero only in places where both of the wave functions have some nonzero value so in this case for the particle in a box both these wave functions were nonzero everywhere except for when the side two function passed through a node in the middle of the box but let's say we had a different problem where let's say psi one had some particular shape and it was zero for part of the range of whatever variable we're interested in and our side two variable a side two wave function has its own different shape and is zero somewhere else so the way I've chosen to draw these wave functions psi one is nonzero only in this region psi two is nonzero only in this region and there's when I multiply psi one times psi two when I do an overlap integral then psi one may be nonzero but it's being multiplied by a zero psi two and vice versa wherever psi two is nonzero it's being multiplied by psi one equal to zero so for this particular case the overlap integral would turn out to be zero so that's an important observation about these overlap integrals they can be zero either because there's no region where the two wave functions overlap one another or in this case even though the wave functions have a high degree of overlap they can be zero for symmetry reasons where the positive contributions from one happen to cancel the negative contributions from another region of the box