 So let's introduce one more bit of terminology that's useful in discussing quantum mechanics and the Schrodinger equation and that terminology is this expression of an eigenvalue problem because the Schrodinger equation in operator form shown here, the Schrodinger equation is an example of an eigenvalue problem. In general, an eigenvalue problem is any problem for which we have some operator but when it acts on the function gives back the same exact function times some constant. So just changed the labels, if the operator is the Hamiltonian and the constant is the energy and the functions are wave functions and what we have is the Schrodinger equation. In the more general case we can say for a general eigenvalue problem if I am provided with some operator the name we give to the function is an eigenfunction, the name we give to the constant is an eigenvalue. So an eigenvalue problem is a problem where usually you're given the operator, someone gives you an operator and says find me all the functions that when I act with that operator on those functions I get back the same function multiplied by a constant. In other words, find the eigenfunctions of that operator such that acting on the eigenfunctions with the operator gives me an eigenvalue times the function. So that will make a lot more sense after we do a concrete example or two. So let's say that our operator is the differentiation operator or the ddx operator. So if I say here's an operator find me the eigenfunctions of the differentiation operator what I'm asking for is what functions are there that I can take the derivative of and get back a constant times the original function. So I'll let you stop and dig in your own calculus part of your brain for just a second and ask yourselves what functions are there that when you take the derivative of them you get back the same function and hopefully you've told me e to the x, right? If I say f of x is e to the x, in fact not just e to the x let's say e to the anything e to the 5x. If f of x is e to the 5x then the derivative of that function the d operator acting on the function in this particular case derivative of e to the 5x gives me 5 e to the 5x, right? Which is a constant 5 times the original function. So we've found a solution to this eigenvalue problem. What functions are there that I can take the derivative of and get back the original function? One such function is e to the 5x, e to the 5x is an eigenfunction of this operator. The eigenvalue of this operator is 5, that's the constant multiplying the original function that we got back. E to the 5x is an eigenfunction and 5 is the eigenvalue and notice those two things are paired. The eigenvalue is the value that gets spit out when I take the derivative of this particular function. If I were to use a different function if I say e to the 7x for example then my operator acting on e to the 7x, derivative of e to the 7x is 7 e to the 7x or 7 times my original function. So in this case e to the 7x is also an eigenfunction of the derivative operator. Its eigenvalue is 7. So notice that each eigenfunction comes paired with its own eigenvalue and that's generally what happens when you have an eigenvalue problem. One will provide you with an operator, you're looking for eigenfunctions in the process of finding functions like e to the 5x or e to the 7x or e to the whatever x you want, each one of those comes paired with a particular eigenvalue that matches it. That function gives rise to that eigenvalue and usually we're looking for not just one solution but the whole family of solutions that obey this eigenvalue problem. So that's terminology that we're going to use when we're discussing Schrodinger's equation. So we have rephrasing Schrodinger's equation. Up until now I've said let's find the wave functions that obey Schrodinger's equation. Another way to say that would be finding the eigenvalues of the Hamiltonian operator. Find all the functions psi, those are the things we call wave functions that are eigenvalues of this Hamiltonian operator. When we find the eigenfunctions we'll also find the eigenvalues and those eigenvalues are going to be the energies of the particles that obey Schrodinger's equation. So now with this last bit of terminology under our belt we're ready to use Schrodinger's equation and tackle a somewhat real world problem for the first time.