 So now our task is to see if we can understand the solutions to the L equals 0 version of the radial portion of the hydron atom Schrodinger equation. So remember we've broken down the Schrodinger equation into a radial piece and an angular piece. The radial version of the Schrodinger equation in the specific case where L equals 0 is this equation right here. So it's possible to treat this formally and solve it as a formal differential equation but we're going to take a somewhat simpler approach right now and we're going to simply demonstrate that a function of this form, an exponentially decaying function, so my radial function is an exponentially decaying function of R. We can just demonstrate that this function solves the radial version of the Schrodinger equation when L equals 0. So that's what we'll do now. Let's start by taking these terms one at a time. So remember what this means is take the derivative of the function then multiply by R squared, then take another derivative, then divide by R squared and multiply these constants. So we'll do that first step by step. So here's our function R. The R derivative of that function, derivative of an exponential just pulls down the exponent. So that looks like minus C, E to the minus C, R. If I then bring in the R squared, so R squared times that result I've just gotten is minus C, R squared, E to the minus C, R. I now have another R derivative to take. So now I have two R's so I need to use the product rule. R derivative of this, derivative of R squared is two R. So the first piece looks like minus two C, R, leaving the exponential alone and then product rule says that leave the minus C, R squared alone and pull down another minus C when I take the derivative of the exponential. So that negative C times this negative C gives me positive C squared and then I still have an R squared and then the exponential. So that's what I get after taking the R derivative of this piece. So that's taking the R derivative of that. Now what I have left to do is divide by R squared and multiply by these constants. So I'll say let me go ahead and rewrite this all now in orange since I'm done my intermediate work. I've got minus these constants times these constants which also have a negative sign. So I'll write H squared over 8 pi squared M. I've got a 1 over R squared multiplying by this R so that's a 1 over R and I've got a 2 and a C. I'll write that here, 2 times C times H squared and I have an exponential. This E to the minus C, R, I'll go ahead and rewrite that now since I'm done taking derivatives. Every time I see E to the minus C, R, I'll just rewrite that as capital R. So I've taken these constants multiplied by the first term. Now I need to take these constants and multiply by the second term. So now I've got a minus H squared over 8 pi squared mass. 1 over R squared now exactly cancels this R squared and I still have a C squared in this term and then E to the minus C, R which I'll write as capital R. So that's the full result of taking these constants times the derivative of R squared times the derivative of my radio function. I can then go ahead and write down my Coulomb term so that looks like minus Z E squared over 4 pi epsilon 1 over R times big R and my energy term minus E R that should equal zero. If indeed this is a solution to the Schrodinger equation, to the radio portion of the Schrodinger equation that's zero, I'll write a question mark because we're still just double checking whether that's true or not. So we can ask ourselves now, is this true or is it possible that this is true? In order for this to be true, everything on the left hand side of the equation has to completely cancel. So I have two terms that look like 1 over R times big R, 1 over R times big R. So those two terms need to completely cancel each other. I also have two terms that just look like my function of R multiplied by constants. So those terms also have to cancel each other. So if I take the first and the third terms first, this 1 over R term needs to cancel this 1 over R term in order for my function to be a solution to the Schrodinger equation. So we need two things to be true. We need the first and the third terms to cancel. So I need the constants 2CH squared over 8 pi squared mass minus ZE squared over 4 pi epsilon not to be zero. So that's the condition I need in order for the first and third terms to cancel. Those certainly don't look the same as each other. And I don't have the flexibility to choose very many of these variables except for C. So H and pi and mass of an electron and the charge of an electron and the permittivity vacuum, those I don't have any control over. However, C is just a constant. I've said, let's see if this exponential solves the Schrodinger equation. It certainly wouldn't solve the Schrodinger equation for an arbitrary value of C, but I can solve for the particular value of C that does make this a solution to Schrodinger's equation. So if I rearrange this equation to solve for C, if I take the second term, the ZE squared over 4 pi epsilon not term and put it on the right side. So 4, that's not correct. ZE squared over 4 pi epsilon not. And then I take everything that's not a C in this term, the 2H squared over 8 pi squared mass and I flip them upside down. So 8 pi squared mass over 2 H squared. So that means the value of C, after some cancellation, this 4 will turn that 8 into 2, which will also cancel that 2. This pi will cancel one of these pi's and what I'm left with is ZE squared times pi times mass over epsilon not H squared is the value of my constant C. So let's pause and talk about that for a minute. That is the particular value of C that makes this a solution to Schrodinger's equation, to the radial portion of Schrodinger's equation. If you notice that this value C multiplies R in an exponential, so the units of C must be one over length. One over length times length makes this unitless like the exponent of an exponential must be. So we know that this collection of constants must be a unitless, I'm sorry, must have units of one over length. So let's define something that has units of length. If I take all these constants and turn them upside down, so I'll say epsilon not times H squared over, let's rearrange these a little pi mass times E squared. I'm leaving out the Z intentionally. So that collection of constants, since Z is unitless, this collection of constants must have units of length since I've turned it upside down. And that is a thing that we're going to call the Bohr radius. So that turns out to be a pretty important quantity in understanding the size of an electron in a hydrogen atom or in hydrogen-like ions, as we'll see more in the future. But this particular collection of fundamental constants defines this thing called the Bohr radius, which has a value of, as we can determine by plugging in these fundamental constants, if we calculated the value of the Bohr radius, that would be about half of an angstrom. So what that means is this constant C that we're talking about, I haven't included the value of Z in the Bohr radius, so all these other constants are one over the Bohr radius. And so the constant C is the atomic number of the atom we're talking about divided by the Bohr radius. So we've succeeded in doing two things. Number one, we've determined that this exponential does, in fact, solve the radial portion of the Schrodinger equation. If the C has a particular value, and we solve for that particular value of the constant C, we have one more thing left to do, which is to guarantee that this second term also cancels the fourth term in order to solve the Schrodinger equation. So the second term, which is constants multiplying by R, needs to cancel the fourth term, which is this constant that we call the energy multiplying R. So in order for that to be true, the other requirement that we have is that minus C squared H squared over 4 pi epsilon naught, nope, that's the wrong collection of constants, C squared H squared over 8 pi squared mass. If I add that to negative E, I have to get zero. So we've determined what the value of C needs to be, what's left here is the energy. If this function solves the radial Schrodinger equation, the energy will have a particular value. That value of the energy will be minus C squared H squared over 8 pi squared mass. But since we now know what C is, C is atomic number divided by Bohr radius, we can rewrite this as minus H squared over 8 pi squared mass, and then C squared will be Z squared over A naught squared. So that is the energy of the value of the energy when this exponential function is inserted and solves the Schrodinger equation.