 Having discussed free particle and particle in a box the field is now set for us to get into Schrodinger equation of hydrogen atom and I hope that you are as happy about it as this big fat smiling happy nucleus that we have here. And the reason why we have this cartoon here is that we should not forget that even though we said that Rutherford model is not tenable the basic experiment that had led to Rutherford model experiment performed by Marston remember this alpha particles shot at a gold foil most of them go through some of them deviate and one in 20000 turn back that experiment was not wrong. So, the basic observation that the almost the entire mass and all the positive charge of the nucleus of the atom is centered at a point that still holds and the fact that the negative charge the electron in the atom is somewhere out in this vast void space in the atom that also holds what does not hold is that this electron goes around in circles or ellipses or whatever. But well it does move how does it move we cannot say exactly what the trajectory is but we can write Schrodinger equation like we did for a free particle like we did for particle in a box. And one more thing that we learned very very important thing in the last module was that we learnt about spherical polar coordinates and we learned a little bit about angular momentum. So, with this now we are all set to talk about hydrogen atom as you will see we are going to set up hydrogen we are going to set up Schrodinger equation for hydrogen atom then we are going to simplify it we are going to use certain assumptions. I will show you the solution of one part of the equations and then we are going to just share with you what the solutions of the other parts are and what sense we can make out of them. But before we start let us remember what we have learned so far the very basic tenets of quantum mechanics are that we are working with Schrodinger equation we are going to write down Schrodinger equation this started off as a classical wave equation for deep ugly waves. But the realization that it is an eigenvalue equation triggered something much bigger the postulate of quantum mechanics came about which said that you should be able to write an eigenvalue equation for every physical observable. So, the wave function contains all the information when the operator operates on it it should be able to give you an eigenvalue equation the eigenvalue that you get is the value of the physical observable we are talking about and it also sort of tells us that for every physical observable like position momentum energy angular momentum we should be able to construct an operator. And then this operator would better be Hermitian because your eigenvalues must be real and the operators are also linear. So, these are all the postulates of quantum mechanics that we have touched upon very very briefly. And we know already how to calculate expectation values and one extremely important information that we have got now from our study of particle in a box is that quantization arises out of boundary condition. So, with this let us now think how we can formulate the problem of hydrogen atom in the language of quantum mechanics. When we try to do that the first thing that we have to write down is the Hamiltonian. Now hydrogen atom is really a two particle central force problem what is the meaning of central force as we said the nucleus contains all the positive charge and the electron is a negatively charged particle. So, the electron moves in whichever way it moves under an attractive potential of the nuclear and there are two particles one is nucleus one is electron and that is always a problem whenever we have a two particle problem we try and reduce it to one particle problems they are much easier to handle. After we do that this hydrogen atom Schrodinger equation is actually completely solvable usually it is not. So, if you want to make fun of quantum chemists usually tease them saying that these people have only one equation and they do not even know how to solve it for most of the cases. Of course, this is a not a very good thing to say and not so valid also, but well when you want to pull the leg of people this is what you do. Now any quantum mechanical formulation of a problem starts with writing down the Hamiltonian or rather writing down the operator. If you are talking about Schrodinger equation we have to write down Hamiltonian which is the involved operator here. So, what are the terms that the Hamiltonian will contain? It will contain a kinetic energy term of the nucleus a kinetic energy term of the electron and a potential energy term for attraction between the electron and the nucleus very simple and we know already what the kinetic energy terms are. We know that kinetic energy terms are like minus h cross square by 2 m del square where del square is del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2. To differentiate between the two we have written one term in capital N one term in small e this means a term that is written in terms this means a term that is expressed by the coordinates of the nucleus and the second term is expressed in coordinates of the electron. The third term of course, is very simple 1 by 4 pi epsilon 0 z and z e square by R E N I think we are all familiar with this very familiar I will ask you what is it called does this law have a name I am sure you know what it is and we have written in Si unit for now that is why this 1 by 4 pi epsilon 0 is there and since one of the charges is plus and one of the charges is minus between proton and electron we have a minus sign here it is as simple as that R E N is a separation between the electron and the nucleus. So, this is the Hamiltonian but we have to solve we have to write it in a little simpler way before we can proceed further because you see if you use that Hamiltonian and write down Schrodinger equation the psi that we have here for now we are calling this psi total even though psi total actually has a little different meaning later on which is not really a part of this course that we are now doing but generally when we talk about psi total we mean a special part of psi multiplied by a spin part of the psi we are not going to get into spin in this course. Here when I say psi total I mean the psi of the entire hydrogen atom nucleus as well as electron that psi is going to be a function of X and Y and Z N X Y Y X E Y E Z E all the 6 coordinates will be there we have to make this a little simpler and we are not going to go through the mathematics of this but it might help if we understand at least what the frames of references are. So, this is how we can formulate the problem let us say this here is the nucleus and this is the electron. So, X N Y and Z N and X C Y E Z E are the coordinates of the nucleus and the electron respectively in a space fixed coordinate system some absolute coordinate system. Now, the position vectors are written as R E and R N now see if I try to write this vector draw this vector R E N how will I get it? So, these are 2 vectors right R E and R N if I subtract R N from R E then I get R which is R E N and that is given by root over X square plus Y square plus Z square. Now, how it comes that we can show easier but I have not even told you what X Y Z R let me tell you that first X is given by X C minus X N Y is given by Y E minus Y N Z is given by Z E minus Z N what am I talking about here where the axes are parallel. So, let us say I will draw this horizontal axis that is Y this is I call it Y dash this let us say is Z dash and this here is X dash I hope it is not very difficult to see that X is parallel to X dash Y is parallel to Y dash Z is parallel to Z dash. So, what is what am I doing I am doing a transformation of coordinates I am moving the origin from the absolute value to where the nucleus is. Now, see what will be the X coordinate of M E X C minus X N in this new frame of reference which I have written as X Y X dash Y dash Z dash Y is going to be Y E minus Y N Z is going to be Z E minus Z N it is a simple as that now it is very simple to see what your R is what is R R is the position vector of the electron in this new coordinate system that I have built right because this is the new origin and this is the point. So, this arrow that you see here that is the position vector of the electron in this new transformed coordinate system. So, this coordinate system is called the internal coordinate system what does that mean that means this talks about the displacement of the electron in terms of the nucleus and the other coordinate system that we do not need to talk about in this course is this these are mass weighted or center of mass coordinates. When you do this is a very standard technique for separation of variables in problems like this you essentially build this coordinate system second coordinate system which talks about the movement of the center of mass. So, the second one capital X capital Y capital Z that talks about movement of the center of mass and this one small x small y small z talks about movement of the smaller mass with respect to the larger mass and here of course we have to use instead of mass we have to use reduced mass I hope you all know what reduced mass is the way I remember it is 1 by mu equal to 1 by m 1 plus 1 by m 2 a question for you the reduced mass suppose I have 2 masses m 1 and m 2 let me give you some numbers one mass of unit 1 another mass of using unit say 1800 roughly proportional to masses of electron and nucleus can you calculate the reduced mass of this system one mass is 1 unit the other mass is 1800 unit. So, I suggest that you stop the video right now pause the video and work out what the reduced mass is if you worked it out you will see that the reduced mass is very very very close to the smaller mass 1 and that is what is important here we will write the you will show you the equation shortly there you will see instead of m we have mu but then in this system mu is practically the mass of the electron and capital M the total mass is practically the mass of the proton because their masses are so very different. Now in this system you can do a formal separation of variables we are not going to do it here just believe me when I say that when I write the Schrodinger equation for hydrogen atom in a relative frame of reference then when we try to do that first of all I get 2 equations one in terms of the center of mass coordinate and one in terms of the relative coordinate. So, what you see here the first term it contains capital M in the denominator that is total mass practically the mass of the proton and this capital R is the coordinates of the center of mass which is essentially the proton the nucleus and this second one shaded in blue that has seen mu a minus h cross square by mu by 2 mu what is this mu practically the mass of the electron and here you have small r which talks about the displacement of the electron from the nucleus. So, this is what we get now how do we separate we separate in this manner well first of all this separation is done in this way I am not going to talk about it whoever is mathematically inclined is welcome to go through all those we are we do not really need to do this let us take it axiomatically. So, we have got 2 equations one like this the first term I will call it h n because that is the only term on the left hand side which is in terms of the capital M mass which is sort of the mass of the nucleus as I have said several times second and third terms together make up the Hamiltonian for the electron again why electron because mu is practically the mass of the electron did I say it too many times perhaps I did, but that is all right. Now how do I write wave function to write to separate this equation what we do is we write the wave function as a product of 2 wave functions one in terms of the center of mass well since it is practically the nucleus I call it chi n and one in terms of the electron psi e. So, why are we doing this remember particle into d box there also since x and y directions were independent we took the wave function to be a product of a wave function in x and a wave functions in y here also the nuclear and the electronic coordinates are independent after separation of variables that is why we can write the wave function as a product of a nuclear factor and an electronic factor. So, when we put that in what will the energy be naturally energy will be a sum of the center of mass that is the nucleus and the reduced mass that is the electron en plus ee and what you see is this color coding is used to highlight which part is associated with nucleus which part is associated with the electron. So, now essentially we collect all the terms in the nucleus coordinates. So, from the left hand side what will I get minus h cross square by 2 capital M del R square that operates on see when it operates on chi and psi what will happen psi is a constant it will come out. So, we are doing separation of variables I will not do it in very much of detail here because we are going to do it in little more detail slightly later for another equation. So, if you do not understand what we are doing here please wait up a little bit let us do that and you can come back and do it yourself crux of the matter is we get one equation as h n operating on chi n giving en multiplied by chi n and h n here is only the kinetic energy operator does that ring a bell have you encountered Schrodinger equation like this actually we have we have encountered it when we talked about free particle. So, this is what gives us the kinetic energy of the atom as a whole and remember this is not quantized en is h cross square k square by 2 m k can take up any values. So, essentially the atom as a whole if you look at the atom as a whole you will see it undergoing translational motion and this translational motion energy translation energy is not going to be quantized this is very important to understand. All right that being said we do not worry about it anymore because we are not interested in the movement of atom as a whole we are interested in movement of the electron with respect to the nucleus for that for the rest of this discussion we are going to focus on the electronic part of Schrodinger equation. Now, if you have not understood anything we have said so far it does not matter if you can believe that we have to formulate this Schrodinger equation for the electronic part and del R square is the kinetic energy minus h cross square by 2 mu del R square is the kinetic energy operator for the electron this one is the potential then I think you can understand the rest of the discussion no problem. Now, this is our Hamiltonian and in fact this is our Schrodinger equation del R square as you know is del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2 and R as we have said already is square root of x square plus y square plus z square. So, if you want you can write it like this you can write it out and this is going to be your Schrodinger equation in terms of x y and z is that okay can we work like this can we just write is equal to e multiplied by psi e and try to solve it actually we cannot because see earlier when we talked about 2D 3D box we could separate it very nicely here the problem is here we have a term in root over x square plus y square plus z square you cannot separate it. So, your Cartesian coordinates will not work what should work well if you go back to the original form of the equation here we have written R and the problem is that we cannot separate R into x y and z. So, if you cannot separate it can you work with it is there a coordinate system in which R itself is a coordinate actually there is and we know what it is that is spherical polar coordinate. So, we need to formulate Schrodinger equation in terms of spherical polar coordinates I will not repeat all these relations and expression for your volume element and all because we discussed already suffice to say that we need to now reformulate the problem in terms of spherical polar coordinates easier said than done because when we try to do that I have removed all these slides usually in my regular classes we have a lot of fun showing 13 slides that we are not going to discuss. There those 13 slides tell us how to go over from del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2 to this big scary expression that we have what we have written here is kinetic energy operated and spherical coordinates you have to believe me on this or you have to actually do the transformation yourself if you want to do it suit yourself I am not about to do I am not about to do it here that being said do you have to remember this no please do not nobody needs to remember anything in a course like this and in case this is used by some colleges by teachers my request to teachers is that we should never ask questions like write down the kinetic energy operator in spherical polar coordinates we need to use our brain not just as a storage device rather we have to use it as a processor right so let us do that let us focus on understanding rather than memorizing things right this is something that is available in many resources we will just use it okay how do I write the Hamiltonian operator this is the kinetic energy operator so to get to the Hamiltonian operator I just have to add the kinetic potential energy term here it is how do I write Schrodinger equation now this Hamiltonian in spherical polar coordinates has to operate on the wave function let us do that this is your Schrodinger equation now so this here is Schrodinger equation for the electronic part in spherical polar coordinates and see what we have these are the terms in R these are the terms in theta and these are in phi the first term is only in R's R second term is a mixture of R and theta the third term contains R and theta and phi so we will have to find a way of separating these equations and this is what we were saying that we are going to do it at least for this part of the discussion how do you do it well first of all let us see how we can simplify further if I multiply by minus 2 mu R square by h cross square what happens this R square gets eliminated right so at least the second term becomes only in terms of theta except for the wave function of course the third term the operator part is only in terms of theta and phi so R is eliminated from second and third terms and then we bring this e psi to the left hand side again I recommend that you keep on pausing the video writing down what we should get in the next step and you come back and see the video after that that way you will understand better I hope you are doing that now I will show you what the answer is if you do this this is what we get first term becomes del del R of R square del psi e del R yeah you have multiplied by minus 2 mu R square by h cross square what does the second term become R square would go 2 mu would go h cross square would go you are left with 1 by sin theta del del theta of sin theta del psi e del theta what about the last term it becomes 1 by sin square theta del 2 psi e del phi 2 and since you have brought whatever was on the right hand side to left hand side the last term that you get is 2 mu R q z square by h cross square what is q we had defined q to take care of all these constants 1 by 4 pi epsilon 0 and all that so that will be plus 2 mu R square h cross square e psi e that is going to be equal to 0 okay so this second last term is the potential energy term the last term is what was there on the right hand side all right now these are in terms of R only are except for the wave function these are in terms of theta and phi there are two kinds of coordinates right R is a line and theta and phi are angles so first as a first step if you can separate R from theta phi that is good enough all right how do I do it again I have to define my wave function in a particular manner so let us define the wave function as a product of an R dependent part a theta dependent part and a phi dependent part we often do not even write this small r small theta small phi in brackets you just write capital R capital theta capital phi what is capital R wave function what is capital theta wave function what is capital phi wave function capital R is a wave function that is written only in terms of small r capital theta is a function of theta only not R not phi capital phi is a function of phi only not R not theta so small letters are coordinates capital letters are wave functions okay how are we okay how is it that we can write it well we are doing separation of variables and these are all independent coordinates okay so let us write that this is what your equation becomes now what what happens when we try to differentiate well this is the step that we had skipped earlier so let me at least draw some arrows here and let me show you let us take the first term when I differentiate see this theta capital theta and capital phi they are not functions of R right so as far as R is concerned they are constants so they are going to come out similarly in the second one capital R and capital phi these are not functions of small r so they are sorry they are not functions of small theta so they are also going to come out and the third term we are differentiating with respect to phi so capital R and capital theta which are not functions of phi they are constants as far as this differentiation is concerned okay and last two terms you do not even have to worry about because they are just products so that is what we get and again please pause work it out yourself and then only see the next step you do that this is what you get capital theta capital phi del del R of R square del capital R del R in fact there is no need to write del here anymore because I do not have a partial derivative any longer is not it so it is absolutely okay and it is desirable that I write dr d capital R dr that is actually better similarly the second term we have capital R capital phi outside 1 by sin theta del del theta of capital theta so again I can write actually d capital theta d theta there is no need of writing that delta any longer and finally in the third term capital R capital theta come out and you are left with 1 by sin square theta del 2 capital phi del phi 2 again it is better to write d 2 capital phi d phi 2 remember d 2 capital phi means d 2 wave function phi dependent wave function and small phi is a coordinate alright so and these are the other terms what is the next step obviously you do not want this capital theta capital phi in the first term you want to get rid of it the easiest way of doing it is to divide by capital R capital theta capital phi do it and see what you get yeah multiply by 1 by capital R capital theta capital phi please do it yourself this is what you get 1 by capital R del del R well d dr of R square dr dr this and this I am not going to read out everything here this is what you get and now have a look well I have written it a little more neatly here now see these are the terms in R no theta no phi these two terms are in terms of theta phi only no R these are equal to 0 rearrange a little bit these are equal to each other but left hand side is in terms of R right hand side is in terms of theta phi they are independent coordinates so there this radial and angular parts the part in radial and part in angular coordinates the part in R and part in theta phi they must be equal to some constant we write it beta why because left hand side is in terms of R small r right hand side is in terms of theta and phi how can they not be constant they have to be a constant very simple mathematical tool that is used universally so when you do that you get two equations the first one is called the radial equation second one is called angular equation and please note that they are connected by beta they are not independent of each other for example in the Schrodinger equation for 2d particle and 2d box the two the equations were independent here they are not independent they are actually correlated by beta and I do not have to do anything more for the radial part it is sorted but angular equation have to still work a little more because they have theta and phi mixture of those how do I do it same thing first of all multiplied by sine square theta this is what you get first term is only in theta second term is only in theta well then I take this I just I sort of interchange sides so now left side is in theta right side is in phi again these are going to be equal to some constant this constant I call m square now why m square why not k because I know what we are going to do later we are not the first people doing it here it is absolutely okay if you write k or whatever you want to write but I know that life becomes simpler and meaningful if I write m square here that is the only reason why it has been written this way all right so we have three equations one a function of r one in terms of r one in terms of theta one in terms of phi they are separated what we are going to do next is that we are going to try to solve this simple one and while I am speaking many of you would have solved it also all right but let us take a break now we will come back and we will solve this and we will tell you what the solutions are of the r dependent part and the theta dependent part