 Before starting our discussion of hydrogen atom it is important that we prepare ourselves a little bit keeping that in mind today we are going to discuss different coordinate system from the Cartesian coordinate that is spherical polar coordinate and also we are going to talk a little bit about this rather important quantity that we must have studied in physics that is angular momentum. So, the question is why is it that we need to talk about spherical polar coordinate why must we work in spherical polar coordinates if we want to build a quantum mechanical description in angular momentum that is what we will learn in this lecture and why at all talk about angular momentum that will become clear when we start talking discussing our Schrodinger equation for hydrogen atom. So, spherical polar coordinates I hope are not completely new to most of us the same point in space that we conventionally described by coordinates x, y and z can also be described by r, theta and phi where r is the distance from the origin 000 point the angle theta is the angle between the z axis and the position vector you can call right position vector is a vector joining the origin to the point in question. So, the angle between z axis and that position vector is called theta and then if you drop a perpendicular on the xy plane let us say we drop a perpendicular on the xy plane then the arrow this blue arrow that we get this is called the projection actually it makes sense for me to join the two tips of the arrow. If you drop a perpendicular from the tip of this arrow to the xy plane then this arrow that we now get in the xy plane that is the projection of the position vector that we are talking with originally the angle between the x axis and the projection vector this is called phi it is almost hidden by the globe unfortunately so I will write again this is called phi. Range of r is from 0 to infinity remember r cannot go from minus infinity to plus infinity 0 to infinity theta goes from 0 to pi 180 degrees. You can ask why is it that does not go to 360 degrees because it is not necessary we are allowing one of the angles phi to go from 0 to 2 pi. So, the moment that happens there is no need for us to talk about theta more than 180 degrees that is why. So, of course I mean one can think that theta should range from 0 to 2 pi and phi should range from 0 to pi but this is the convention that is used everywhere we should not build our own convention unless it is absolutely necessary we go with whatever is the existing convention. So, remember range of theta is from 0 to pi range of phi is from 0 to 2 pi the relationships are very simple z equal to r cos theta quite simple right because if you draw a perpendicular from this point to the z axis something like this this is perpendicular what will be the length of the arrow on the z axis that is a z component that is z and what is this angle theta. So, it is not very difficult to see that z is equal to r cos theta r is missing here for some reason but here it is written r is the hypotenuse z is the side adjacent side. So, z is equal to r cos theta what is x all right. So, this length is r so and this angle is theta this angle is 90 degrees minus theta. So, I hope it is not very difficult to understand that the length of this arrow the projection that is r multiplied by sin theta r multiplied by cos of 90 degree minus theta that is r multiplied by sin theta. Now, when we look at this triangle here you drop a perpendicular from the tip of the arrow on the x y plane to the x axis is a perpendicular. Now, what we get is this here r sin theta this is your hypotenuse and this is the adjacent side. So, you get r sin theta cos phi. Similarly, y turns out to be r sin theta r sin theta is same in both because r sin theta is the length of the projection here r sin theta sin phi quite simple. You can work out the inverse relation here x y z the Cartesian coordinates are subjects of formula you can turn it around and make r theta and phi the subject of formula I will not do that you can do it yourself what we get is r equal to square root of x square plus y square plus z square that is of course very obvious right this is sort of the body diagonal r and the lengths of the three sides of the cuboid r x y and z. So, of course x square plus y square plus z square is r square or r equal to square root of x square plus y square plus z square what is theta actually what is theta comes from the first relationship anyway cos theta equal to z by r. So, theta of course is cos inverse z by r what is phi the easiest relationship is that what is sin phi by cos phi that will be y by x so that is tan phi is not it so phi equal to tan inverse y by x I said you do it I would not but then I ended up doing it because it is so simple anyway. So, these are the relationships and well I think by and large you can remember it but if required you can always derive it is not such a big deal the point I want to make is that spherical polar coordinates are things that we are sort of introduced to very early in our lives when we are in school if you think of a globe we have all studied geography at some level and we know about latitude and longitude what is latitude what is longitude are they related to theta and phi r of course is constant for the earth that is the surface of the earth but what about theta what about phi theta is not exactly latitude it is 90 degrees minus latitude because latitude remember is defined from the equator. So, what we call theta equal to 90 degrees in our system is written 0 degrees as far as latitude is concerned and there the range is plus 90 to minus 90 right. So, theta is related to latitude and phi is actually longitude phi goes around like this right in the equatorial plane. So, phi is exactly longitude. Now we have been already introduced to this d tau business we know that psi psi star d tau is the probability of finding the particle at some point r within a small volume element and we had said that if you work in Cartesian coordinates x, y, z then d tau is simply dx, dy, dz when we work in spherical polar coordinates it becomes a little different and this is something that we will use so we better learn. So, here the diagram is drawn already but I will still try to draw it so that we understand little better. So, let us let me draw here maybe so suppose I want to define a volume element at this point what do I do I increase all the coordinates a little. So, I can increase r by an amount dr of course the way I have drawn it this is a large amount actually dr is very very small compared to the length of r here I am exaggerating so that we see better all right. Next what do I have to do I have to work with theta how do I work with theta change theta a little bit so make it theta plus d theta. So, now I can draw another line where this small angle here where smaller than what it seems in this diagram is d theta. So, now see what will be the length of this r since d theta is a small angle this length is going to be r d theta. So, two sides of the volume element are defined already one side is dr the other side is r d theta what about the third side how do I define the third side. So, drop this now I will increase phi a little bit like this. So, now our projection will go somewhere here now I can think of this arc this is the third side of the volume element just draw a perpendicular like this this arc is the volume element what will be the length remember this length here we already said is r sin theta and I have increased by increase the angle by d theta d phi. So, this is going to be r sin theta d phi I am not writing in a nice place this is going to be we will write here r sin theta d phi. So, the three sides of the volume element are defined now just to see whether we understand it or not I will draw the other sides of the arc as well let us hope I have not forgotten which color I used this is one and now I will draw r this is r again you have to increase this as well and just because I am a little bored of changing colors I will complete the volume element is the same color anyway. So, I hope you see the volume element here the volume element is not exactly a cuboid it has curved sides but the curved sides are almost linear because they are small segments of a large sphere you can think and in case you have had trouble understanding my poorly drawn diagram here you have a little better diagram here also you see the sides are dr r d theta and r sin theta d phi. So, the volume element volume of the volume element is going to be r square dr sin theta d theta d phi this is very important and we are going to use this later on in our discussion. Now, let us talk about angular momentum as you will see when we discuss hydrogen atom angular momentum is going to be a very important property that we will worry about. I think once again we all know that angular momentum is defined as a cross product of r and p r cross p what is the meaning of cross product r is a vector p is a vector l is also vector. So, l is the vector product of r and p what is the magnitude of l then magnitude of l is going to be r p sin theta and what will be the direction? Direction will be perpendicular if this to the plane of r and p r and p will always be in some plane. So, direction of l is going to be perpendicular to that. Now, let us work out what this is and while doing that it makes sense if you work with x y and z components to begin our discussion. So, we will write r like this i j k are the unit vectors along x y and z respectively and x y and z are the scalar values of well x y and z respectively. So, r vector is given by i vector multiplied by x plus j vector multiplied by y plus k vector multiplied by z and p vector is given by i vector multiplied by p x plus j vector multiplied by p y plus k vector multiplied by p z. We want to work out what is l equal to r cross p. And when we do that, this is what we get. First of all, why do we have 6 terms and not 9 terms? We should have got something like i cross i x square also. Well, the thing is we need to remember this i cross i j cross j k cross k everything is equal to 0. Because remember what the magnitude is going to be, it is going to be sin theta, i square sin theta that kind of thing would have happened. The angle between i and i, j and j and k and k, this angle is 0, sin theta is 0. So, cross product of the same vector with itself is 0. We do not have to worry about that. What about the others? i cross j is equal to k. Remember these are all perpendicular to each other. So, k is perpendicular to the, so k is basically along z axis, it is perpendicular to the xy plane. And j cross i is minus k that is important to understand. So, if you change the order in this vector product, the direction is going to change even though magnitude will remain the same. And it is very easy to remember also i cross j equal to k, j cross k equal to i, k cross i equal to j in alphabetical order, go in alphabetical order. If you are going in alphabetical order, you are going to get positive. And if you go in anti-alphabetical order, j cross i that will be minus k, k cross j minus i, i cross k is minus j. It is as simple as that. If you just want to remember it, of course, you can work it out and see what it is. So, let us see. In this expansion that we have performed, let us collect the terms in i. Remember i is obtained by taking cross product of j and k that is i and cross product of k and j is minus i. So, j and k, j cross k y p z, this becomes i y p z. And we have k cross j z p y that becomes minus i multiplied by z p y. So, if I collect the terms in i, I get i multiplied by y p z minus z p y. This very nice symmetric expression and the other two terms also turn out to be like this plus j multiplied by z p x minus x p z plus k multiplied by x p y minus y p z. So, see what is y p z minus z p y? That is the x component of the angular momentum, because it is multiplying the unit vector along x direction. What is z p x minus x p z? It is the y component of angular momentum. And what is x p y minus y p x? That is the z component of angular momentum. This is conveniently written in the form of a determinant. i multiplied by y p z minus z p y minus j multiplied by x p z minus z p x, which boils down to plus j multiplied by z p x minus x p z plus k multiplied by x p y minus y p z, y p x. It is a very simple determinant form in which we can write this. So, this is the classical description. Now, let us think how we are going to write the same expressions in the language of quantum mechanics. Remember, in quantum mechanics, we have an operator for every physical observable. Operator for x, y and z are just position multiplying the wave function. Operator for something like p x would be h cross by i del del x. Operator for p y would be h cross i del del y. For p z, the operator is h cross by i del del z. So, all we have to do is in this determinant, we have to substitute x y and z by x hat y hat z hat respectively. And we have to substitute p x hat by h cross by i del del x and so on and so forth. When we do that, this is the angular momentum operator that we generate. If you know position and momentum operator, you can virtually generate all operators that we require. So, L hat operator is given by h cross by i multiplying this determinant i j k x y z del del x del del y del del z. This is what it is. Now, we will have something to say about the total angular momentum and z component of angular momentum starting from here. Basically, we want to write down the operators for total angular momentum and z component of angular momentum in terms of spherical polar coordinates. You will see why. The problem actually becomes simple. And why are you worrying only about z component? We will say that as well, even though it might be taking things a little too far for this particular course. So, in case you find that discussion to be too much, you can just skip that part. It depends on whatever is being done in your college. So, L x hat as we said earlier is y p z minus z p y L y hat will be z p x hat minus x p z L z hat will be x p y hat minus y p x hat. So, and then when we write the expressions for the p q hat kind of operators where q can be x or y or z, this is the expression that we get. Now, you know the relationship between r theta phi and x y z. Using these relationships, it is possible to convert from this del del z del del y to things like del del r del del phi del del theta. It is a little longish. So, we are not going to do it here. If you are interested, it is not difficult. It is just a little too long. You can do it yourself. So, it turns out that in spherical polar coordinates, L x hat operator is minus i h cross multiplied by minus sin phi del del theta, multiplying minus cot theta cos phi del del phi. You could also write h cross by i here instead of writing minus i h cross. For L y hat, the operator looks very, very similar. The only difference is wherever you have sin phi for L x, you have cos phi for L y. L z hat is the best of them all, minus i h cross or h cross by i, you can write minus i h cross or plus h cross by i multiplied by del del phi. Very, very convenient. That is what we are going to work with. And the other operator that is very important is not L. We generally do not work with L. We work with square of angular momentum. And that operator is obtained by taking L x square plus L y square plus L z square. And to cut a long story short, this is the form of the operator. This is something that will keep on coming back in our, in our discussion later on. Do you have to remember this for God's sake? No. We should provide this whenever it is required. There is no need to remember too much here. If you remember del del phi multiplied by h cross by i, I am very happy. L square, please do not try to remember. It will be provided in our discussions wherever required. So, these are the two operators that we are going to work with. L z hat and L square hat. Now, why is it that we are not interested in L x and L y? Because again, we are not going to do the derivation. If you are interested in the derivation, I recommend that you see my lectures that are available on YouTube for a more elaborate quantum chemistry course. You have to write Anindya Datta Quantum and search within YouTube, you will see 68 lectures and they are named. So, you go and see, go through the angular momentum lectures, you will get answers to whatever questions you might have at this point. For now, let us take it axiomatically that in quantum mechanics, if two operators commute, what is the meaning of commute? Meaning, you take a wave function, make a hat operate on it first and then make b hat operate on it. You get something. Then you reverse the sequence of operation, make b hat operate on it first and then make a hat operate on it. You get the same answer. So, another way of writing it would be this. What I am saying is you have some wave function phi. I will just write phi here, make a hat operate on it. You get something. Let us say a multiplied by phi. Now, make b hat operate on it. So, basically b hat is operating on a hat phi. So, this is what you get b hat operating on a phi. The other sequence, let b hat operate on phi, you get let us say b multiplied by phi. Now, let a hat operate on this b hat phi. This is a phi, sorry. Then you get a hat operating on b phi. It can be proved. We are not going to prove here that b hat a hat phi is equal to a hat b hat phi. Or you can write like this, a hat b hat minus b hat a hat phi equal to 0. That in the convenient notation that we write, we write like this. a hat, b hat within third bracket means a commutator, a hat b hat minus b hat a hat. b hat a hat, what is b hat a hat? b hat comma a hat, the sequence of operation is just opposite. Of course, in this case it will be same. But the crux of the matter is that if the commutator is 0, then they have a common set of functions. And properties associated with a hat and b hat can be determined simultaneously. It turns out again, we are not going to prove it here. It is there in those angular momentum lectures. It can be shown that L square hat and L z hat commute. That means, you can determine the square of total angular momentum and the z component of angular momentum simultaneously for a system. However, L x and L hat, L y hat and L z hat, L z hat and L x hat do not commute. They do not commute. So, L x and L y, L y and L z, L z and L x, they cannot be determined together. So, the only hope you have is to determine the total angular momentum and its component along one direction that we call z direction. Here, students usually have this question, what is z direction? I am talking about hydrogen atom, let us say. How does the atom know what is z? Atom does not know. Remember, wave function collapse? Remember that everything is, your system is in an entangled state before you make the measurement. Only when you make the measurement, wave function collapses into something. Suppose, I want to measure the z component of angular momentum. How do I see it experimentally? I apply a magnetic field, is not it? Zim and effect. So, the direction of the magnetic field becomes the z direction. What we are saying is, if we have an angular momentum, we have an angular momentum like this. When we apply a magnetic field, now this z direction is defined. So, theta is defined with respect to the direction of the magnetic field. Now, we can determine the length of this arrow, which is the angular momentum itself. Actually, we can determine square of it. We cannot really determine whether it is pointing up or down, just like just from here. L square is what we can determine. And the other thing that we can determine is a z component. So, from here you get L square, from here you get L z. Combining the two, you can say whether the arrow is pointing up or down of course. This is what we are going to discuss in a little more detail, when we talk about hydrogen atom.