 Hello there we have seen in our earlier class how a magnetic resonance transition takes place today we are going to see the same thing from a quantum mechanical perspective what we had seen earlier is something like this we keep a magnetic movement in a magnetic field magnetic field pointing along z direction then the magnetic movement precess around this magnetic field these are the coordinate system and then we apply another magnetic field let us say B1 which rotates in the xy plane and then if the rotational frequency of this is the same as the frequency of its precession then the magnetic movement vector turns from this to this and that is the transition. So, here the frequency of precession is given by the larman frequency omega L gamma electron by B is the field. So, when this frequency is same as this we get the resonance condition. So, this is the classical description of magnetic resonance today we will see what happens quantum mechanically. So, to get into the magnetic resonance from the quantum angle perspective let us review some basics of quantum mechanics in quantum mechanics this the wave function which is a function of let us say all coordinates and time this is a 1 x coordinate x, but one can have all the possible coordinates of the all the particles there this has information that is that you can think of about the system and this follows this Schrodinger equation this is the time dependence Schrodinger equation where H is a Hamiltonian operator it is a kinetic energy operator plus potential energy operator. Now when the potential energy is not a function of time it depends only on coordinates then this wave function psi can be written as function of only the space coordinate say phi of x and the energy term comes in this fashion this is the time independent part here and this phi x satisfies the same type of Schrodinger equation. So, this is called the time independence Schrodinger equation this is the basis of quantum mechanics now this is a particular case when the potential energy is independent of time then you get an equation of this kind and the total wave function is a product of the space part and the time part and it appears in this fashion in E is the energy of the system. Now the other concepts we need to revise is the angular momentum and it is and there and it is allowed values in quantum mechanics that is a L is the operator correspond to the angular momentum this is written in terms of its coordinate components i j these three components of angular momentum operator the magnitude of this is given as the product of this which is Lj square it so happened that the property of these three components of the operator and the total angular momentum square there are sort of restrictions that restricts the allowed values in this fashion that we can have only the total angular momentum and one of its component. So, you can choose whatever component we think of Lx L0 I am sorry this will be Ly Lx Ly Lz and the total magnitude of this is given by L square and that and one of the components can be measured at the same time or these two can have well different value at the same time that is a very important restriction here for example, when you have a picture of this kind it shows that any instant of time the magnetic moment is pointing is this direction. So, it has a well different value of the magnetic moment in all possible direction it is because Mx, My and Mz and these in turn should be related to the corresponding component of the angular momentum. Now this restriction say that that is not possible you can have only one component of the angular momentum and the total value of this. So, that puts a limit so this picture is not therefore, quite applicable if we strictly follow the quantum on mechanical principles ok. So, conventionally Lz is taken to be the direction of quantization. So, we can therefore, get the L square values and Lz values these are well defined for orbital angular momentum the corresponding wave function or rather the Eigen function of this operators are written symbolically by their allowed values let us say L and Ml such that L square operator gives the value of L into L plus 1 this is the short hand notation of saying that this represent a particular Eigen function for a given L and given Ml. Similarly, Lz gives the value in the restriction comes in this fashion that L can take value of this kind integral values and Ml takes again value of integral values L. So, if L is equal to 1 then Ml will be minus 1 0 plus 1. So, if you are dealing with spin angular momentum exactly similar properties hold there and we could write that S square is the total angular momentum square and I can write S MS gives S into S plus 1 and SZ gives here the restriction is that the counter number S can take 0 half 1 3 by etcetera. So, here half integral values are also allowed and then for L this will be strictly integral values MS similarly it change from minus S to plus S in steps of 1. So, in general if the angular momentum is let us say called j then j square will give a value j into j plus 1 h square Mj and a z gives Mj. So, this is a very general expression. Now we define another pair of operators in for angular momentum for example, L we call L plus minus is defined to be Lx plus minus i Ly similarly for spin angular momentum S plus minus defined to be Sy. These are useful when you need to find out some integral that involves Lx Ly or Sx Sy. Now these are property of these are defined in terms of their behavior with respect to the Eigen function of L square operator let us say I have got L ML this will give plus minus 1 similarly S plus minus S MS will be plus minus 1. We will find this expressions very useful in deriving certain expressions. One important consequence of this form of the wave function that is when the potential energy does not depend on time then the functional form of this is a product of the only space part times the energy part in this fashion. This special property is this that if you want to measure the physical property of any type that the system has in the analytical variable let us say kinetic energy potential energy some other properties like let us say nipple moment for molecular boundary length boundary angle all these properties each of them there is a operator to find out what the possible value that this particular system has the prescription is that we put the wave function in this fashion and then evaluate this integral by the operator is in the middle of that this is called the expectation value of the of the physical quantity described by this operator. Now here if the wave function of this kind then you see what happens this will look like now this is just too many times here yes this is better. Now here this is a complex concept of this so this can be taken out of the integral this will be exponential plus i this is a allowed operation I can take this term out of the integral because this integral is with respect to only the space coordinate it does not involve integration with respect to time this is a function of time this is a function of time. So, this can come out of the integral to give rise to this term now here see this are complex concept of each other. So, this give us just one so this is actually equal to therefore ix. So, here therefore this expectation value of any property that you can think of that becomes a function of only this kind and when you integrate the respect to all spatial coordinate this becomes a quantity which does not have any more time dependence. So, that means that all physical properties of the system that you can think of for a wave function which is described by this becomes independent of time. So, that means the system does not evolve it does not change we call that it has it is in a stationary state. So, all the properties therefore become independent of time the key to that is that wave function which is function of course space and time is a product of space part and time part. And this is possible only when if you go back this potential energy is independent of time then only such thing is possible ok. With this background now we try to see how much we can describe the magnetic resonant transition in terms of quantum integral principle. We start with this that magnetic moment this mu is placed in magnetic field B. So, its energy is given as minus mu dot B. So, here for quantum mechanics we replace all this by their corresponding operators. The Hamilton operator for this interaction could be given as similar to this. Now the magnetic moment of a let us say electron spin arises from its angular momentum and that is given by minus G E B E S. So, here then the Hamilton operator becomes G E beta S dot B. So, this is the Hamiltonian operator for the electron spin S put in a magnetic field. If there is only one electron then it is allowed component of the spin angular momentum is S equal to half and M S is equal to plus minus half. Now if the magnetic field is applied along the Z direction then B is let us say B 0 key this is our Ziemann field then Hamiltonian becomes G E beta E B 0 S Z. So, this is the Hamiltonian. So, here since the two component of the angular momentum is given by plus minus half I could designate the state of this by this symbol alpha which corresponds to plus half and beta which corresponds to minus half. So, here therefore, S Z alpha gives plus half H cross alpha S Z beta corresponds to minus half H cross beta. Then the energy can be now found out easily by operating H on this two states. So, alpha gives this and similarly beta state gives minus B 0 by 2 beta. So, this is of course, something like a spatial part of this space part symbolically spin is not quite related to any physical space, but we take it to the spin coordinate of the electron. So, to the total wave function which is of this kind will therefore, look like let us call it psi of alpha subscript alpha will be this alpha into e to the power minus i e alpha by T by H. Here what is the energy? Energy is given by this one. So, here I can write e of alpha is equal to similarly energy for the beta state. So, this is the total wave function for the alpha state. Similarly, total wave function for beta state will be given by this is the subscript beta. I hope there should be no confusion between the subscript beta and this beta e this is Bohr magneton beta subscript e. This is the this alpha and beta corresponds to the spin state plus half and minus half. Just keep in mind that these two should not cause any confusion and no mix up takes place. So, these are the wave function that we found out for spin which is kept in magnetic field and whose Hamiltonian is given by this fashion. These are Eigen state of the Hamiltonian and also these are stationary states. Nothing pretty much happens there. That is system does not evolve. So, we can write this in terms of the energy level diagram. This is by psi alpha psi beta corresponding to m s equal to minus half m s equal to plus half and this energy is given by this is alpha here beta here and e of alpha is G e beta e minus and here this is alpha e beta is G e beta e by 2. So, we have got complete description of the spin in a magnetic field. Now, let us apply the small magnetic field along the x y plane. This x y z this is b 0 applied along the z direction. Now, we apply a small magnetic field which is moving in this x y plane. This was magnitude is given by b 1 and b 1 is very very small compared to this one. How do I describe that? Suppose at time t equal to 0 the b 1 was exactly along the x direction. Then after some time it has come here. See this angle is theta and this angular velocity of b 1 around this z axis is omega then omega will be omega times t equal to theta t equal to 0 this was here now after time t it has come here. So, the component here is x component and this is the y component. So, x component becomes b x becomes b 1 cos omega t y component b y is b 1 sin omega t. So, this rotating magnetic field can be described by this b vector. So, b 1 vector is b 1 cos omega t i unit vector b 1 sin omega t j unit vector. So, this is the vector which describes the rotation of this b 1 field in the x y plane with an angular velocity omega. So, when this is present what will be the Hamiltonian of the system? We go back to this again. Here the b now total magnetic field that the spin system experiences is given by say b 0 in the z direction the z 1 field here plus i times b 1 cos omega t plus j times b 1 sin omega t. So, the Hamiltonian becomes same way as this. So, this Hamiltonian becomes new Hamiltonian now in the presence of this b 1 field will be g e beta e b 0 s z which is a product of this and the z component of this plus b 1 cos omega t x component and then g e beta e s of x some omega t g e beta e s of y this comes from the x component and y component of this one. So, this is the magnetic interaction that gives rise to this Hamiltonian of this kind. Now, here as I said earlier that b 0 is the z 1 field which is much bigger than b 1 very small. So, we can treat this as a perturbation and this is the main Hamiltonian. So, this could be written as therefore, a 0 plus let us say h prime a 0 is therefore, g e beta e b 0 s z h prime is g e beta e b 1 a 6 cos omega t s y sin omega t. So, this is the perturbation this is the main unperturbed Hamiltonian and we know the solution to the time dependence Schrodinger equation gives this psi alpha is equal to alpha these are the solution for this Hamiltonian unperturbed Hamiltonian these are the unperturbed wave function therefore, now we want to find out the wave function that satisfies this total Hamiltonian h equal to h 0 plus h prime. So, here see this is a perturbation and these are the unperturbed wave function we can think of the wave function which satisfies this Hamiltonian that this could be written as a linear combination of these two. In other words we take these to be the basis wave function and we expand the wave function corresponding to the total wave function total Hamiltonian in terms of these two bases. So, this becomes this is c 1 c alpha these and these are these and these. Now, this being a now time independent Schrodinger equation and in particular you see there is a time independent part which is present here now we expect this to be in general function of time. So, how to solve this we simply put this here and see what happens put it here. So, this will give me h 0 c alpha t plus is equal to the right hand side here now we notice that this unperturbed wave function psi alpha and h 0 the unperturbed Hamiltonian these operating on this gives h by 2 pi i del psi t operating on this directly. Now, this is a function of time this also could be a function of time. So, this we can write in this fashion that h this too much down. So, what we get here is 0. So, this time there it will operate on this as well on this one. So, this gives here. So, here this term is exactly equal to this term this comes from the unperturbed Schrodinger equation and unperturbed wave function. Similarly, this term is exactly equal to this term. So, we can therefore cancel this from the two sides of the equation. So, this gives me now you are aim is to find out this coefficients c beta t and c alpha t that tells me how the system is evolving with time. So, to do that suppose you multiply on this side by psi alpha here. So, what I find here is that because of the orthogonality of this two wave function psi alpha and psi beta this term goes to 0 and this is equal to 1. So, what I get is that minus h by 2 pi i this one which is this can be taken out of the integral. Now, this psi alpha and psi beta we know their form which we have it here this can be now simplified to write in this fashion. Similarly, the coefficient d beta time returns c alpha y. So, these are very general form of equation where is c alpha is coupled to c beta c beta is coupled to c alpha. In our particular case now the Hamiltonian h is given as. So, you put it here and then evaluate the integral with respect to the alpha and beta state. So, that gives me this in the first beta. Here you see that again that these are two couple differential equation and the way it is two energy levels alpha and beta comes is the energy difference that comes into picture here. This omega is the frequency of the b 1 rotating around the x y plane. Now, in a to solve this we can use the perturbation technique such that we assume that at time t equal to 0 the system was in the beta state that is equal to 0 it continues here. So, in other word this c alpha at 0 was 0 and c beta time t equal to 1. So, we want to find out therefore, what is the probability that this will make a transition from here to there that is c alpha becomes non-zero. So, we will continue this derivation of this one in the subsequent lecture we stop at this moment now.