 In the last lecture I made some preliminary remarks about linear response theory, a topic which we will study in some detail and today I would like to set the stage for it by discussing how time evolution can be viewed of dynamical observables, can be viewed both in classical and quantum mechanics in a kind of unified manner at least as far as the formalism is concerned and then we can proceed with linear response theory itself. Now remember that our target in linear response theory is to try and understand how observables change with time to first order in a perturbation under finite temperature conditions in general. So both in the presence in the presence of thermal fluctuations as well as an external perturbation we would like to find out how dynamical observables change on the average under this perturbation to first order in the external force or perturbation. Now it turns out to be extremely convenient to recast the usual Hamiltonian dynamical evolution in terms of an operator called the Liouville operator for the advantage of which will be that we have at least formally way of writing dynamical evolution which is independent of whether the system is classical or quantum mechanical. So let us start with the following observation and I will do both the classical and the quantum mechanical evolution kind of side by side so you begin to see what the commonalities are although of course we know that classical dynamics and quantum dynamics are very different things altogether. So time evolution and the Liouville operator is one we shall have in mind always a Hamiltonian system. The evolution we are talking about is Hamiltonian evolution and the Hamiltonian as you know is a function of set of generalized coordinates and the generalized momenta conjugate momenta in classical physics we would write this as Hamiltonian h of q, p where I use q and p as symbols to stand for the full set of generalized coordinates and the full set of corresponding conjugate momenta. And as you know in classical mechanics for which I will use abbreviation Cm in classical mechanics the evolution of these dynamical variables is given by Hamilton's equations. So you have q dot is delta h over delta p and p dot is minus delta h over delta q and in principle if you specify the initial values of all the q's and all the p's one is supposed to solve this set of equations to get the values of q and p at any arbitrary instant of time. So for instance the solution of this would be qt at time t will be some function of the initial values q naught, p naught and of course time itself and similarly for p some p this is pt of q naught and p naught. So given the initial conditions notionally one is supposed to solve the equations so that you can write what the q's and p's are at any instant of time subsequent to that. Quantum mechanically this is not how things go, quantum mechanically this becomes an operator so in qm you again have a Hamiltonian operator q and p. I would not bother to write operator symbols on all these objects but it is understood to be the case and then one asks what the equation of motion is for any dynamical variable any function of all the q's and p's for instance if you had some function a of q's and p's and observable which is a function of the q's and p's then we know that the Heisenberg equation of motion says ih cross da over dt is equal to the commutator of a with h. Classically the analog of the Heisenberg equation of motion of course is the equation for the time variation of any a which is a function of q and p this quantity is equal to the Poisson bracket of a with h. So this is an active picture of quantum mechanics where we say that the observables are changing with time and the time rate of change is given by this Heisenberg equation of motion which is a quantum analog of this equation of motion in terms of the Poisson bracket here okay. Now is there some way in which we can write both these equations in exactly the same form the answer is yes. So we introduce an object called the Liouville operator L which says that da over dt in this case is equal to i times L A C m and q m. In both classical mechanics and quantum mechanics we write it in this form because you can see here that the equation is linear in a and it is linear in a here too. So one could write it in this form with an i here for a purpose which will become clear in a few minutes and this operator L is called the Liouville operator. By definition this i L A is equal to a Poisson bracket with h out here. So it is immediately clear that L acting on any a is equal to I bring the i down out here to write da over dt is i L A. So I want i L A to be a with h and therefore L A is a with h divided by i and I get rid of a minus sign so it is i times h with a. This is true in classical mechanics and the quantum analog of this is again from this equation I write this as i L A on the right hand side and therefore in quantum mechanics da over dt equal to a h divided by ih cross and if this is equal to i L A then it says L with a is equal to 1 over whatever it is so L with a is equal to 1 over h cross Poisson bracket of commutator of h with a quantum mechanics. So formally I introduce a super operator this L should be regarded in quantum mechanics as a super operator because it says you perform something you take the operator a which acts on state vectors in the Hilbert space concerned and what you do is when L acts on this operator a it is equal it is the same thing as commuting the h with a a with h in this fashion okay. So once we have this written down in this form then the formal solution to this equation the formal solution is like this formally a of t equal to e to the power i L t a of 0 that is what this equation says here. What do I mean by a of 0 well you already familiar with this in quantum mechanics more than in classical mechanics because we are working in the Heisenberg picture here and as you know the solution to this this thing explicitly is simply that a is the Schrodinger picture operator and this fellow is equal to for instance e to the power i h t over h cross a of 0 e to the minus i h t over h cross in quantum mechanics that is the formal solution to the Heisenberg equation of motion. So it is really saying now you see why it is a super operator it says e to the i L t acting on the Schrodinger picture operator a of 0 is equal to operating with this operator on the left of it and that operator on the it is conjugate unitary it is a Hermitian conjugate on the right of it okay. In classical mechanics in classical mechanics the formal solution to this equation would again be this is q m so in classical mechanics it would correspond to saying that a of t which is really a q t p t where you supposed to solve the Hamilton equations of motion for the q's and p's and then substitute those as the argument in this a so by definition I call that a of t this quantity here is again equal to e to the i L t a of 0 or this is a function of q 0's and p 0's here and by this I mean the exponentiation of this Poisson bracket operation. So that is why it is so formidable looking in classical mechanics because you are supposed to take this operation the exponential of a Poisson bracket and act on that of the Poisson bracket operation and act on this function here. So in principle one can write the solution down formally very simply but in practice it is very complicated as you can see. Whatever it is the properties we are going to need in what follows are that L this thing here is a Hermitian operator and once we have established that it is a Hermitian operator then matter becomes matters become very simple because the exponential of i times a Hermitian operator is a unitary operator. So while we are familiar with the fact that Hamiltonian evolution in quantum mechanics with a Hermitian Hamiltonian is a unitary evolution in classical mechanics that is not so familiar in this language but it is still a unitary evolution all the same in a sense which will become clear. So let me first try to establish that this L is a unitary operator is a Hermitian operator both in classical and in quantum mechanics. Now as you know when you define when you have a Hilbert space and you want to define a Hermitian operator net you would like the following so we are going to establish the Hermiticity in both classical and quantum mechanics. Just to remind you in quantum mechanics in the usual cases when you have a linear vector space with elements psi, phi and so on which are ket vectors then you would say that an operator is Hermitian so let me this thing. If for any pair of elements of the space Hilbert space you have b psi on phi equal to psi d phi where this is an inner product defined in that space. Now we have to answer two questions right away what do I mean by a scalar product or inner product of in a function space of operators because remember in quantum mechanics we are talking in the Heisenberg picture so the elements are actually observables their operators. So first thing I have to do is to define a suitable inner product the proper inner product definition for operators and I have to do the same thing in classical physics as well okay. In classical physics matters are a little simpler because my dynamical observables are functions of the face of the independent phase space variables here. So they really have space of functions in quantum mechanics we have a space of operators. Now how do I define an inner product in this and once I do that then I would impose this condition to show that an operator like L is a Hermitian operator. So what is the definition of an inner product here so let us do classical mechanics the observables in this problem are functions of phase space so let us call these functions f of q p g of q p and so on and then the natural definition of an inner product f with g is a summation or an integration over all the independent dynamical variables d q d p f star of q p g of q and p that is the definition so it is a normal usual kind of function space and in this function for instance the same sort of thing applies when we talk about wave functions in quantum mechanics they are representatives of the state vectors in some basis like the position or momentum basis and then you impose inner product of the scalar product of 2 such state states by in the position basis by integrating over all positions f star g but here we are talking in classical mechanics in phase space so q and p are both independent observables and therefore the definition involves an integration over all q and p in this fashion. We need a corresponding definition in the space of operators in quantum mechanics and here is what it is so in quantum mechanics the inner product of 2 operators a b by definition remember these are operators which act on a Hilbert space of states of state vectors and you can find a basis set of vectors in that Hilbert space then the inner product is defined as the trace with respect to that basis or any basis of this quantity a dagger b this has all the required desirable properties of an inner product one can check that these inner products have satisfy whatever is needed for an inner product in norm in linear vector space theory okay. Now given this we have to establish that this operator L is a Hermitian operator namely it satisfies this condition here so let us do that in classical physics we therefore have to show that L times f g must be equal to and I put a question mark here we need to show this must be equal to f with energy that is what has to be shown and out here we have to show that L is a Hermitian operator which means that we need to show that L a b is equal to a with L with this definition of the inner product and if you do that then L is a Hermitian operator and e to the power ilt which governs time evolution both in classical and in quantum mechanics is a unitary evolution with conservation of probability and so on okay. Now let us start with this we would like to show this and let me do that by the following so f with L g equal to integral dp dq integral dp f star of q and p L with g but remember that L with g in classical mechanics L with L on g is equal to i times the Poisson bracket of h with g that was our definition of L times g so we need to put that in here and you are going to get an i outside the Poisson bracket of h with g which is equal to delta h over delta q delta g over delta p minus delta h over delta p delta g over delta q okay. This is over all the phase space n degrees of freedom or all the nq's and np's and let us assume the simplest case that all the q's and p's run over the entire real line namely minus infinity to infinity in all the variables although that is not important in this case so this thing here becomes equal to i times in the first integral integrate over p you use this fact to integrate over p so you have a dq but then inside you have let me not use a curly bracket because it is confusing f star of qp delta h over delta q times g so let me not write the arguments of these functions f star delta h over delta q times g I have integrated over p so that is gone and I have this over the boundary values so limits in p so let me notionally write this as p equal to minus infinity to infinity the boundary points in the range of p okay minus and now the whole thing is inside a bracket minus integral dp that is still sitting there and then the derivative of this thing with respect to p so we have this thing has been integrated so we differentiate this delta over delta p of delta h over delta q times g f f star sorry derivative of f star that is this thing delta h over delta q and then a g because that is been integrated over so much for the q integration of this term and in this term do the integral over q so you are left with minus i times integral dp and then a boundary term which is equal to f star delta h over delta p this time that still remains times g you should write it like this q equal to minus infinity to infinity minus and now comes the integral again over q delta over delta p since we integrated over q in this term so differentiate with respect to q sorry q differentiate f star delta h over delta p and then multiplied by g put a bracket outside okay now we expect our functions to be normalizable functions to be well behaved to be a member of some function space perhaps l2 or something like that so the functions these things would vanish at spatial infinity plus or minus infinity both in p and in q and all the variables so the surface terms drop out in the normal conditions and you are left with this term and this term and of course here you can see that if I differentiate this I get delta f star over delta p delta h over delta q and then a second term which is equal to f star times delta 2 h over delta p delta q but exactly the same term appears when I differentiate this term here with respect to q and those 2 cancel out because there is a minus sign out here so I might as well replace this term by delta p delta f star over delta p delta h over delta q so I am going to write it like this and the same argument applies here and this is delta f star this delta h over delta p times g in this fashion and you are left with this minus that and as you can see this is straightaway equal to i times an integral over dq integral over dp and then you have delta f star over p h over q with a minus sign and with a plus sign you have f star over q h over p so it is equal to the Poisson bracket of f star with h and then there is a g outside but this is equal to integral dq integral dp let us put a minus sign and put h with f star so this whole thing is equal to that which is equal to i times h with f the whole thing star on g and by definition this guy here is l on f so this is equal to integral dq integral dp lf star g which by definition is equal to lf with g so we started with f with lg and we ended up showing that it is exactly the same for all f and g with of lf with g so this immediately implies that l equal to l in other words l is self adjoint I have been rather loose in saying things are Hermitian one has to distinguish in these infinite dimensional spaces in general between Hermitian symmetric and self adjoint operators what we really shown is self adjointness and I have not explicitly said so but I have used this loose terminology that it is Hermitian because in quantum mechanics we very often use that in lieu of a more rigorous statement that it is self adjoint this is really shown itself adjoint but for that you need to show what the domain of l dagger is and as compared to that of l and show that they both have the same domain I must slurring over some of those nice cities but l is a Hermitian operator or a self adjoint operator in the sense we want it to be what about quantum mechanics this looks something totally different establishing this but not so because all we have to do is to use this expression here so if you took the left hand side for example in quantum mechanics l a with b by definition equal to trace of l a dagger with b in this fashion so let me use this bracket to put everything inside for the trace the argument of the trace this is equal to the trace of now l with a we already had a formula for this it was 1 over h cross the Poisson bracket of h with a so that 1 over h cross comes out and you had a Poisson bracket of h with a dagger let's put a round brackets so that's the quantum mechanical definition of the Louisville operator but this is equal to 1 over h cross trace of the dagger of h a so this is equal to the trace of h a minus a h dagger b the whole thing inside this is equal to 1 over h cross trace of well h a dagger is a dagger h h is Hermitian and I multiplied by b that's here minus the trace of so 1 over h cross is outside a with h Hermitian conjugate is h dagger a dagger but h dagger is the same as a so a dagger so you have this and you have this which is equal to 1 over h cross trace of a dagger h b minus trace of now I use the cyclic property of the trace so I argue that this is a block and can be moved to the right and therefore you have b h a dagger then mess up something oh sorry write this more simply as a dagger I use the cyclic property to move this to the right and therefore this is a dagger b by the cyclic property of the trace but this is the trace of a dagger with the commutator of h with b so we have this whole thing therefore l a with b is equal to 1 over h cross the trace of a dagger commutator of h with b that by definition is equal to the trace of a dagger l b that was the definition 1 over h cross h with b is the definition of the Louisville operator acting on b so it is this which again by definition is equal to a with lb we started with l a, b and we have ended up with showing that it is equal to a with lb so again it follows that l is l dagger this establishes the unitarity or the it establishes the hermiticity of the Louisville operator both in classical and in quantum physics quantum mechanics what does that gain us that tells us immediately it says immediately that e to the power i l t which if you remember is in fact the time development operator u which takes you from the state at t equal to 0 to the state at time t under this evolution governed by the Hamiltonian h this thing here is such that since this is hermitian this is equal to implies that u dagger u is equal to the identity which is also equal to u, u dagger okay. I have slurred over details here u is unitary this implies that this thing here is unitary evolution now it is got consequences the fact that this is unitary is essentially in quantum mechanics a statement of the fact if you like that the Schrodinger and Heisenberg pictures are unitary equivalent to each other you go from one to the other by unitary transformation generated precisely by this governed by this u the transformation of time development operator access the transformation operator okay. Now can we say this in a slightly more precise manner well for this we need to have the notion of a density matrix we need to have a notion as to what is the expectation value of a quantum mechanical or classical function of the dynamical variables so let us get to that so expectation values you see in classical mechanics when we have deterministic Hamiltonian mechanics we are not used to writing down things in terms of distribution functions in phase space but you could equally well regard classical evolution as evolution according to a phase space distribution function where the distribution function satisfies an equation called the Liouville equation which is essentially the way the dynamical variables change in other words you have a phase space distribution row of Q P and T which is if you like a delta function at the values given by of Q's and P's given by the solutions of Heisenberg of Hamilton's equations of motion what I am trying to say is that whatever you say in quantum mechanics you could say equivalently in classical mechanics as well and you could define the expectation value of a dynamical variable a of Q P you could write a in classical mechanics as equal to as equal to the integral d Q d P of some distribution function row Q P let me call it zero here of a of Q T P T this would be the analog of the Schrodinger picture in classical mechanics once I have tell you what this quantity is which tells you how this variable changes the expectation value changes as a function of time is given by averaging over all the phase space with this weight function this distribution function which is fixed once and for all it is like the Schrodinger picture in quantum mechanics where the state vector does evolves and I am sorry the other way about the Heisenberg picture where the state vector does not evolve and the dynamical variables evolve but you could also equally well put the blame of time evolution on the distribution function and leave these alone so you could also write this as d Q d P row of Q P T a of Q zero P zero but you need an equation of motion for this row to do this we have an equation of motion for this it says d a over d t is the Poisson bracket of a with H but what about this thing here well that is called the Liouville equation in classical mechanics and it says delta row over delta t is equal to minus the Poisson bracket of row with H this is the same as plus H with row this thing here it is a probability distribution it is not a physical observable so it does not satisfy the same equation of motion as physical observables to there is an important crucial minus sign here which makes this equation different from that for any normal dynamical observable like a or b or whatever similarly in quantum mechanics you would replace these formulas by writing a trace formula because this is like an inner product as you can see so the whole thing can also be written as equal to row inner product of row with a so I should really be precise row of zero with a of t or row of t with a of zero it does not matter they are equivalent to each other whether I take an active viewpoint and say the dynamical observables change for a given distribution function or the distribution function changes by this rule Liouville's equation keeping the dynamical observables unchanged under time they give exactly the same result okay. Now what is the corresponding statement in quantum mechanics as you know when you have pure states when you have states described by state vectors this thing here this row would be given by very specific form which I will come to in a minute and you have here on this in this case you have the Heisenberg picture and this case you have the Schrodinger picture does not matter how you evaluate this average here now the quantum mechanical equivalent of it is again a trace precisely the same thing so in all cases QM and CM the expectation value of a can be written as trace row a where this quantity is called a density matrix or density operator and it is Hermitian as we will see in a minute I will assert this there show that this is Hermitian this operator here so this is the definition of the inner product assuming that this is normalized to unity otherwise you are going to put this trace row the denominator we will see in a second that this is the corresponds to the class usual formula in quantum mechanics for what you write by mean by an expectation value of an operator in a given state here but this is the classical equation and the corresponding Liouville equation is this fellow here now quantum mechanics change a slight in quantum mechanics you are used to writing a thing like a of t is equal to in the conventional simplest form of quantum mechanics where the system is described by state vector in a Hilbert space at any instant of time which you can normalize this thing here is psi of t a psi of t divided by this quantity psi of t psi of t I will assume this to be equal to 1 this denominator the state is normalized you do not have that denominator factor and just the matrix element of this a between the state vector on either side okay this is the Schrodinger picture but in the Heisenberg picture you really write this as equal to psi of 0 a of t psi of 0 and we know what the psi of t is if you solve the Schrodinger equation for the psi of t this is equal to e to the power minus i h t over h cross psi of 0 so if you put that in here this on this side it becomes psi of 0 bra e to the power plus i h t over h cross so in a in between and this is what you are used to calling as a of t in the Heisenberg picture so that is this statement here okay so it does not matter which picture you use but the point is you need to use a density matrix in a more general context where you have temperature fluctuations and therefore the system need not be described by a state vector what does one do in this case the answer is obvious which is known to you perhaps you use a density matrix which is not writable necessarily in the form of state vectors but has a certain there is a Hermitian matrix which describes the analog of this quantity here and that is the following this formula is still valid but in quantum mechanics so this is C m and in quantum mechanics you have an equation says delta rho over delta t equal to once again a thing like this is equal to h with rho with an i h cross and this is called the von Neumann equation for the density matrix okay in the case in which you have state vectors describing the state of the system then density matrix rho in that case is just psi with psi and this is called a pure state where this is some state vector in the Hilbert space of the system not necessarily an energy eigenstate or anything like that some state vector which could be expanded in a basis of energy eigenstates in general but more generally when you have statistical fluctuations like thermal fluctuations what you really must do is to use a matrix which is a superposition of various possible states with different prescribed probabilities and that is done as follows we start with the following what you do for a single pure state is to write psi as equal to summation over n C n phi n where these are the states of a basis in the Hilbert space for instance and then it is very clear what this expectation value is in terms of rho if you were to write this rho in that fashion then it is clear that let me let me see the simplest way of writing this yes I would like to write it as trace whatever it is so what happens to psi a psi this is equal to summation over m summation over n C n C m star let us put this let us expand this in terms of m so it is C m star is equal to summation over m summation over n C m star C n so I have these two and then I have a sandwiched in between phi m a phi n that is the expectation value when you normalize the whole thing this quantity is what you call a m n with an m on the left and an n on the right and this quantity if I call this so I can write this as summation over m summation over n a m n rho n m where rho n m is equal to C n C m star which is equal to a sum over n and then I sum over m as well so this is equal to trace a that is precisely the formula that I wrote down first saying that you can express the expectation value as the trace of a times the observable times the density operator here where this density operator has these matrix elements in the base is concerned now when you have a mixed state a so called mixed state the whole thing goes through as it stands except that if you have a superposition of mixed states labeled by some quantity J some number J this is equal to C n J here when your density matrix in general would be of the form rho would be of the form summation over this J and if the state J pure state J appears with the probability P J which is specified may be from statistical mechanics may be in the canonical ensemble then this is P J psi J psi J in this fashion for psi J for each of the psi J's you have to make an expansion of this kind and now it is easy to see that you have exactly the same kind of thing going through except that rho n m. So once again expectation a is equal to trace rho a that is the same as trace a rho by cyclic invariance of the trace where rho n m is not equal to C n alone at C n for this pure state J C m J star time summation over J with the probability factor P I P J so instead of just one term like this one product you have a whole superposition of such products and this quantity is called the density operator or the density matrix here. So this is a slightly more general formalism than the usual one that you have in quantum mechanics because we are going to allow for the fact that these quantities may be specified save the finite temperature in the canonical ensemble by maybe e to the power minus beta times some appropriate energy that kind. So to allow for that one needs the more general foundation formulation and this is the general formula here. Now the equivalence between the Schrodinger and Heisenberg pictures is exactly this it is like saying that either I compute a of 0 in a density matrix which goes like e to the minus i l t rho of 0 this is rho at times t. So either I do this with respect to this density operator or by a unitary transformation you can also write this as equal to expectation of e to the i l t a of 0 with respect to rho of 0. So this is the Heisenberg picture and that is the Schrodinger picture and the two are equivalent by unitary transformation and this kind of puts both classical and quantum mechanics at least the formalism of time evolution formally on the same footing and we will see next what happens when you introduce the liberal operator in the presence of a perturbation that will be the content of linear the responsibility of linear response theory which will take up next time.