in the current lecture we are going to discuss a very important concept of quantum statistics and this is regarding density Matrix actually this density Matrix is just a classical analog of the density distribution function you have studied in the earlier lecture the density distribution function denoted by the symbol rho which in general is a function of the position coordinate Q the conjugate momentum p and probability time t two in case of a stationary system this time is not present here so this is a actually called density distribution function or probability distribution function this density Matrix is actually the classical analog of this density distribution function okay as the name states that this will be expressed in terms of a matrix and you know actually in quantum mechanics all the quantum mechanical operator are expressed as a matrix we can express a quantum mechanical operator in Matrix form and so that actually this density Matrix is actually a quantum mechanical operator and that's why this operator is also called a statistical operator or it is also popularly known as density operator okay and as I have told you that this is a very important operator used in solving the quantum mechanical problem so actually this density Matrix you can say is a powerful and versatile tool for describing and analyzing the statistical behavior of quantum mechanical system okay in fact this formalism of density Matrix allows us to calculate the various properties of quantum system and among those various properties I have mentioned here some of the important properties that can be dealt that can be described that can be analyzed or that can be evaluated by the use of this very important operator which is called density Matrix let us see ah what are the important applications where this density Matrix can be used you can use it to find the expectation values of observables you can also use it to find the in symbol average Evolution under time dependent hamiltonians calculation of entropy and entanglement measures okay these are some of the important applications where this density Matrix Finds Its application okay now actually ah this density Matrix you know is a quantum mechanical operator and most of the quantum mechanical operator you know are hermitian so actually this density Matrix is also a hermitian operator and actually this hermitian operator is positive and semi-definite Matrix okay and you know that in quantum mechanics we always Define any operator in terms of the associated wave function or the state vector or the eigenvector so actually this density Matrix is also defined in terms of the state vector or you may call it eigenvector or wave function PSI okay now as you know that quantum mechanical system may be in two types of states the first one is called Pure State and another is called mixed state while you are talking about the statistical mechanics in general we will talk about the mixed state actually this pure state in general is a theoretical concept normally our system or quantum mechanical system are found in general in mixed state pure state is just an idealization but theoretically we will consider this pure state 2. so we will Define this density Matrix considering our quantum mechanical system in pure State and also in mixed state ok so let us see how this density Matrix will be defined when the state of our quantum mechanical system will be a pure state when you say that if a quantum mechanical system is in a pure Quantum State what does it mean actually a pure Quantum state of a system is represented by a single eigen vector or I or a state vector let us say that single Vector is PSI Alpha actually Alpha is just an indication of the state you can say that our system is but is particularly in a state denoted by this kit Alpha this is a single state so if the quantum's state of our system is represented by this single state vector k alpha or PSI Alpha then you can say that various state of the system is a pure Quantum State okay and if a system is in a pure Quantum State then how this density Matrix is defined the definition is very simple actually in this condition the density Matrix is simply the outer product of the state Vector which itself it means if this pure state has the state Vector PSI alpha or k Alpha then you have to find to take the outer product of this gate Alpha with itself and that will give the density Matrix of that pure state so if we denote the density matrix by the symbol rho then rho is simply equal to Outer product this is just order product actually this sign here has a touched one another but but Mark this symbol like this that these two angle brackets are actually not touching one another okay so you can see or I am just erasing that symbol you can see okay now see here this is like this okay and this is you know this is the symbol of order product so when our system is in a particular State denoted by the state vector k Alpha then the density Matrix is equal to the outer product of K Alpha with itself and so this equation one defines the density Matrix when our system is in a pure Quantum State ok now we will see the case of mixed race State one thing here I have mentioned that actually when the system is in a pure Quantum State then density Matrix you can see is just a an outer product of the state Vector with itself so in this case you can say that this density Matrix is a rank 1 projection operator actually when we take the order product of two State vectors we get a projection operator so in this case density Matrix is simply interpreted as rank 1 projection operator okay now we consider that our quantum mechanical system is in a mixed state it means there are so many systems you can also say and this and the state is not pure but it is just a combination of different Quantum States in statistical mechanics there is a very good example of this mixed state and this is statistical in symbol okay so I have mentioned here that a mixed state is described by a statistical Ensemble of different pure States okay you can consider that there is a large number of pure Estates in The Ensemble and so that is called a mixed state okay in fact when there will be a combination of different pure States and our aim is to define or find the density Matrix how we can find actually you have to take the order product of the different pure States different pure States let us consider that our different pure states are Kate Kate I k j j okay and so on okay in this condition if your aim is to find the density Matrix what you will do we will actually simply find the outer product of the different pure states with itself and multiply them so and and some of them not multiply so you can write it like this this is outer product of I sorry this has been just written wrongly so see here order product of kit I with I Palace outer product of J with j plus one but one thing here is also remarkable that what is the probability of a particular state in fact our system may be in different Quantum state but the probability of finding of the system in different Quantum States will be different so actually we have to take in this condition the weighted sum not the simple sum ok so let us consider that the probability of finding of the system in this particular a Quantum State denoted by the gate Vector I is let us say p of I then you have to multiply this outer product by p n so finally you can say that in in the case of mixed state the density Matrix is actually a weighted sum not the simple sum of the outer products of the state vectors of the different pure States okay and so let us consider a statistical Ensemble with probability P of alpha for which each pure state is Alpha or you can say get Alpha in this condition when we will sum up the weighted sum of the outer product of different pure States what will be our result you can see that will be sum of P Alpha Times order product of PSI Alpha with PSI alpha or you can say that this is sum of the order product of P Alpha times outer product of a kit Alpha with Ketel okay so rho is in fact in this condition defined as sum of P Alpha Times order product of K Alpha with ket Alpha okay so in this way in case of mixed state we can find that density Matrix okay now actually when you talk about the state vectors PSI of alpha which is for a pure State you know that actually a state Vector PSI Alpha which represents a pure State denoted by the symbol kit Alpha that can be expressed as a linear combination of the orthonormal eigen functions Pi n different file so you can write this PSI Alpha is equal to sum over n c n Alpha times Phi n okay and if you will take the complex conjugate of this PSI Alpha you can write PSI Alpha star equal to summation over m c m alpha star Phi investor actually here I have changed the index in to M and this is just a state change okay and actually this CN or cm are actually the coefficient of expansion of this PSI of alpha okay so as this PSI of alpha which represents the eigen vector or the state Vector of a pure state are expressed in terms of the linear combination of the orthonormal eigenvectors Phi n so in this condition from by the use of this equation number two if your aim is to find rho what will be the result in this condition now we write the row as rho MN okay this is actually M by n Matrix so the components are are written here in by the symbol rho MN and this rho MN will be now what at the place of PSI n PSI M of alpha PSI Alpha star you can write this P Alpha times CM Alpha star and at the place of PSI Alpha we will write this c n Alpha okay times it and so I and this is the a scalar product of this Pi M star and Phi n okay but you know that this Phi n or Phi m are orthonormal these are Ortho sorry here the sign will be like this like this now since Phi n is an orthonormal eigenvector so this outer product will be simply equal to 1 okay this will be equal to 1 and so I have omitted it and you can write that this rho MN is equal to sum of P Alpha times c m alpha star c n Alpha and this is actually the basic definition of the density Matrix in case of the mixed state of our Quantum system okay now we will see some of the important properties of this set density Matrix so the first property as I have told you earlier that this is just a hermitian operator so this density Matrix is an operator and this acts on the Hilbert space of the system okay now as you can see that this Matrix has been expressed by a m times n Matrix so its components can be written like this this is Row 1 Row 1 1 Row 1 2 so on row N and similarly row two one Row 2 2 Row 2 N and if in this way you can write rho m one row M two so on row MN okay in most of the cases this m is equal to n okay actually the diagonal elements of this Matrix these are the diagonal elements the diagonal elements of this density Matrix provides information about the probability of the system being in a particular State okay what is the probability of the system of finding in a particular state that is determined in terms of the diagonal elements while the off diagonal elements of this Matrix contain the information about the coherence and Quantum correlation between different states okay depressed now as I have told you that it is a hermitian operator and as you know that Hermes any hermitian operator is self-adjoined so you can say that this rho MN is equal to rho M and Digger this is its hermitian conjugate or in other words you can write rho MN equal to rho MN star okay as I have mentioned earlier that the it is positive and semi definite when you say that density Matrix is positive and same indefinite what does it mean it means that all of its eigen values are non-negative all of its eigen values are non-negative and when you will take the trace of this density Matrix what do you mean by Trace of the Matrix trace of Matrix means some of these diagonal elements so when you will take the sum of the diagonal elements you will get the trace of this density Matrix and and actually the trace of a density Matrix that is some of the diagonal elements of the density Matrix is always equal to 1. actually this result implies the normalization of probability okay normalational probability so in brief I have described the meaning and the important properties of this density Matrix but now in the further lecture we will see first of all one of the important application that is that time dependent time evolution in this application I have mentioned it here you can see we will discuss particularly the this application Evolution under time dependent hamiltonians actually when we will consider this application you will get a very important result which is also known as quantum mechanical version of Leo Village theorem so in the next lecture we will see the time evolution of hamiltonian or time dependence of hamiltonian or in other words you can say in the next lecture we are going to deal with the quantum mechanical version of Leo Billy theorem already we have studied it's a classical version in in the previous lecture okay thank you very much