Density Matrix Theory (Part 1): Building an Intuition

hey everybody how you do I hope you're having a great day today I will be introducing you to density matrix theory and since this is a very difficult concept to gain a firm grip on I'm gonna try to make it intuitive for you to understand and explain in pretty relatively simple terms why it's useful to use things such as the net city matrix to describe the universe or the system plus its path so first of all when we perform an experiment we either consciously or unconsciously divide the universe up into parts that we know and parts that we don't know or maybe instead of parts we know parts we have information about and that we're interested in and that part is called the system and the system is connected to the bath and it interacts with the bath now we don't know everything that goes on in the bath the bath is a bunch of random variables that fluctuate in a way that we don't know about so an example would be a chemist in his lab performing a chemical reaction and inside of that flask is his system and he presumably knows everything that goes on inside of that flask he might say you know he's adding some hydrogen and some oxygen and he's light and well he's putting heat into the system and from that he's making water so in that way he says he knows everything that happens inside his system inside the flask but the flask might be in an oil bath or something like that and well I already said bath it might be an oil bath so everything that's outside the flask is called the bath so a system can be described by a single vector just like we described a wave function with a vector so this is important we should write this down system can be described by a vector if the system can be described this way then it's in what's called a pure state if it cannot be described by a single vector or a linear combination of actors if not then it's in a mixed state so this peer state in this mixed state is very important terms density matrix theory and initially they don't make all that much sense but let's take an example of physicist laser physicist and he wants to know everything about his laser light such as the polarization well polarization can be divided into two parts the horizontally polarized part which we can represent as a single vector say 1 0 and the vertically polarized light which we can represent as the vector 0 1 so for instance if you have horizontally polarized light which is like this well it's made up of two components this part here which is H or if you want to make it longer it's a some constant times H and this part here which we'll call beta V some constant times V and in that way you can describe any vector let's call it D for diagonally polarized so we can represent D as alpha H plus beta V now D u it's kind of weird because if I ask you if it's a pure state or mixed state you might want to say what's a mixed state it's a mixture of H and V but look at the definition D can be described by one vector so if we write this out we got alpha times 1 0 plus beta times 0 1 which we can just write as alpha beta so D is a pure state and the physicists can describe everything about the polarization of his laser light by the vector D and these alpha and beta I mean they might not be numb they might be complex numbers so in that way he can describe circularly polarized light so let's imagine that physicist and initially he knows that his light is in a pure state D described by this vector well that light interacts with the bath in this case the bath might be a R or sunlight or something so initially at time T equals zero the physicist knows physicist describes polarization by D by some vector D but after a time T not zero then he doesn't know what state his systems in and he also cannot describe his system well his laser light polarization he can't describe that by a single vector after some time he has to describe it with something else let's imagine the case of on polarize light but imagine is made up of purely vertical polarized light and purely horizontal polarized light now if you take one photon at random you know that it's gonna be a 50% chance that you have a horizontally polarized photon and 50% of the time is vertically polarized so now you can't describe that with one vector you have to have a matrix to describe it and that matrix has to give information such as well the population of each state for instance how many photons are vertically polarized how many photons are horizontally polarized and it has to give information about the interaction between the system and the Baths and that information about the populations is given on the diagonal so this diagonal here gives you information about populations but the populations of states the information on the diagonals gives you information about the interactions interaction with the Baths will say so now let's imagine or horizontally polarized light but let's imagine describing net in this matrix terms and by the way you can probably guess the matrix for describing here's the density matrix it's used interchangeably with the term but the distinction is not that big of a deal so remember we had our matrix H and to make a matrix out of just this one vector we can multiply the column vector of H by well the transpose of H or just the row vector of H and if you want to use direct notation since really we're thinking about quantum mechanics here you would write that so we write 1 0 times 1 0 or in other words we would write this so this density matrix here describes a pure state because we can just we have all the information that this matrix contains in this vector there's no really reason to have a matrix here because all the information is in a single vector and similarly if you do the same case with V you get 0 0 0 1 and this is also pure state for the same reason but let's imagine back to that case where we had unpolarized light but it's just made of pure or horizontal and cured vertical components well then we know that the there half the time you're gonna have horizontal polarization half the time you're gonna have vertical but polarizations don't really interact with each other so that's why these off diagonals are 0 and note that this used for on polarized note that this on polarized light density matrix I shouldn't rope the vector thing here that was dumb ok there's a matrix U or an operator and notice that you can't write this matrix U as a single vector you can try it it ain't gonna work so therefore u describes a mixed state and off diagonals are zero because for all we know the horizontally polarized light is not interacting with the vertically polarized light there's just a 50% chance of picking each polarization in the same way that there's a 50% chance when you flip a coin you're gonna get heads 50% chance of tails the heads state and the tails State don't really interact with each other it's just a statistics thing and now I'll let you consider the state of diagonal polarization and what the density matrix would look like that so you would write diagonal polarization X I'm calling it as half half and if you want a normal and say you have to put 1 over root 2 there so I'll leave you to consider what that density matrix looks like it's probably not initially what you expect but I'm sure it will give you some insight anyways thanks for watching if you liked the video give me a thumbs up if you didn't give me a thumbs down but besides that have a great day

Channel: Heisenberg's Dog

Views: 9,326

Rating: 4.9272728 out of 5

Keywords: density matrix, density operator, density matrix theory, quantum mechanics, pure state, mixed state, spectroscopy

Id: hJrlLEbkbrM

Length: 13min 21sec (801 seconds)

Published: Sun May 13 2018