very good so it's time to start so today I want to talk about general features of quantum mechanics quantum mechanics is something that takes some time to learn and we're going to be doing some of that learning this semester but I want to give you a perspective of where we're going what are the basic features how quantum mechanics looks what's surprising about it and introduce some ideas that will be relevant throughout this semester and some that will be relevant for later courses as well so it's an overview of quantum mechanics so quantum mechanics at this moment is almost a hundred years old officially and we will here you know this year we're celebrating 2016 we're celebrating a centenary of general relativity and when will the centenary of quantum mechanics be I'm pretty sure we'll be in 2025 because in 1925 shredding there and Heisenberg pretty much wrote down the equations of quantum mechanics but quantum mechanics really begins earlier their roots that led to quantum mechanics began in the late years of the 19th century with work of Planck and then at the beginning of the century with work of Einstein and others of as we will see today and in the next few lecture so they thought the puzzles the ideas that led to quantum mechanics begin before 1925 and in 1925 it suddenly happens so what is quantum mechanics quantum mechanics is really a framework to do physics as we will understand so quantum physics has replaced classical physics as the correct description of fundamental theory so classical physics may be a good approximation but we know that at some point is not quite right it's not only not perfectly accurate it's conceptually very different from the way things really work so quantum physics has replaced classical physics and quantum physics is the principles of quantum mechanics applied to different physical phenomena so you have for example quantum electrodynamics which is quantum mechanics apply to electromagnetism you have quantum chromodynamics which is quantum mechanics apply to the strong interaction you have quantum optics when you apply quantum mechanics to photons you have quantum gravity when you try to apply quantum mechanics to gravitation and I why laughs and that's what gives rise to string theory which is presumably a quantum theory of gravity and in fact the quantum theory of all interactions if it is correct because not only describes gravity but it describes all other forces so so quantum mechanics is the framework and we apply to many things so what are we going to cover today what are we going to review essentially five topics one the linearity of quantum mechanics to the necessity of complex numbers three the loss of determinism quar for the unusual features of superposition is and finally what is entanglement so that's what we aim to discuss today so we will begin with the number one linearity and that's of a very fundamental aspect of quantum mechanics something that we have to pay a lot of attention to so whenever you have a theory you have some dynamical variables these are the variables you want to find their values because they are connected with observation if you have dynamical variables you can compare the values of those variables or some values produced from those variables to the results of an experiment so you have equations of motion so linearity we we're talking linearity you have some equations of motion e o m and you have dynamical variables if you have a theory you have some equations that you have so for those dynamical variables and the most famous example of a theory that this linear is Maxwell's theory of electromagnetism Maxwell's theory of electromagnetism is a linear theory what does that mean well first practically what it means is that if you have a solution for example a plane wave propagating in this direction and you have another solution a plane wave propagating towards you then you can form a third solution which is two plane waves propagating simultaneously and you don't have to change anything you can just put them together and you get a new solution the two waves propagate without touching each other without affecting each other and together they form a new solution this is extraordinary useful in practice because the air around us is filled with electromagnetic waves all your cell phones send electromagnetic waves up the sky to satellites and radio stations and transmitting stations and they millions of phone calls go simultaneously without affecting each other a transatlantic cable contain that millions of phone calls at the same time as much and as much data in video and internet it's all superposition all these millions of conversations go simultaneously through the cable without interfering with each other mathematically we have the following situation in Maxwell's theory you have an electric field a magnetic field a charge density and a current density that's charge per unit area per unit time that's the current density and this set of data correspond to a solution if they satisfy Maxwell's equations which is a set of equations for the electromagnetic field charge densities and current density so suppose this is a solution that you verify that it's so Maxwell's equation then linearity implies that the following you multiply this by alpha alpha e alpha B alpha Rho and alpha J and think of this as the new electric field the new magnetic field the new charge density and the new current is also a solution if this is a solution linearity implies that you can multiply those values by a number a constant number a alpha being a real number and this is still a solution it also implies more linearity means another thing as well it means that if you have two solutions a 1 B 1 Rho 1 J 1 and E 2 B 2 Rho 2 J 2 if these are two solutions then linearity implies that the sum P 1 plus e 2 B 1 plus B 2 Row 1 plus Row 2 and J 1 plus J 2 is also a solution solution so that's the meaning the technical meaning of linearity we have two solutions we can add them we have a single solution you can scale it by a number now I have not shown you the equations and what makes them linear but I can explain this a little more as to what does it mean to have a linear equation precisely what do we mean by a linear equation so a linear equation and we write it schematically we try to avoid details we try to get across the concept a linear equation we write this L u equal zero where u is your unknown and L is what is called the linear operator something that acts on you and that thing equation is of the form L on u equals 0 now you might say okay that already looks to me a little strange because you have just one unknown and here we have several unknowns so this is not very general and you could have several equations well that won't change much we can have several linear operators if you have several equations like L 1 on something L 2 on something all this one's equal to zero as you have several equations or you can have several use or several unknowns and you could say something like you have L on U V W equals 0 or you have several unknowns but it's easier to just think of this first once you understand this you can think about the case we have many equations so what is a linear equation is something in which this L the unknown can be anything but L must have important properties as being a linear operator will mean that L on a times u where a said number should be equal to a L U and L on u1 plus u2 on two unknowns is equal to L u 1 plus L u 2 this is what we mean by the operator being linear so if an operator is linear you also have L on alpha u 1 plus beta u 2 you apply first the second property L on the first plus L on the second so this is L of alpha u 1 plus L of beta u 2 and then using the first property this is alpha L of U 1 plus beta L of U 2 and then you realize that if u 1 and u 2 are solutions which means L u 1 equal L u 2 equals 0 if they solve the equation then alpha u 1 plus beta u 2 is a solution because if L u 1 is 0 and L u 2 0 L of alpha u 1 plus beta u 2 is 0 and it is a solution so up this is how we write a linear equation now an example probably would help if I have the differential equation D u D T plus 1 over tau u equals 0 I can write it as a equation of the form L u equals 0 by taking L on you to be defined to be be u VT plus 1 over tau u now it's pretty much I haven't done much here I've just said look let's define L active on you to be this and then certainly this equation is just L u equals 0 the question would be maybe if somebody would tell you how do you write L alone well L alone probably we should write it as B DT without anything here plus 1 over tau now that's a way you would write it to try to understand yourself what's going on and you say well think when L ax or the variable U the first term takes the derivative and the second term which is a number just multiplies it so you could write L as this thing and now it is straightforward to check that this is a linear operator L is linear and for that you have to check the two properties there so for example L on a you would be V DT of au plus 1 over tau au which is a times the u D tau plus one over tau u which is a lu and you can check i've asked you to check the other property l on u 1 plus u 2 is equal l u 1 plus L u2 please do it