 Okay. The topic for today is time evolution in quantum mechanics, which we haven't talked about so far. Let's suppose the state vector of a system will be speaking here of a pure state. So it's described by a state vector. And some initial time is psi of t zero. The state vector is evolved in time, so at a later time there's a different state vector that's called psi of t. Now, what's the relationship between the two? Now, we postulate that there is a linear relationship between the two, so that the later time is given by some operator acting inside the earlier time, called the operator u, linear operator. The operator u must be parameterized by the two sides, the initial and the final times. So we'll edit this by t comma t zero. Actually, I put the final time first, t, and the initial time second in the two parameters of u. It's convenient way of ordering the two times. So this is a postulate, actually. You can add this to the other postulates of quantum mechanics that we've talked about so far. In other words, it's being confirmed by a parallel experiment. Now, about this operator u, it has certain properties which are reasonable properties that we can write down. There's three that I want to mention. The first one is essentially an initial condition. It says that if the two times are equal, so let's say u of t zero comma t zero, then you must be equal to the identity operator because if you don't have any time evolution then nothing happens to the state. The second property that I want to mention is if u is unitary. And we require this because unitary operators preserve probabilities and this is what we expect in our time evolution. If you have a particle somewhere in space with probability one, initially, then you expect that to be true in a later time also. Well, that is at least the much you're creating and destroying particles which you can do in relativity theory. But in the non-relativistic theory that would be, you know, you don't do that so part of the number is concerned. Actually, even in the relativistic theory, the probability is concerned and so even there you have unitary time evolution. Alright, the third property that we require review is a kind of a composition property. It says that if we take u with final time t one and initial time t zero and we multiply it by the u with final time t two and initial time t one you see this corresponds to starting at t zero evolving for time t one then you sort of stop thinking about things and you take whatever state you have at that point as a new initial condition for a new evolution starting at t one and then going at t two the answer is not to be the same thing as going straight from t zero and t two coming t zero. It's a plausible property that this operator u should have. So those are our three principle requirements on the time evolution operator, unitary time evolution operator u. Now, let's take a look at what happens over a short time evolution in concept of infinitesimal time. Let's consider the operator u of let's say t plus epsilon comma t where t is now the initial time and it was just a short time later as the final time. Let's look what happens on the small time evolution. We can expand this in a particular series in powers of epsilon. The leading term is just given by setting epsilon equal to zero to t t which by the initial conditions is one, so initial term is one and for the correction term allow me to write it in the form of minus i capital omega excuse me, minus i epsilon has to be the initial time epsilon minus i epsilon capital omega t and then there's the higher order terms of order of x1 squared and so on which I don't care about right now. So this is just where I'm writing the first order term but of course by Taylor's theorem this operator omega is related to the derivative of u and in fact the negative t we can see is equal to i times the partial derivative of u is a function of let's say t prime and t with respect to t prime where the derivative is evaluated at t prime equals t. This is just a way of saying it's the first derivative which appears in the Taylor's service. Now you'll notice I factored off a factor of i for this first derivative in the definition of omega and this is just gone for convenience but the reason for doing it is that it makes the omega is an operator, it makes omega a commission operator. The reason omega is a commission is because the original operator u is unitary. If you write down u and u dagger you multiply them together you'll find that omega is equal to omega dagger so it's a commission operator. I won't go through the proof of that because it's the same proof as what we did earlier in the momentum operator I think in the last lecture. But in any case omega is equal to omega dagger and it's a commission operator. Now I've written omega depending on time and the reason it does is that in this derivative although we differentiate with respect to t prime what we've done is that t prime equals to t so the right hand side depends on t and nothing else. In other words omega depends on t because the right hand side depends on t as well. There's actually a special case where omega in fact turns out not to depend on t and I'll go into that in just a moment but in general it does depend on t. No. No. Alright. Because of this relationship here we say just as just language we say that omega is the generator of time evolution what it means is that it's a commission operator which is responsible for carrying out an infinitesimal time evolution. You want to move the skateboard for a small time you just need an act of this operator and one of course is trivial only omega here is permission. That'll take you inside at one time to a small increment in time later so the language is omega's regenerator. Now in classical mechanics the generator of time evolution is the classical Hamiltonian. This is all understood in the framework of classical mechanics involves Poisson brackets and things like that but I don't want to go into it because this is of course the quantum mechanics and not a classical mechanics. But in any case the classical Hamiltonian plays a similar role. It can be used to advance the observables of the system from one time to an infinitesimally small later time by forming the Poisson brackets and Hamiltonian of the observable. So for these reasons we suspect that the operator of omega which appears here should be closely related with what we would consider the quantum Hamiltonian. We haven't find what a quantum Hamiltonian is yet but if I call it h just to give it a name the idea is that h ought to be related to omega somehow or maybe it depends on t here. Since it does we expect h will depend on t also. So the suggestion is that the proper definition of the quantum Hamiltonian is the relationship between these operators. Now however it can't just simply be h equals omega because the dimensions aren't right. h has dimensions of energy and omega has dimensions of inverse time. We fixed this up by assuming that there's a proportionality factor in here which has to have dimensions of action and we call this proportionality factor h bar. I went through a very similar means of introducing h bar in the last lecture and we identified the quantum Hamiltonian, excuse me the quantum momentum p is being h bar k where k was the generator of translations. So the idea here is that the quantum operator of momentum is emerging as a generator of translations and the quantum Hamiltonian is emerging as a generator of time translations. There's a question about what these two h bars should be equal based on the logic I'm presenting here, why should they be equal? Well, one compelling reason just to suppose that they are is that it's clear that this is a space and time version of a relativistic transformation but one is forming translations both in space and time, relativity would unite these two together and there's no way that you could achieve relativistic covariance unless these two h bars are the same. So we'll take them to be the same. There is only one h bar. In any case, this thing gives us the definition of the quantum Hamiltonian as the generator of infinitesimal. Apart from the factor of h bar, it's the generator of infinitesimal time translations. All right. Now, this means that I can rewrite this expression for the infinitesimal time evolution operator as 1 minus i epsilon over h bar times the Hamiltonian of h just to rewrite the first term of this series. Before going on with quantum mechanics, let me say something about what it means when a Hamiltonian depends on time. Describe this in classical mechanics. In several cases where we have forces that are derived from a potential, a force can be written as, a particle can be written as minus the gradient of a potential which depends on position. Sometimes the potential also depends on time. We talk about time dependent potential. Two examples of this. Let's suppose we have a caster plane here like this. Let's suppose we put a DC voltage across it. Let me look at a charged particle electron in there. In this case, you have a static electric field here. I guess it's pointing like this. Let's call it E0. It's a static electric field. But it's a corresponding potential which is equal to... Let me do it this way. Let me go the other direction. Let's call the vertical direction z like this. When the potential is minus the magnitude of the electric field times z times the charge of the particle minus q like this. This is what you call a time independent potential. The potential only depends on where you are, the z-coordinate, but not on the time. This would give you the force according to this formula here. On the other hand, if we put an AC voltage on the capacitor, and don't make it too high a frequency, then this potential can get replaced by what I had before, but multiplied by a cosine of omega t factor. This means, of course, that the force on the particle depends not only on where the particle is, but also on the time, the phase of the wave. This is an example of a potential. It does depend on time. This kind of a time dependence wouldn't occur. It doesn't have to occur because you could have a DC situation. First, this is called an explicit time dependence. That's the usual terminology used for it in classical mechanics, but also in quantum mechanics. It means that the observables have a dependence on time, as we're saying, explicit dependence on time. All right, there's one example of it in classical mechanics. Here's another one, which I'll draw pictures, so I'm going to make my hands. You probably know that when they send space probes, they try to get gravity moves from the planets, when they launch these space probes that can let the solar system completely, they use Jupiter to get a gravity boost, because it's the biggest planet, that's the biggest gravity. Jupiter is really big and the space probe is really small, so to a very good approximation, the space probe is not effectively working with Jupiter. This means that Jupiter is just following some given trajectory around the solar system, and therefore it generates what is a time-dependent gravitational field. The force that the space probe feels, or any object feels in the solar system depends not only on where it is, but also on where Jupiter is. So I know it depends on time as well. So it's a time-dependent gravitational field. It's an example of this kind of thing. Now, there's a lesson in this, because you get an energy boost at the time of the plane. The lesson is that a time-dependent force field and also a time-dependent Hamiltonian is that energy is not conserved. It's the only conserved way that Hamiltonian is time-independent. You can see this in the example of gravity boost because the space probe gains energy from Jupiter and flies out of the solar system. That's what you get in a time-dependent force field, like I see in the electron field, where you gain or lose energy in a time-dependent field. Now, that's a lesson of time-dependent Hamiltonian. These are two examples of the classical mechanics of time-dependent Hamiltonians. We must expect that something similar to this happens also in quantum mechanics, and that's the reason why this H of t, or this time-dependence, that appears there. Nevertheless, the case in which the Hamiltonian is time-independent is, of course, very important in practice, because you optimize that easy field, and this is not there. And there are some simplifications that occur in that case, so I'll be talking about that later on. But right now, we're general, and assume that H has a time-independence. All right, all right. Now, all right. So this is the introduction of the Hamiltonian operator. Now, let's work out a differential equation for the unitary time evolution operator u. First, what we're going to be interested in is the derivative of u with respect to the final time in saying, hey, u is a function of t and t0, and consider its derivative as a function of the final time. In terms of the derivative, the final time is variable, and the initial time is fixed. Well, let's only look at a formula like this. By the definition of the limit of the derivative in calculus, we can write this as a limit as u evaluated in t plus epsilon comma t0 minus u of t comma t0 divided by epsilon. However, the first u that appears in the numerator by the composition law can be written as u of t plus epsilon comma t multiplied by u of t comma t0. That's this composition law in the number c up there. And so you now see there's a common factor of u of t comma t0 on the right-hand side of the numerator. And so this cannot be written as the limit of epsilon goes to zero, of u of t plus epsilon comma t minus the identity over epsilon. That whole thing multiplied times u of t comma t0. Now, this remaining derivative is precisely what is precisely this derivative. I can write this as partial respect to t prime, of u of t prime comma t evaluated in t prime equals c. That's just the meaning of that. There's the derivative there. Times u of t comma t0. However, this derivative must appear in the numerator somewhere. It's almost the same thing as omega. It's minus i omega, as you see. In fact, we can write it in terms of h bar. This becomes minus i over h bar times the Hamiltonian, minus the function of time times u of t comma t0. The result of this is that if I bring the i h bar over the other side, we get a differential equation of the inventory time evolution operator. With respect to the final time, considering the initial time fixed, which is given by the Hamiltonian multiplying onto the time evolution operator. This equation can be regarded as a generalized version of the Schrodinger equation, as it gives the time evolution. It's a differential equation describing the time evolution of the system. All right. So this is equations motion. Mostly related to this is another version of the Schrodinger equation, which is obtained by, and if we calculate the let's call it by i h bar, let's calculate the time derivative of the evolution of the state factor, psi of t. Well, mind you, the board psi of t is the u operator multiplied times psi of the initial time. So the time derivative, which adds only the final time, is going to act only on u and not the initial value of psi. So if I do that, I'll write this over here. What I'm going to get is i h bar partial of u with respect to t times psi of the initial time t0 to the right-hand side here. However, since i h bar of the unity is h times u, this becomes h times u times psi of 0, t0. However, u times psi of t0 is the same as psi of t. So putting this all together with that i h bar of the psi of t is the Hamiltonian, which generally depends on time multiplying over the psi of t. And this is an alternative version of the Schrodinger equation, which is probably more familiar to you because it's expressed in terms of these two things are really typically related to one another. They can be regarded as two different slightly different versions of the Schrodinger equation. The time-dependent Schrodinger equation is time-dependent. Now, one final remark to make about this is to consider the special case in which the Hamiltonian is a time-dependent. If this way, this way it isn't the h dt is equal to 0, then there are some simplifications because in that case if h doesn't depend on time, then this differential equation is easy to integrate. You get the answer in terms of an exponential. It implies that u of t comma t0 is equal to e to the minus i t minus i t t minus t0 times the Hamiltonian divided by h bar. This becomes the unique solution for the unitary time-evolution operator subject to the initial condition which is part a up there. What you see is that in this case this is a special case but an important special case that happens frequently. You see the time-evolution operator now depends only on the elapsed time. So then we just write it this way as u of the single parameter which is 2 minus 2 0. In fact usually in cases like this we take t0 equals to 0 when we reinterpret the differential t to mean not the clock time but rather the elapsed time and if you do that you're going to need an even simpler formula that says u of t is equal to e to the minus i t h over h bar. That's the formula in this case of the slight reinterpretation of the symbols. It's important to remember that these formulas here and here don't only apply to cases in which the Hamiltonian is time independent but the Hamiltonian has a time dependence then you've got a much less trivial problem in trying to solve this because that can solve a differential equation as best as you can. Next I'd like to tell you about the the two pictures, the Heisenberg illustrator, Schrodinger and Heisenberg picture. These are two different the equivalent descriptions of quantum mechanics the picture that we've been using so far well really to only talk about time dependence today so I say so far this morning but it means so far in the course is the Schrodinger picture in fact the Schrodinger picture is what we'll use for most of the course nevertheless the Heisenberg picture gives insight into certain things, it's important for certain things conceptually and it's also well adapted to the treatment of problems such as permanent oscillator very nicely solved in the Heisenberg picture. In any case let me now tell you what the difference is between the pictures. In the Schrodinger picture it gets involved in time as I just explained since I want to distinguish between two pictures let me start putting subscripts on things and I'll put an s on it if I mean the Schrodinger picture but let's have a convention that if I don't put any subscript on it at all I mean the Schrodinger picture by default so this is just what we were talking about in this slide. The cats in the Heisenberg picture already has psi h with an h subscript and this is the definition of it it was given by the Unitary Time Evolution operator U of t comma t0 dagger applied to the cats in the Schrodinger picture so this is the definition of the cats in the Heisenberg picture which is the definition but you see what the definition does is that since there's a dagger on this view what it does is it strips off the time evolution that's in the Schrodinger picture and takes it back to the initial time so as an aside this is also equal to the Schrodinger cat at time t0 and since this doesn't depend on time it's just taking the first time it means that in the Heisenberg picture the cats don't evolve so there is no Schrodinger equation for state factors in the Heisenberg picture now we can also transform operators into the Heisenberg picture suppose I've got an operator in Schrodinger picture which I'll call as and as I just explained it may have may, some frequently doesn't but sometimes it has an explicit time dependence which I'll indicate here as as of t so the corresponding operator in the Heisenberg picture by definition is equal to the operator U of t comma t0 dagger on the left d comma t0 with no dagger on the right it's a conjugation of the Schrodinger operator in this manner by means of the unitary time evolution operator, again this is a definition now the Heisenberg operator always has a time dependence because even if the Schrodinger operator was time independent so there was no function of time in the middle, these two U operators depend on time so the Heisenberg operators always have a time dependence in fact this is what's called the explicit time dependence and the time dependence that comes from the two U's is considered to be an implicit time dependence these terminologies perhaps for transparent classical mechanics I'll refer you to my notes on classical mechanics if you want to understand that but in any case there's two different reasons why Heisenberg operator might have a time dependence but the U's are always here so they always give you an implicit time dependence well, now the so this is just definitions in these two pictures but since the Keck-Stohl involved in the operators doing what we now need is an equation of evolution for the operators using the Heisenberg equation in motion so let's work that out just using this definition let's compute i h bar times the derivative of the Heisenberg operator with this definite T, what's that equal to well, by the chain rule there's three terms here there's an i h bar partial of the U-battery with respect to T times a shorter times U and then there's a second term which is i h bar U-battery times partial h, let's write it as D h shorter dT times U and then there's another term which is U-battery times a shorter times i h bar partial U of respect to T using the three terms chain rule for the three terms now by the Schrodinger equation the the derivative of U of this last expression is the same thing as h times U this is h in the Schrodinger picture so I put an s on it and if I take the permission conjugate of it I get a minus sign because i goes to minus i and it reverses all these factors U goes to U-battery so in fact this first term that we have here and that's minus U-battery times h h Schrodinger I don't know if it's Schrodinger that because it's permission alright and so combining all three of these terms together I get if we look at the let's look at the first let's do the third term first let's write it this way we get U-battery h Schrodinger h Schrodinger times U the third term but the first term we get minus U-battery h Schrodinger h Schrodinger and then for the middle term I'll just copy it i h bar U-battery d a Schrodinger d t or I'll add a partial derivative a partial derivative d t U I guess there's a question about U-battery in any case these are the three terms now in these first two terms let's make an assertion of U times U-battery which of course is just the identity but we do that because then we see the combination U-battery h Schrodinger U and U-battery h Schrodinger U which is the rule for converting to the Schrodinger to the Heisenberg picture so the first two terms become a Heisenberg times h Heisenberg minus h Heisenberg times a Heisenberg plus the third term of this copy and the result is you see there's a commutator here in the first term so I'll write this over here as we get an equation of evolution i h bar d a Heisenberg d t is equal to the commutator of a Heisenberg with a Hamiltonian Heisenberg picture plus this last term the last term is obtained by taking the operator and assuring the picture differentiating with respect to the explicit time dependence and then conjugating by U-battery U which is what you need to go for a Schrodinger picture to a Heisenberg picture so allow me just to write the last term as an i h bar in there as well as partial a with respect to t by this and I'll put the h parenthesis of the h subscript on it which is just this combination right here that it's something as the transform of the Heisenberg picture so the result is that this is the equation of motion for the operators in the Heisenberg picture and this replaces the Schrodinger equation because it's not between the evolution of the operators one of the advantages of the Heisenberg picture is if there's a closer relationship to classical mechanics than does the Schrodinger picture in fact in classical mechanics there's an evolution of an operator which you call the total time derivative of an operator along an orbit and this is one of the mechanics here just to mention classical mechanics so this is our classical observables the time derivative is given by first of all a Poisson bracket of the observable in the Hamiltonian and then the derivative with respect to the explicit time dependence and so you see this strong similarity parallel between in one of the classical equations of motion if you use the Heisenberg picture in quantum mechanics so this is one of the reasons to be interested in the Heisenberg picture as it gives you the basic connection with classical mechanics what's that's apparent it's not the only reason for interested in the Heisenberg picture another reason for the importance of the Heisenberg picture I haven't mentioned this yet is that in relativistic field theory it was necessary in order to obtain an event as covariance of the fields so this is something we may see at the end of the second semester it's a question you guys this DAET can you write down a subscript that's in particular the Schrodinger right here yeah this is DAET computed in the Schrodinger picture but then the derivative is then converted to Heisenberg picture that's what this formula means here so in other words just the DAET and the 8th semester just another way of writing this of course in many cases the operators have no explicit time dependence in which cases last term vanishes and then the Heisenberg equations of motion are just the commutator of the Hamiltonian alright now some further remarks about the Heisenberg picture the first important remark is that the two pictures are physically equivalent is that you can make the predictions of what experimenters will observe on the basis of theory using either the Schrodinger picture or the Heisenberg picture they're both completely equivalent in that respect the reason this is in all predictions experimental predictions ultimately come down to computing matrix elements of operators let's say in generally in two different states is that if you can compute these things then you can make all possible physical predictions let's look at this first one the Schrodinger picture in which the cats are the Schrodinger picture and so therefore they depend on time but the A's are in the the A is also the Schrodinger picture let's do this in the Schrodinger picture well let's compare it to the Heisenberg picture in which the inside the cats don't depend on time but the A's do let's do the Heisenberg picture first if we take the definition that the Heisenberg cats cats here were brought by taking permission conjugate of it and likewise for operators which is rather here what you get for this last line here Heisenberg line is that first of all in the right this is going to turn into the Heisenberg picture first if we take the definition that the Heisenberg cats cats here are A Schrodinger times U and then the Psi H draws going to turn into the Psi Psi S Schrodinger times U like this and so you seem to use them to do the A's cancel but what you're left with this becomes equal to the Heisenberg Schrodinger picture and so in particular exactly the same time dependence so all the probabilities that when measures in quantum mechanics are swear so make the settlements like this suppose it's the same in both directions no there's one final remark about the yeah maybe maybe a couple, one, two remarks one of them is, I give this one remark it's a special case in which in which the Hamiltonian is time independent the HBG is equal to zero because in that case the arbitrary time evolution operator U of t comma t zero as we've seen is E minus I t minus t zero times H divided by H bar and the main point I want to make is this operator U is now a function of H and so in a special case in which the Hamiltonian is time independent the operator U commutes with the Hamiltonian and as a result we take the Hamiltonian and this is the Schrodinger Hamiltonian we're talking about here and as a result we take the Hamiltonian in the Heisenberg picture which is U dagger times H Schrodinger times U H Schrodinger commutes with U I can bring U dagger past the H Schrodinger and what I can cancel out with U on the other side and I just get H Schrodinger so when the Hamiltonian is independent of time the Schrodinger Hamiltonian and Heisenberg Hamiltonians are actually the same that's the right simplification this by the way is related to the fact in classical mechanics that if the Hamiltonian is independent of time then energy is conserved I just mentioned in the case of the Jupiter situation where you have time dependent Hamiltonian and it is not conserved but in fact when the Hamiltonian has some explicit time dependence then the energy is conserved and it's equal to the Hamiltonian this is an analogous state of mechanics that when the Hamiltonian is independent of time the Schrodinger and Heisenberg operators are the same this is a way of saying there is no evolution of the Hamiltonian itself in fact you see let me take this equation here let's interpret A as being the Hamiltonian itself and let's suppose it has no explicit time dependence so this term goes away well what do we have then we have IH part D and the Heisenberg Hamiltonian with respect to time is equal to itself which is zero so in the case of the term you have been in Hamiltonian and the Heisenberg Hamiltonian is conserved in time here is another one more quick remark about the difference between these two pictures the historical remark the vast majority of the work that people do in quantum mechanics is certainly not in the Schrodinger picture this is certainly true in the nominal ballistic theory anyway but nevertheless the Heisenberg picture came first historically Heisenberg's paper on quantum mechanics which was the first time real quantum mechanics was ever presented was in 1925 but it involved the Heisenberg picture that is to say Heisenberg's paper concerning the time evolution of operators not of wave functions or states in fact he didn't even know that there was wave functions and he certainly knew that there were operators but he didn't really know exactly what they acted on nevertheless it was possible to solve quite a few problems using this formalism basically these equations of motion working with commentators the harmonic oscillator is pretty easy hydrogen atom is much harder than was done by Cowley and really a brilliant paper shortly after that and solved the Heisenberg picture it was only some months later in 1926 that the Schrodinger published his wave equation basically presented the Schrodinger picture it didn't take very long for people to realize that the two were physically equivalent and most physicists were happy to prop the Heisenberg in those days it was called matrix mechanics because the operators were seeing the matrices but in any case most physicists were happy to forget about the Heisenberg picture because they liked to solve wave equations and they were familiar with doing that but nevertheless I won't argue that the Heisenberg picture was not only came first but in some senses it's actually more fundamental anyway that's a little bit of a mystery no okay now so far I've been talking about Hamiltonians but I haven't said what the Hamiltonians are so there's a question about how do we know what the Hamiltonians are if we want to get it the way of guessing what the Hamiltonians are is to borrow them from classical mechanics this is sometimes called the process of quantization if you know what the classical description of a physical system is but not the quantum one you may use the classical description as a stepping stone or springboard for going over to the quantum description because I say it's called quantization however I want to emphasize that this has always involved some kind of a guess it's not a logically deductible the deductive process in fact there can't be any such process for a quantizing classical system and the reason is is that quantum mechanics has more physics in it than classical mechanics does in the case of problems in the time people like it or condensed matter physics things of that sort we know by now how to quantize the corresponding classical systems and we know where the errors are and what additional things have to be taken into account most importantly we know that spin phenomenon has no analog in classical mechanics makes its appearance in quantum mechanics there's new physics there in other words due to spin there's other cases such as the gravitational field where at the present time the classical description is very well known in Einstein's general relativity but as we get nobody somebody will think they know but it's very questionable what the correct quantum description is so the process of quantization is not necessarily trivial but in any case in the kinds of common problems that occur in animal holistic physics we simply borrow a Hamiltonian from classical mechanics so in classical mechanics let's start by saying some things about classical mechanics the forces derived on particles are derived from a potential that is minus the gradient of potential which they depended on time as I explained a few minutes ago then there is a classical Hamiltonian described in the system and it's the sum of the kinetic energies plus the potential energy and the kinetic energy is written in terms of the canonical momentum p and a problem like this the momentum p is the same thing as them x dot m v so the canonical and the kinetic momentum agree for problems of this sort but this p squared it really means the square of the vector the three vector squared v squared over two m is another way of writing one half of m v squared which is kinetic energy so that's the first term is the kinetic energy and the sum of the potential energy and so it's a guess that in problems of this sort we might just borrow this Hamiltonian reinterpret the symbols p and x instead of being classical observables as being quantum observables that act on our ket space and this would describe quantum mechanics similar situations where potential is known a whole lot of questions arise in the process of doing this one of them is is it correct does it give us the correct Hamiltonian before you ask that you could ask was it correct and even classically that the motion could be described by a potential the fact is that in many physical circumstances in practice people use potentials phenomenological potentials phenomenological means that the potential energy is just some it could be a fit to experimental data without peering too deeply into the deeper meaning of where the potential came from or what fundamental force laws would lie behind it giving an example one could consider the collision of two atoms as you know it's repulsive maybe it's repulsive in large distances it may be attractive to fewer distances and you can hope in that force to be represented in terms of the potential well books talk about it all the time as if it is representable by potential but is it really the answer is only in a certain approximation you find this out if you go to a more fundamental theory this involves more or not the higher approximation and so on and then you find that there actually are corrections where the force is not completely discriminated by means of the potential for another example the interaction between a neutron and a proton for example the deuterium nucleus which you know is a bound state of a proton and a neutron and to some degree you can describe by a potential but it's actually only about the proof description turns out to be rather strong I sticking my deck out a little bit I think it could be something like the 10% level it's a significant level but stem interactions that are not described with my means of the potential so the deuterons can only be described proofed by such a potential one case in which the potential seems to be fundamental is that of electrostatics in which you write the electrostatic potential down you know how to do this for the hydrogen atom it's just the one over the distance between the proton and the electron isn't that a fundamental potential energy well because it's based on the electromagnetic theory so it's not phenomenological the answer is well maybe you could say that but it doesn't elect a lot of effects it's important if you look at things more carefully it's really based on the electrostatic approximation the electric magnetism the electric magnetic fields that are produced by charges and I'm just speaking in class from the electric magnetic theory electric magnetic fields that are produced by charges are only approximately given in electric fields are only approximately given by Coulomb's law it's only when the charges are stationary that Coulomb's law is valid if they're moving then not only are there magnetic forces but the electric force is not given by Coulomb's law it's more complicated especially well charges are moving and atoms and molecules so anything that involves electrostatic potentials that you write down and you're going to do this all the time it's done all the time the time of the molecular condensed matter physics those are based on what I will call an electrostatic approximation and one of us always remember that it is only an approximation and there's other corrections that need to be added so anyway so this is this is a process of quantization here it involves gas and it also involves approximations as to the terminal physics and the use of the potential but anyway having done that we now reinterpret this as a quantum operator then we know what P and X mean as operators on wave functions X is a multiplication by X and P is a a differential operator so if you get the usual shorting your equation as a differential equation for a particle limiting a few dimensions which you all know this is a differential operator acting on the wave function inside the X and T and the right hand side of the time depending on the shorting of the equation looks like this previous lecture we worked out the action of momentum on the wave functions in the configuration representation of this is what it is what you do it in a Kepp language also it's going to be T squared over 2M once V applied to psi of T is equal to my h bar partial to psi of T psi of T in some sense in some cases people regard this as the shorting of the equation it's the one shorting of a real Kepp original figure but the race did not but the equation this version of the shorting of the equation in some ways Kepp version is actually better because it applies to many generalizations of this multi particle systems all kinds of other elements I'm assuming you know about separation of variables in time about energy and eigenfunctions this complete set of states and if you do this and by the way I'll mention that the separation of the shorting of the equation in time only works when Hamiltonian is time independent so I need to drop this C if I want to do that but this would lead to an eigenvalue problem which would be this back along psi n of x gives us an area of the eigenvalue E n of x E n times psi n of x this is the eigenfunction, eigenvalue problem for a time independent Hamiltonian which is the only case that it's really meaningful and this is for this comes out of separation of variables and is used for expanding the time dependent solution as a linear combination of the energy and eigenfunctions alright, now next I'd like to introduce or take some parts about the probability density of current in quantum mechanics by the way, just to go back one step as I emphasize going from the classical to the quantum Hamiltonian how do you know if the guess is right? you work out the physical predictions and you compare it to experiment that's the only criterion obviously that can be used now, if you do that you'll find stuff to be exactly right because other things play along alright, now about the probability density he explained I believe in the last lecture the square of the wave function this is some of you well know the probability density of the configuration space regarding the measurement of the position of particles across the ensemble today we're talking about time dependent wave functions so this means the probability density is also a function of time don't confuse this row here with the density operator this is a completely different object it's an ordinary function of the configuration space one thing to say is that the normalization of the density if we integrate it over all space is independent of time and in fact it's equal to 1 for all times and this just follows by the unitary of the time evolution operator the unitary concerns the norm and that's just what this integral is it's the norm of the state psi alright now if you've got a quantity whose integral over all space is constant in time such as this probability density it doesn't mean that the quantity in question sort of moves around continuously in time it's conceivable for example you could have two months of some quantity in different parts of space and then abruptly some of it disappears here and it reappears over there now you wouldn't expect this on the basis of relativity theory which gives you a limit on how fast signals can propagate in the same way conservation of something could happen that way usually in physics when some quantity that's integrated over all space is conserved it actually evolves continuously if something moves from one place to another in a continuous manner and that means that there's a current that's associated with it it's a function of position in time and when taken together they satisfy a probability equation in a continuous manner in which the quantity moves around and also the total conservation and so the question is what is the current, is there a current? what is it for the case of the Schrodinger equation? well the answer is for the case of the probability one could write this expression down quite generally this is really only a part of the moving in three dimensions but this is a regardless of the Hamiltonian issues or the potential of one of those magnetic forces this is the expression of the density however the probability current depends on the Hamiltonian in a more specific manner for the case of the Hamiltonian it's up here a kinetic plus potential the probability current looks like this well first let me define a velocity operator velocity operator is in the momentum operator divided by the mass which is what it's supposed to be for the Hamiltonian kinetic plus potential is p equals mv plus the psi h bar over m times the gradient well j is equal to the real part of psi star times the velocity operator times psi so the velocity operator acts on psi that is the real part of that if you were going to compute the expectation value of the velocity operator with respect to some state psi you would of course integrate over all space dq of x when you have psi star times the velocity operator times psi this is what the expectation value would be when you see the current as you see it's the real part of the integrand of this expectation value we don't do the integral so the current is not the same as the expectation value of the velocity operator and moreover it's real it has to be because the current has to be real so now it's just an exercise using the Schrodinger equation to show the continuity equation is satisfied with this definition of j ok so this much I think should be familiar to you for your first course in quantum mechanics let me now make a few comments about this then we can go there is a question of the physical interpretation of the density you may be recall that I said that although we frequently use language such as the way function of the electron in fact what the way function describes is the statistical properties of an ensemble more specifically than the single system in fact it doesn't tell us much about the single system so in particular this row here is giving us statistical properties of an ensemble for example in the case of hydrogen you know that the electron is going to have a probability density that just concentrates strictly symmetric dies off exponentially and concentrates around the proton the hydrogen atom now does this mean that the actual electron is smeared out over space in a hydrogen atom in that manner well that's a single system the answer to this is no because if you measure an electron measure a particle you never find part of a particle in one place in another part in another place and in particular if a particle is charged you never find a fractional charge it's always a whole charge you always find a particle somewhere in space so this row is properly interpreted as a density which is determined by going across the ensemble not by looking at a single system well let's look at this a little more carefully if we did make a measurement of a single system then and it says you're going to find the electron in a particular place what about the fact that that electron produces an electric field if I have a proton and an electron out of here that produces a dipole field which marks distances falls off as whatever our Q and if I had the charge smeared out in a spiritless symmetric manner with a proton in the center then the electric field with that charge distribution would go to zero in the first place if you just move several core radii away they're essentially outside the charge cloud since it's spiritless symmetric by Gauss's law the field is just given by the total charge inside the sphere which is zero because the electron and proton cancel so in other words there's a question now to fall off as whatever our Q where it doesn't fall off much faster than that essentially zero outside the charge cloud which is the answer also if it's a dipole you'd expect it to be a time dependent dipole because the electron is whizzing around okay just make one comment and I'll let you go and then as I say there's a subtlety in this because the electron is whizzing around pretty fast it's the time scale is 10 minus 16 seconds something like that when you measure the position of the electron you've got to use some measuring device that acts on a time scale that's a tall order because it's a very short time scale if you use that measuring device and it takes a longer time to make the measurement then you're not going to see where the electron is but you're going to see it's averaged out and even classically if you average it out what you're going to get is a spiritless symmetric charge distribution you'll get something that looks like an ensemble and not a specific system so part of the moral is I'll elaborate this a little more next time but part of the moral of this is is that time averages especially if this is true for energy I can state is that time averages are equivalent to ensemble averages and so in that sense you can speak of a single system that isn't described by density but it involves an adiabatic as an estimate of time scales and involves an adiabatic approximation so before you go I want to make sure that everybody got the email I sent yesterday because I'm really trying to straighten out this investment