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Advanced Quantum Mechanics | Lecture 1

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the Stanford University so
like this I feel I have to start with a review because I do understand 2 quarters since had lectures from Ramirez quantum mechanics the the other he and they can't spend the whole quarter going that way I want to spend the quarter is the applications of quantum mechanics not that technology but the physical problems and the basic physics problems that I want to go through this quarter of of those fully quantum mechanics were originally was originally invented discovered invented our body height Einstein ties emerge from Inger polio was both namely Adams electrons Amadou Adams 1st How do Adams 1st before I do electrons electrons part of Adams but will wear will do a little bit about electrons just almost nothing but just enough to be able to write down a basic equations of the hydrogen atom discuss the hydrogen and a bit and then move on to further investigations of the electron and want them we think the electron was actually studying art class of particles not just collect charms a wide class of particles which behave like electrons which include new particles quart of things like that and then the folk time so that the Adams electrons photons but 1 of the themes that will recover is symmetries symmetries of nature and how cemeteries Our realized inquiry Canucks how symmetries are represented in quantum mechanics and what they tell us about the the quantum mechanical system symmetries in quantum mechanics are extremely powerful tool which I'm going to tell you a symmetry is if you symmetry it took 8 at the moment but let's begin with very very light review are the last quarter of course 1st of all a whole story begins by asking how you represent the state of the system we go back areas that what is a system that I assume your system is and how represent states of the system it classical mechanics the states of the system are represent points in phase space in quantum mechanics they represented by state vectors the idea of there is not a vector like a pointer and spaced pointing in the direction space it's an abstract notion Ucon and that vectors Puca multiply them by constant and the mathematical structure of the state vector it is a mathematical vector In a vector space vector space is now three-dimensional except in special cases it could be any dimensional solely represents state by state vectors and the symbol for a state vector it is directs kept symbol kept KTT which is the latter half all of the turned black like Brad jet Will Brown jet every state can be easily represented by had vector or by but corresponding brought together we often put them together performer Brock cat or bracket on these broad sectors are one-to-one correspondence with the cat sectors and you should really think of them roughly as being the complex conjugate same census complex conjugates of complex numbers are complex numbers are ubiquitous in quantum mechanics and saw complex vectors so state vectors are represented by states states are represented by state vectors state elected as well a Castillo bit more but a 3rd say the whole thing of course we would then go through the the entire quarter and in addition there are observables things that you measure measure balls observables and classical mechanics are also durables them things you measure or we don't spend a lot of time talking about their mathematical structure we just assume you know what I'm talking about war position of a particles measurable momentum of a particle energy of a particle or ever In quantum mechanics we have to be very precise what we mean by an observable and observe rules are represented by a linear permission operators are label them with what what's is a Greek letters look of of those English Roman weapons my wife yes it's true
it's capital after a cake but
Latin letters representing observables and they are all are for me she operators that act on the vectors permission for our purposes translates or is he quantum mechanical equivalent of real real as in real or complex or imaginary our they are the real operators which are equal to their own permission conjugate but every such operator in the space states is called an observer and presumably there is really a way to observe it we II get values and eigenvectors of these permission operators play a special role Saunders just remind you what it and I give value and eigenvectors if you'll have a Harnisch and operate a Yukon search for sectors let's label the vector what's label it little aII or a big better yet Alpha to run away but with the Greek index what's label it with an alpha In a meeting in operator can act on alpha and alpha is eigenvectors of 80 if the action of aII is just multiply the vet the Alpha my number in this case on user redundant notation saying patience for the Eigen value and the eigenvectors this as the Eigen value the eigenvectors Alpha is an eigenvectors of 80 with the Eigen value out and the significance of that equation is that 1st of all the set of ideas values of an operator of an observable other set of possible results of measures if you have a quantity which would you measure it can take on the value of 1 3 and 7 then the Eigen values of that operator all 1 3 and 7 the eigenvectors of the state vectors of the system for which if you make a measurement we answer is a definite not statistical determined determined stick and the answer measurement is out for a precise Sumitomo remember this summer doing is mostly just refresh your memory now be particularly systems were going to be discussing mostly in this quarter a systems of particles particles or largely characterized by saying things which have a locations based on their relocation space for example of spaces one-dimensional then a particle is the thing which has a coordinated X located some exit basis three-dimensional the particles move around they have x y z himself what but to write formulas for the case of 1 dimension but you would 2 4 3 dimensions a moment old started that's going back there were 1 point that I need to emphasize that every Braun and every cat but called the brine much codified it was not necessarily the particular brought goes together with siding with the kids sorry that every player like that there isn't in a proper was a number that numbers in general a complex number and it is represented as the inner product in a proper Edison number it's a number for every pair of vectors after it represents the kind of product of the 2 vectors its is similar in its mathematical structure the dot product between 2 vectors of ordinary space but it's a more abstract object at in a park between 2 vectors and plays an important role in the logic of quantum mechanics in particular if 2 states are distinguishable uniquely by the properties that somewhat observable that you could measure which would be different in the 2 of them but supposes some quantity that you could measure and it is definite but they answered in the state side is different than the
state fiery then owed them would say those 2 states of distinctly different there's no chance of confusing them is a measurement you could do to distinguish them in those states are said to be orthogonal orthogonal means physically identifiably different end mathematically it's the a statement that the 2 and inner product equal to 0 so orthogonality is a fundamental property of relationships between vectors which says that they are different you can't confuse them but you can confused but you shouldn't confuse them OK that when it comes to particles the most important observable is the position of a particle so let's just discussed particles moving or white wine is the X-axis the location of the particle is just the value of X an obviously X you should be thought of as an observable in particular there are states which are labeled by for our value of position but court action north this is the point x North over here I'm just using anorchia they indicate of particular position there are states are which labeled by X which have the property that if you measure the particle particles position you will definitely find it at text want next concept Oh and of course state vectors for different values let's call them X next prime of exit next prime are 2 different position X and X prime clearly those 2 states Our distinguishable operationally and by measurement distinguishable of eastern states father now take any state any state whatever of this particle citing its inner product with the state representing the particle at X is called the wave function of the wave function of particle and threatened sigh of X it's a sign that goes into the Schroedinger equation strutting sire believe Schroeder was the 1st for it to work the court sorry sorry and the meaning of the wave function is closely related to but not the same as the probability that the particle is at position x is an arbitrary function well a completely arbitrary but some functions What does it represents sigh of X Siletz is closely related to the probability the fine the particle the position next but it's not quite that the probability itself for its P E of X is the product of the wave function plans its complex conjugate wave function times its complex conjugate is positive for any number of any complex number at times its own conjugate is always positive probabilities are always positive sign Amex's are generally complex numbers they positively commune negative can be imaginary vacant B O every light sold it would not make sense to say that sign is a probability but it does make sense to say terms stars a probability that I Pepsi most important profitable particle that has a position it has eigenvectors which represent particles of known position and we can construct of course a position operated a a position operated just multiplies sigh of acts by exits itself ex of the idea position important observable for particle Romania certainly observables for particle but the other a particularly important 1 is called the you what the momentum the moment classical mechanics ending quantum mechanics positions and moment the come together all before we do that I should decide should discuss the EIB issue of a particle moving not just in
1 dimension 1 happens of a particle is moving in 3 dimensions Nova words are a real particle end it has the real observable positions the 3 components of the positioning the ex-wife if you like but you could rotate taxis our you could just think of the position as a point in three-dimensional space and that case who just think of X he was appointed three-dimensional space we could fill out we fell out the equations by saying there are state which represent particles at 9 locations In 3 dimension x y and z this represents a particle located at a point in three-dimensional space heater ex-wife as they enter the rules will be similar very similar the inner products of ex- prime why primacy crime that would be zeroing if is this not if X is not a Quebec's Prime and why not a good 1 but if the entire polling x y z is not the same as the point that far more prime G. prime then bizarre observably different positions for particle in the states are orthogonal to each other I'm not I'm going to suppress the wine industry but keep in mind that the position of a particle depends on the dimensionality of the space that would talking about and in the real world spaces three-dimensional OK for each component of positions there is also a component of momentum and classical physics but nonrelativistic classical physics the momentum is just romance times that component of velocity a quantum mechanics the momentum is also blew observable is also represented by an operator by mission operate permission operators be thought of In the abstract as objects which act on mathematical that vectors there's the space of the states although they can be thought of more concrete Lee as operations on wave functions he the 1 you can think of them concrete as operations that you watch what they know about X itself what is that represents be operated X well it's just taking the state vector sigh of taking the wave function and multiplying it by X if sigh have X as a function represent things beat and the probability amplitude sigh of Texas called probability amplitude away functional particle then if you want to apply the operator represent big positioned the size of eaters multiplies sigh of expects of our the eigenvectors of X on a wave functions which are highly P very very narrow Dirac delta function in the same way we momentum operator no I'm not going to explain the details of this you go back of lecture broached the world or whatever lectures themselves what lecture was it that we talked about momentum everybody remember part this case a yet A 8 8 8 it was the track and remember with great the grief Charita it OK so these momentum particle is represented by an operator called P a key also does something what does the size of actors it differentiates it it differentiates a with respect X but not quite the real thing is that multiplies it by minors all right the complex number there's a factor of monks constant age Bora and differentiate Beebe IDX that is being operator that acts on wave functions that represents these are the observable momentum Everybody remember this did we refer it OK now what about the eigenvectors of momentum the eigenvectors Of position a Dirac delta function their functions which are Zero everywhere
except look I mean value position another words the eigenvectors representing a particle known to be a X nor is a function which is 0 everywhere except the next marked with very high in a narrow but not worry about precise mathematical definition of scalded Iraq built a fortune 1 and it's not too hard to believe that the wave function particle is so concentrated at a particular point of the probability for finding it is 0 everywhere except at that point that of course makes sense that what I these states for which the momentum over what wave functions correspond to particles with definite momentum and also got by solving the eigenvalue equation the eigenvalue equation is that P whatever it is that we know what it is what's right performer what what's right beyond the abstract equations 1st appeal on society it is equal to Littlepine signed the adding value being wheeled not the Philippine nor is the value of momentum that because particle has end in this particular state leader the momentum of the particles known to be penalized this equation translates just before placing P minus age body by DX minus each broader side by X is equal to pee not the number the article number forums sigh of exercises from genomics diuretics and this is easy to solve this issue whether these equations which says that derivative of something is proportional to that same day derivative side apart from a numerical number the numerical number we can get mulled during the ice age promised side will reappear on this side after this the equation of the type that represents an exponential derivatives something is proportional to something itself and the solution of it is easy for the R P naught X divided by plant constant now I often give very tired of writing parks constant and often just equal to 1 was sometimes do that In fact from road while tried attracted for the time being but later on Monday just drop it end I think by the time I do drop it will done enough times it you know where the right there right place where question that in the 2 television disrupted the would With say that would say those are a message I did Victor got at the eigenvectors but that's ideas and eigenvectors of P argue that not the blue little holds any think anyway functions eatery IPA piano were here no mostly functions don't satisfy this equation don't exactly which refer basically this way from ROK that you notice of course that is a huge difference between the eigenvectors of position which of his narrow little spiky functions and the eigenvectors of momentum in particular let's take the probability that's associated with each the VIP nor X that's multiplied by its own complex conjugate what happens if you multiply this by its own complex country you want Ito the something firms minor size some things just plain 1 so this wave function in the 1 which represents a particle at it would definitely momentum its probability distribution smeared out over the entire wine but complete contrast the eigenvectors of the particles located the a definite position is highly concentrated in infinitely narrowly and this of course is a manifestation of the year of the uncertainty principle our if you know 0 rout if you know the momentum of a particle then its position is completely uncertain and likewise if you know the position of a particle its momentum is also completely I'm certain refer you back to lecture notes our for that purpose as basically quantum mechanics and that's and it is what this is just momentum them is our only if they expect fog delayed probability is no probably at the probability for position what given that the momentum is pinot are now an experimental is going to do an experiment not the check its momentum was going to do an experiment which measures its position different kinds of experiments In that 5 5 is the fact that probably aptitude for that Fletcher if he argued that right now there is also a motion of and the amplitude for finding different momenta and I'll just remind you that connected the Fourier transforms promote would do that now if we need Fourier Transform will come back to it at the moment our Fourier transforms and the momentum representation this is called a position representation representation of state vectors by sign of X is called a position representation it was also a momentum representation but I I would get to pick us too far afield right now yes all of the position of the outside of acts he said we side with what is that was there it was it looks like the square of dope function which is higher than the dealt but concentrated at the origin of the concentrate the origin in our we have not talked about normalizing wave functions making the area under here equal to 1 that's a thing that we wanted to end herb but as I said I hesitate to go into any given thing too deeply because who warned the doing the whole class again so he days that although the call you can you can represent them by column vectors words but there be a little awkward to represent expire combat the X takes on a continuous infinity of values where's the column vectors useful quantity which takes on some discreet the you could think of it the formerly head is some kind of continuous column right there that's the but for the cat sectors and the broad vectors you can think of as rogue sectors OK but turf works only need that our I'll remind you but did you will have gone back studied the lecture you won't be OK now saying how things change with time things do change retiring the evolution of the system with time as a special case of transformations that you couldn't do well on the state of a system that glasses also touring classical mechanics media over most of the system is represented as a transformation in phase space a transformation of the phase position in quantum mechanics me evolution of a system is a special case of a transformation that you couldn't do on the space states taking
every state to some of state of course but the idea is very straightforward if the state had 1 incident of time is some signed at some later instant of current issues something else and that's something else there's a transform transformation of the original size there some Rosen quantum mechanics about the nature of that transformation all that transformation the on the amount of elapsed time 50 amount of elapsed time is very very short then you would expect this transformation of practically BUT identity trans fat Hrvatska back if amount of time that elapsed is 0 and of course the transformation is completely trio it is gives back the same state you could say in that case the transformation was just be identity operation on the euro on the state vector but more generally these time evolution is characterized the time evolution operation call operator is an operator in a sense of operators that's an operator but an operator which depends on the amount of elapsed Korean it's given name is usually called you would like you stands for unitary but will come back in a moment for unitary means our end its parametritis by an amount of elapsed time that is such that when it acts on a stay at a given instant of time it takes you to the state and the amount of time t Leader so for example if it acts on the state where the state happens to be at times equals 0 then it would give you sigh at time to eat it just updates state if acts on some state which happens to represent things other times 0 but let's say act time T-1 1 then it takes should at trying 1 plus a another words it represents the evolution of the system from T-1 T-1 plus that's the notion of you that's the motion of a year time evolution operator it's a linear operator but it is not a her mission operator what kind of operator is well known linear postulates and the postulate is a pass to it very easily it goes back to the minors 1st law of physics that I hop on all the time that information it is conserved in quantum mechanics what it means is that if you start with 2 tool vectors to states which are observably different which cannot be confused another words which are orthogonal annually both of them evolve they will stay orthogonal it will not happen to state that Our observably different Willie evolved into was interstates which are not observably different that says is that orthogonal vectors remain orthogonal in fact you could do a little bit better you control from that statement that meet in a product between vectors is unchanged by time and so they just to explore a bad idea a little bit where they knew is it preserves the relationships between pairs of sectors so for example if we start with site and fight that is they may be orthogonal they need not be of Fargo Cicero this in a product Zero with enough orthogonal it is whatever it is that in the past so it is that if we transform society and we transform society now funny is a bra vector and it transforms with permission conjugate view that would be you dagger so far this equation said is that the trans of society when you take it in a product of the trans form of 5 is the same as the product of sigh find the beginning it says that the inner products between vectors don't change with time that can only be for arbitrary pair signed is viewed dagger times you is of operator that's called a unitary operator you dagger times you use the unit operator of the identity operator just 1 that's the notion of a unitary operator and the time evolution in fact all transformation all interesting transformations that you do on the space states and will go talk about others besides time pollution all interesting transformations ways of transforming vectors and other factors are unitary you better you is equal to 1 that's different than saying You was equaled one-off course you was a unitary operated by up let's facility out really side but I thought of a mail fraud stated that that it's not you could prove that a few was operated and maintained orthogonality maintains organ apart is that they say the idea of body builder is effect and the rest of that that's which is saying is correct but I would say it follows from the orthogonality is preserved with time and then it's possible to prove that in a product of preserved through exercise their 1st of all think about it just ordinary three-dimensional space supposing you have some operation on vectors ordinary vectors which preserves orthogonality OK what kind of operations operators on pairs of vectors will preserve the fact that the orthogonal rotations right rotations also have the property that they preserve in the or the bar products this is a mathematics Mississippi mathematics so this is the definition of a unitary operator mini unitary operators are many possible things that represent the evolution of the system but all units OK but is but it example of what you Chester boat position operators are usually not unitary except the momentum is now in the carrier operate the position was not a unitary operator on Yawkey operators all operators who's I did values freezes everybody in order phase means and eatery on a real number a point on the unit circle of phase represents on the complex plane point on the unit circle 1 is a point unit circle is a point and the unit circle minus 1 of my decide our world square root of 2 times 1 plus is a phase he unitary operators on analogs for operators of phases the
Eigen values or all phases of have a unitary operator and I look at eigenvalues they will be points sprinkled around on the unit circle that's why the cold Unitarian because some sense there for their the magnitude of her Eigen values is unit OK now let's but see what more we can learn about you from making some reasonable assumption the 1st reasonable assumption is that he is equal to 0 0 the EU was just equaled the 1 quality says if you don't allow any time to elapse the vectors does comes back for self and you'll is equal to the you move operator so you'll of Xerox His equal that I be identity operator the operator was matrix representation is just 1 1 along the back which is the operator which does nothing it just takes a thing back to itself that was you'll unitary shirt is leveraging conjugate are of you is permission can't get of identity operator which is just too identity operator itself and identity cards identity is just identity so yes 1 it is 1 order you have operators unitary and it's the time evolution no evolution happen OK what about evolution by a very very small amount of time so let's instead of writing you 0 0 let's right you will Epsilon and take you will Epsilon it is not 1 something happens over time evolution but it is close to 1 of its cluster 1 that of course is an assumption that see assumption that systems evolve continuously but they don't make radical jumps over very very short periods of time our is asylum assumption assumption ultimately which is justified by experiment so that says that you of a small-time epsilon if we were to expanded in small and small epsilon would be 1 plus something proportional epsilon times an operated where is a finite the operator the smallness of Epsilon was represented by epsilon he finds something which our temporarily called age but wrong identification and will change it Butternut Court H what's another letter demeanor lure quick quick no normal G G G F G 8 . 3 so you was of this form is nothing wrong with this this is correct thing the only question is where we know about G. so let's right then on you conjugate permission conjugal Repsol on that's equal to 1 plus epsilon pens G country permission conjugated J. ballots multiply these together and insisted the answer the leading order in epsilon is just 1 or say saying that 1 plus Absolon G dagger times 1 plus epsilon GE is equal 1 but so queer one-time forms 1 sorts leftover was left over is epsilon kinds GE dagger plus GTE and that must be 0 Toyota epsilon working toward a epsilon the condition that you is unitary is a condition that G plus its own volition conjugated is equal to 0 and other words getting rid of the epsilon that they can't give G physical Tom minus G itself operators with this peculiar property Dino numbers that have that property a number that has a property that it's my visit from complex turned that is true but a wider class of pure and legendary so this is the analog of GE being pure and legendary if we multiplied by I see it becomes permission this is called in died permission is when dagger equals Judy rewrite that someplace yacht equals a daggers and observable Hey Eagles miners G weakened thinks that very simply just defined G dagger to be warm plus or minus I always forget to restrict them to work through to harm the right 1 plus I Epsilon a witch but I think the rioters minus that that's a that's a you put a plus sign or might assigned here's a convention complete convention defined H Sorin becomes warning plus I Epsilon age age diver and the condition just becomes that H equals age that another words if I put a minus Ali Epsilon h plus I epsilon that mind by epsilon 8 each than the condition they use unitary is just the conditions that His Felicia another was that age is an observable young passenger data set test of the order Epsilon we could work toward a epsilon squared but then we would have to be consistent about it and put epsilon squared terms here the epsilon square Turner want to put here would be minus epsilon squared over 2 times in each square and the but we consistently worked toward power in Epsilon and the Sumit that epsilon is small enough that we can ignore epsilon square that consistently read them over and over and over our but Virginia pedigree right we do after combat and check the formulas makes sense the higher order and they do not discuss them a little bit but working systematically Epsilon dropping anything of higher order Repsol ordered that is what will follow the lead of the gets a the box yes not well OK with this definition you would say choose from for approximately permission but London allowing a notice that factors out of both sides of the equation wants is factored out just as the age not think the thing which is approximate is this relationship you're got not welcome back and tell you what to do well the right formula for while at some point OK cell where we learned we learned that they that they Terry time evolution which containers of U. of he also is specified in terms of a certain observable but I've called h h since the Hamiltonian but also stands for her mission of stands the Hamiltonian it is a Hamiltonian of love of quantum mechanics and so we come to the conclusion that there exists an H we will call it the Hamiltonian and it is related to the time evolution for small time intervals by 1
minus i Epsilon H was the general equation epsilon was not small incidentally of epsilon is not small we might want to quality and might give up calling at epsilon epsilon is usually reserved for a very small quantity our we might try writing you was of tea but this wouldn't be right a large time it wouldn't be lodged were from large firms but so let's see if we can improve it a little bit well you know I think I won't spend time now I'll just tell you the answer the answer the fist really should be written and E I H T now what the media for me just say a word or 2 about exponential if you have an exponential he some small number and let's just call the small number of Epsilon where epsilon small kind an ordinary number and nickel ordinary number little H epsilon times H is a small number of very small epsilon eagerly epsilon age it is 1 plus epsilon H. that that formal Taylor series expansion of keeping only the 1st assuming his 2nd term higher terms or negligible cake now suppose you want to take you wanna make epsilon little bigger instead of just Epsilon I wanted each of the 2 epsilon times in which each of the 2 epsilon times age it is just the square Evita the Epsilon H. Ito the tomb time something square so it's equal to 1 plus epsilon each square supposing you do this Over and over and all we which incidentally is equal to 1 plus tool epsilon H plus epsilon squared each quarter toe but this is still an approximation this was an approximation of this just below approximation of it is a theory that if if you do this all marine over enough enough times let's say in that time he epsilon times In The times such that epsilon times in his mind that no words epsilon times In epsilon isn't infinite personally small number but I do enough times so the 18 times epsilon is a finite number mainly T dues over and over what have excuse me your back but still it this way 1 plus Absolon H and I do this a number of times a large number of turned in such that in times epsilon equals fit well I get going that's a binomial it's a binomial expansion but also a construction of the exponential function Our it's a rigorous theorem that as epsilon gets smaller in a gets large situated in times of storms kept fix this just become each 30 team so Major but G for me 1 person thought that I think it's right abuse right right and the way you could just understand it is by saying that the small Absolon arms 1 plus epsilon H is the same as eatonii epsilon age and take Italy and power the Epsilon just multiplies the end and becomes OK so I will tell you right now these generalization of this for a finite kinds 1 my assigned here 1 minus H T for our side he really becomes Ito the miners HP and this is an important factor but a small intervals small epsilon we could just be consistent and work toward a epsilon will find out everything we need to know OK so there's a concept now of a Hamiltonian let's see if we can understand this equation a little bit better Armed let's say take the wave function society of state vector state vector size time times keep plus Epsilon I am starting at time t with a certain wave function and I'm allowing it to change over kindly time interval What is the answer to this to this In that is he called me for the H Epsilon in times of society at time t see what I did minus plus sorry my legacy Starting at time t let up to date by a small little time Epsilon the role it is we multiplied by you are but are you off epsilon word updating by an amount of time Epsilon and that's equivalent the multiplying by Ito the miners died at age epsilon website so this is the rule for going from a time when neighboring card for now let's right that down as 1 minus I epsilon times 5 teams and that's right beside America forgettable OK but and sunny areas society he plus so what now multiply out 1 time sigh of sigh of key but let's take them all with the left hand side I don't give a sign of plus Epsilon my sigh of key just term coming from 1 of the term coming from 1 will give us the difference between sigh of T plus Epsilon Miners 5 we still adorned with building a differential equation for signed Sciotti plus epsilon might decide if is just equal minors I H Epsilon and finally the bye bye Epsilon OK what's the left inside left-hand side is the kind derivative of the state vector site and the right inside His just might decide each side focus of that leads us finally to assuring a equation the Schroedinger equation says that side by side by sees this could be a wave function or it could be the state vector I will leave the notation a little bit abused I won't bother
writing we kept vector here it stands for both the equation for the state sector or the equation for the wave function itself decide by sea is equal from minors I each side if you are for sectors in 2 of turned it into Eddie equation for a state vector of kettle stations there otherwise it's just the equation for a heat wave function this is the Schroedinger equation this is the kind of deep pending shorter Schroedinger equation OK this is Schroeder tying depend TBS time-dependent Schroedinger equation what it tells you is how the state vector changes as a function of time if you are so lucky the and work the Hamiltonian Hamiltonian is permission operated and you can make all kinds of quantum mechanical systems just to play with choosing different ages we did this last quarter Okayama was about time independent assured equation the kind independent Schroedinger equation is just a statement that h is permission operator it has eigenvalues eigenvectors its Eigen values are the observable Our values that you would get a few measured age recalled and another name energy levels energy levels incidentally I promise you that I would keep around plugs constant I've already drop it shaft goes over here goes I don't promise by any means will be consistent about plants constant yes I'll be completely consistent by choosing point constant always Beagle toward sometimes a sometimes won't put in but I will never confuse it with ideal the square root of 2 OK so that's the time dependent the time independent Schroedinger equation that's just be ideal value equation for the ah Eigen values of 8 and all it says is that on and I didnt vector of the energy of a state with a definite energy it is equal to 80 times the energy E being Eigen value H being the Eigen value of H so this equation here is the equation which tells you how defying the states and the values of energy that go with them that correspond to states of definite energy you solve this equation and it gives you a whole bunch of possible Eigen values and eigenvectors those are the eigenvectors American values of the energy operator again we went through this we did some examples last West quarters ago hand this equation tells you how the state vector changes with kind and as you can see it sustained hm often both equations so we question energy levels of system are intimately bound up with the way the the system changes with time OK so there and that's a that's quite a character generator said before was quarter character nutshell slightly bigger not but there but that's it but that's where you need not a summary of the quarter's worth quantum mechanics would any questions it doesn't plan equation our seemed to lack defended reticulation place site all case cable dedicated wrote kit but that is because just on a quick b the HST ages he said not forward your reward dropped Richter to search we've this leave the capsule into cancer for the moment so we can do is we can ask how 80 eigenvectors changes with time that we can do so we can plug in here and we can ask how the eigenvectors change of time and the ensuing years the they eigenvectors will change with time by just multiplying it by times the energy level that's what tells us How eigenvectors change with time limits remember the answer however argue that the change of and just gets multiply by each items e energy times time a place where we can go through that we don't need to Europe's go back to to return check it out right now they time evolution is 1 example of transformations that you could do on a system Park which preserve certain facts about the system in particular which preserves the inner products between vectors another words which preserve logical relationships between vectors preserve the notion of orthogonality preserve the notion of road vectors being same vectors and that kind of form their mania all the kinds of transformations that you may want to do on the system In particular among them are symmetry transformations we will need to talk about symmetry transformations now I will tell you will go in 1 of the most important symmetries of nature the occur over and over and over again in all sorts of context but in particular In the fury of Adams is rotational symmetry Adam particular a hydrogen atom hydrogen atom is an electron moving in the central force field the equations of a hydrogen atom a rotationally symmetric and that's because the potential energy Coolong potential is rotationally symmetric to central forest and so hydrogen atoms and basically all systems in nature really hydrogen atoms particularly enjoy a rotational symmetry if you wrote take awhile it means is that if you wrote take your coordinates the description of a hydrogen atoms doesn't change are another way to say it is if you rotate the hydrogen atom it's still hydrogen atoms it me on that may not be the same state but it's still hydrogen atom described by the same set of equations some of the most of the symmetry it is Central in quantum mechanics it was also in the classical mechanics but did comes up and really hit you overhead in quantum mechanics and we need to explore largely for the purposes of understanding rotational symmetry we need to to understand the concept of a cemetery like those spend 15 minutes explaining exactly what a cemetery here keep in mind if you wanna keep 1 my dear to deepen keep in mind that Ferrer just have something in your head 1 of them is rotations cemetery where it just as the take a system which satisfies a certain set of equations and you rotated physically rotated physically rotated about Max's you may or may not change it I can tell you this 0 take me no change me you if foreign here and I was in outer space in free space the year the Leonard equation whatever the equation governing years would
be the same for the rotated or the on rotated another way of saying it is I couldn't tell whether I was rotated out Europe they mean just go away I can't however was rotated because the equations that govern my metabolism my internal structure and so forth are exactly the same as have and that rotate rotated OK so replaces symmetry is 1 example another example as translations symmetry to take me in outer space and you sit back and say how I feel find you take the same state but you translate me over by a meter U.S. me how I feel I say exactly the same as before 2 symmetry on the other hand there are situations where translations may not be a cemetery if there was a year of furnace over there fire fire and you put me over here and said have you feel if you'll find translate Leonia field fine so the presence of an object may break translations of trees on the other hand if you think about a minute later what if I really translate Everything and defenders then we restore the translations of tree so symmetries our operations are there all our operations that you could do want states of systems and if the operation doesn't change the equations doesn't change the properties of the system doesn't change the way you described the system then that transformations called asymmetry for a free particle fervor NEC ordinary system without outer space far from anything else and you translated that's a cemetery you rotated Baptist Street what his queasy change its dimensions by squeezing it that's generally not circuitry detector crystal over rock salt or something new squeeze down compressive by a factor of 2 doesn't behave the same way are so the all kinds of things which not trees nevertheless they still may be operations that you can define but they may not be cemeteries OK let's see let's talk about the condition but 1st of all private quantify idea we have state let's Janek typical state of the system that were interested in his core side will operation on that operation might correspond to rotating system and my correspond to translating the system or my correspond something Alderlea than that like squeezing stretching it but doing something through it and represent backed by a In the operator V. I use V because I've already use you but intended to be a unitary operate a white military apparatus well if I really have the symmetry and I have to states which are different in each other I apply the symmetry operation I expect them stay different if there are 2 states all of meet happy me and said the different from each other they're observably different there's a visit observable Duke measure the happiness of operator and the different you then wrote take the state out in outer space the distinction should remain so that means that orthogonal states should remain orthogonal when you do a symmetry operation and that says that they operate a V should be unitary Of these unitary represents vehicles a dagger a 1 and I'll use you will because I'm saving you for a special case Of kind translation of evolution Okatie now let's suppose Our job is Wilbur tricky soulful as you lose and his it's it's quite tricky picks me half an hour each time I tried to do this to get right but it's also very very simple wet suppose that a wave function or a state vector them interchangeably but scored sign 1 under kind evolution become this site to or if I thanks I won and I allow it to evolve with time that represent that will you hear your represents time evolution does not obvious something else and if I take so I won't allow it to revolve for a certain amount of time it will become size of the U. another way to say that over here is that you'll on sigh 1 is equal to site as evolves I won from time and you get site now let's imagine the transformed version of this transformed it could be by rotation it could be by translation by some symmetry operation What does this equation safer transformed things well let's call is what's called a side prior to many the keep in mind exactly what's going on here at 1 end to represent some initial state and some final state evolution from 1 to another than under Prime represent the action of the symmetry operations which could be rotation could be rotational could be translation OK here's what I maintain if there really is a symmetry and warned Prime if we were talking about rotations and sigh 1 Prime is just the rotated state of sigh Warren sigh 1 Prime will evolve through the rotated version of side too but the that says if I of all myself over a period of time and I'm in some state may get some other state I go from happy cannot so happy then when I wrote myself I should also find under the same evolution that I go from the rotated happy state of the role of the rotated not so happy state that really means that have a symmetry that found that she evolution of be transformed state behaves the same way as the usual Stitzel Buttrick OK let's see if we can make equation nervous now arm yup 1st the 1st here is just be equation site to use EU sigh 1 of Africa's 2nd equation let's go to the 2nd equation the goal from unkind the prime you multiplied by V so the 2nd equation here says that v it serves that I'm sorry is always confuses me wrong but but if says the flight 2 prices is equal to and sigh want go from
wanted to which is kind of erosion the girl from trying the prime is V uses symmetry operations are I time it he I want a story however you will site 2 0 is you times sigh 1 Prime and sigh warned Prime is NYTimes that's the right hand side the left-hand side is V on-site tool but site low His you'll I 1 so left a perpetual confusion that the that a is sums me every time I tried to do this on the blackboard what it says is that of a thing the symmetry another word preserves the way he ever evolution takes place the transformation which preserves be kind relationships between sectors then it says that he kinds you must be equal to you times read this of course must be true affect every state of its Toby a truce symmetry of nature the true symmetry of nature rotation symmetry for example is not something which only applies a firm stand vertically up bright should apply Frank any particular state of the system and so if Vita sends you on any sigh 1 is equal to you times V on any sigh wonder what it says is the symmetries of nature of their operations ease unitary operators which continue with the time evolution operator world won't brawl yesterday to thank for all time if you will see its enough food to be true for 1 time and will be true time yes that's correct OK so what you Intel a operation things you could do to the state vectors which preserve orthogonality which commute with the time evolution operator aura symmetries that preserved the evolution of EU evolutionary relationships as it is today exactly the world's of very simple and they make perfect sense Endo yes us exactly words OK so how we identify symmetries we look for operations which commute with the kind of Aleutian operator now remembered that the time evolution operator is itself take the the limit of very small kind evolution just analyze a little bit says were 1st of all says that he can't 1 plus Epsilon H 1 miner said former each 1 9 aside for each is equal to 1 minus epsilon H. This is 1st order again Wednesday and just expanding about are the 1st daughter was good enough accuses everything we really want not 1st of all V times 1 is equal to 1 times was canceled and what it says Is that minus i Epsilon each equals Midas Epsilon agency are canceling out the minus side epsilon it says that he commutes with the Hamiltonian OK would cancel out these things might decide Epsilon a symmetry is a unitary operation which commute for the Hamiltonian any military operation which commute the Hamiltonian is a symmetry in symmetry is a unitary operation which commute but very abstract extremely abstract and ought to appreciate it we have to do a couple of examples will do some examples of a moment but let's just focus on that Terry operations on the space of states which commute with the Hamiltonian our symmetries how that will be limited but that it will require please help yacht a symmetry whatever symmetry is it preserves logical relationships between the vectors with the assumption that assumption that the symmetries of nature take orthogonal states for Cardinal states that bro seems plausible a symmetry should not take 2 states which are distinctly different from each other and act on them to give states which are not distinctly different from each other you wouldn't call rose tree not mutually exclusive thing should stay mutually exclusive OK so we don't have India abstract mathematical definitions of any symmetry operator of any cemetery but with this overcome much clearer when we do some examples cemetery try me a trader after each with D is equal to 0 we remember from 2 quarters ago what the commutator of they were the Hamiltonian is kind derivative your soul with this there is so whatever the operator it's conserved conservation of a certain quantity in this case V mean that a commuter the Hamiltonian is the seen as saying there's a cemetery is seems before classical mechanics what's the connection between symmetries in conservation laws called in classical mechanics 12 lithosphere right it's much simpler and court the mechanics much simpler in quantum mechanics are we don't need any fancy Lagrangian not anything else always need to know is that time evolution commutes with the symmetry operation itself and that to be ready in 2 different ways they can say that doing this symmetry operations well it says works can be red 2 different ways but the rebuttable saves and OK that's now let's go 1 step further there are different kinds of cemeteries for me give you some examples of 2 different kinds of secretaries discrete symmetries and there are continuous symmetry is a good example of a
discrete symmetry on think about reflection in the a mirror reflection Ameritech's aright he and poor left here on risk because that's the mirror image of left-handers right here that's a discreet operation thanks left the right but there's nothing Sullivan between just a discreet operation left to right give it a go another example of a discreet operation on disposing of a pair of electrons or pair of particles a pair of particles and they happen to be the same part of the same kind of particle a pair of electrons are a pair of proton then if I take the 2 particles of light to change their wherever the state is eyeing interchange 1 particle with the other particle that's a symmetry why because the particles are exactly the same kind of proton over here and a proton over years as a pro time over here for power he'll just interchange of those of discrete operations are discrete operations means the storm nothing in between or what's a heads and tails just a heads and tails ahead and tale if the calling it happens to have markings on whether it's heads of tales and you turned the ball over the physics of a coin flip cornea exactly the same those are discrete operations by discrete I mean in distinction the continuous operations a continuous operations are like rotations space rotations space you could do a very small rotation small means 1 by a very small angle and fighting go further you could say that every rotation about taxes could be built up by tiny rotations about that same Max that means that the rotations have a continuity to them that the flip of a coin does not have a flip of a coin is either tales and is nothing in between classical coins are 4 systems in three-dimensional space Yukon roar two-dimensional space for that matter you could erupt take them to corrupt them by 90 degrees that's a big that's a transformation but you could build up that 90 degree rotation by lots and lots of little info test track I was a cold continuous symmetries or about translations retreat translations symmetry continuous or discrete wealth continuous you could take any translation of system and think of it as being compounded have lots of small translation the continuous symmetries are the ones at and particularly the rotation cemeteries and translations of the trees are the ones I wanna focus on what really know about continuous trees we know we can build them out of lots of little elementary small transformation of us focused on small transformations we're going from rule basically the same manipulations that we went through all of it he kind translation is a continuous thing you could build up a time translation by 10 seconds by thinking of it as lots of little time translations by microseconds and that led us to the conclusion that you'll lose are related through her mission operators through this relationship here exactly the same as true off every continuous symmetry exactly the same thing is true every continuous symmetry can be thought of as being built up by Will infinitesimal ones which have performed 1 minors vii Epsilon not times a Hamiltonian but something permission operator let's call G GE spins the generator GE stands the generator nor was thinking about now is a symmetry transformation but 1 of them which is very close to being the identity of such a transformation can always be represented as something close to the unit operator and close to the you operator means just shifted by a small amount by the same argument that told us that H is her mission GE is also mission symmetry operations which are continuous Comey represented in terms of infinitesimal transformations which are generated by generated G of reply the center here with if the comet that other 8 each with 1 minor supply epsilon GE is equal 0 was at say what the comic each with 1 connected with a thing with 1 0 everything commutes the simple the apparatus we II epsilon that factors out what tells us is that symmetry operations are generated by this is what it means to be generated by generated by and be built up by lots of little ones they generated by things which commute with the Hamiltonian so if you wanna find all the symmetries of a problem you stop looking around for all the things which commute with a Hamiltonian visit another feature of things which commuter the Hamiltonian member from last quarter strong both in the art world was out they're conserved self conservation and symmetry are closely connected if you want to find all of the symmetries of a problem it's equivalent of finding all the things which are conserved viruses very abstract and so we need to do with an example or To get the idea let's do it example what's begin With translation translation of a particle from 1 place to another translation of a wave function who want to take away function of wave function aluminum although a bed where translate it through new wave functions I would want to find out what operation does that if we have a sigh of ethics what operation do we do to translate it a new position we're going to take that no positioned only deviate by a small amount by America epsilon but we know what the answer is the answer is sigh of X goes to 0 sigh of X Epsilon and so on now it could be a large number which is translating by a large number but specifically I want to think about the case of a small translation translation by a little bit
X another words we can write translated wave function Vatican right this way whatever whatever is we have to get our views yet aware of is on sigh of X to give us sigh of X plus a as easy that that's what every year that he does in this case and act sigh acts and it translates the wave function by amount that so what now epsilon the small we can also write that this is equal to sigh of X plus a derivative of sided with respect to X Front Epsilon Saleh where he does Ford V onside drugs is it gives back the same size plus something proportional to the derivative that suggests suggested proves the Olivier's is the unit operator 1st turned here Exxon sigh of Exeter givebacks ibex plus a small change in the small changes epsilon tying the derivative operation I haven't written on extra Carrai on X V vii is an operator it's the unit operator plus at the 1 times the derivative operator we can act on any wave function where will give us is the wave function infinitesimal we displayed now about decide by X What about Beebe IDX Debye IDX is an operator it operates on wave functions a linear operator it's connected with the momentum operator remember relationship the momentum operator he was minus I age body by DX riots over where we were doing right multiplied by applying here and abide by each differentiation is I over age Barre Times momentum so what's right wait 1 plus Epsilon Phi Epsilon over each times the momentum we don't have an example of infinitesimal symmetry Operation Infinite testimony despair will be chef and we found that when generator is the GE where was we have to get is 1 letter-size epsilon g in this case I would have much preferred that I defined things of this came out we might this year but lost his that a results were right on the last strike and if it doesn't want tried so almost say I substituted I think I got this right away are yeah that's right do you disagree with this contract that now wore on laughter I think shift the wave function variety which is still a function of the right going minors I'm here now the I made a mistake In defining a shift to the right the shift to both suck it should X plus epsilon but sigh of Expo society X minus 1 so this is actually a minus sign here if you track that rule will find our as a young a minus sign here OK so it identifies for us what these symmetry generated G for translations what is it it's some momentum divided by plant constant momentum the buyer but plants constant Wickham right the generator of translations use the language generators are generator extra lotion if this PT sub X P Sebec strictly speaking divided by party we drop Clark's constant there will drop plants constant and as a matter of fact the definition the official definition really doesn't have formed constant but calm as an example as an example he of X now is Felix is the momentum along the X axis conserved pens on the Hamiltonian it depends on the Hamiltonian so let's write down Hamiltonian Hamiltonian for a free particle particle moving in empty space with no forces acting on it H In that case it is just a squared all the away at the moment the square divided by twice the man if this is a one-dimensional problem than a really is just P extras the x component the momentum as the Expos beat Hamiltonian commute with PEX what becomes equivalent to the question of whether PEX commutes with PX squares yes every operator commutes with every function of itself POX commutes with POX squared POX commutes with any function of p of X sold for a free particle moving along the x-axis we 1st of all find that the momentum is concerned and of course we also have a translation salutatory the translation symmetry is just a fact of a free particle will behave the same way wherever you start started here and get sticky and then starting the starting point just everything would translate OK so that that's an example P is both the conserved quantity and generator and translation was more a space if there are more directions a space than there are several cemeteries you could translate along the x-axis you could translate along the y-axis you could translate along the z-axis in that case the Hamiltonian becomes PX square was PY school in order to land to land and the translations symmetry along the x-axis is still just PX translation along the Y axis generators P Y and so forth in each of them commute for the Hamiltonian services an example of the connection between symmetries in conservation laws which is very simple the next time you sucker about out of time the next time we will do rotations trees cases symmetry is much much more interesting it's much more involved does and it will bring us the subject of group theory some group theory but the group theory is completely trivial next time we will talk about rotations of space what's interesting about rotations in space is they don't commute with each other a rotation about X followed by a rotation by wine is not the same as a rotation my wife and that will give us the idea although none commuting cemeteries symmetries collections of symmetries which don't commute with each other and refined those they have power they have real power tell us about all kinds of physical properties of poker from external pressure for more
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Metadaten

Formale Metadaten

Titel Advanced Quantum Mechanics | Lecture 1
Serientitel Lecture Collection | Advanced Quantum Mechanics
Teil 1
Anzahl der Teile 7
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15095
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 7, 2013) Leonard Susskind derives the energy levels of electrons in an atom using the quantum mechanics of angular momentum, and then moves on to describe the quantum mechanics of the harmonic oscillator.

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