Lecture 3 | Modern Physics: Quantum Mechanics (Stanford)

this program is brought to you by Stanford University please visit us at stanford.edu all right we were developing the mathematics of linear operators as I've emphasized the mathematics of quantum mechanics is linear algebra linear algebra consists of a set of concepts beginning with a vector space the vectors represent the states of a system in ways that will become clear if that doesn't ring a bell doesn't make sense to you now that's ok just as long as you understand the mathematical rules of vector spaces at this point are the states of a system or of course not what you measure what you measure is observable quantities things like momentum things like other quantities that are available to detect and to record and to measure and the measurable quantities are represented not by vectors in the vector space but by linear operators hermitian linear operators so we need to develop that concept we started I don't know how many times we started but let's just review very very quickly a linear operator I want to I won't explain to you again what linear means a linear operator you find a piece of chalk this is not chalk yeah a linear operator I will represent by a capitalized letter and I'm going to put a little hat over it just to indicate that it's a that it's a linear operator it acts it operates it does something to vectors in particular it does something to ket vectors and gives you new ket vectors I'm going to call this one C all right now that allows us that together with the idea of the dual vector space and the notion of inner product allows us to define what are called matrix elements of operators matrix elements of operators kind of are numbers they're all numbers they're a collection of numbers a number for each pair of states that you can write down each pair of states let's say a and B and they're simply defined let the operator act on a it gives you a state it gives you another vector let me put a bracket around it to indicate that K has acted on a to give a new state that I could call C but let me just leave it this way and now take the inner product of that with another vector B that is usually simplified the notation for it is usually simplified just to be K a but the way that you read it is K acts on a to give a new vector K times a and then you take the inner product with B and the result is called the matrix element of K I should put some hats over this shouldn't I indicate that it's an operator this sometimes just more briefly called K BAE it's called the matrix element of K between vector B and vector a and it's a notation that's very closely related to matrix locate notation now remember the idea of a basis of vectors a basis of vectors is a collection of vectors which I can label n if the dimensionality of a vector space is whatever it is D for dimension then a basis has d independent vectors all of them orthogonal to each other and all of them of unit length so just pictorially our basis it's hard to draw vectors in a complex vector space the best I can do is to draw them in a real vector space but for example these would be a basis of vectors in ordinary real three-dimensional space there are other bases another basis for example would be let's see if we have another color of chalk another basis might be tilted relative to this basis these are intended to be all mutually orthogonal vectors all of unit length the blue vectors in three dimensions and these are two different bases the blue basis and a black basis let's pick a basis of vectors so let's erase the blue basis and simply enumerate the basis vectors one two and three for example and that's the collection of ket vectors in we represent red states and blue boy Michael you're on tonight we could be all home listening to the president give the last State of the Union address has everybody decided who they're going to vote for in the primaries you hadn't I bet I know which party you're going to vote for anybody who has vote the other party out of here now okay yeah so given a basis of vectors we can tabulate or record all of the matrix elements of K between vectors N and M and that's a table it's a kind of table a square table of elements n by n matrix and n by n matrix K 1 1 K 1 2 and so forth let's put K 2 1 down here and we tabulate them write them all down and make some matrix and endcap a a square matrix whose size is purport as equal to the dimension of the space okay so those are the matrix elements of K and they really do describe the entire operator K they describe it in some detail they're uniquely describe it yes I I mean N and M to be basis vectors drawn from the same basis right ending them our basis vectors drawn from the same basis alright so this could be N equals 3 M equals 2 and so forth so since there are how many basis vectors there are D where D is the dimension of the space the basis vectors and therefore d squared matrix elements n can run from 1 to D and confront run from 1 to D ok now let me remind you about something else I'll write it over here about a basis a basis has the property that you can expand any vector in it that you can take any vector whatever let's call it a and write it as a sum over n of a set of coefficients let's call them a sub n times the basis vector n with sum over the basis vectors I'll include the summation signs so that any vector can be represented in terms of them that's just a statement the animal at the analog of the statement and in three dimensions any vector can be represented in terms of three basis vectors okay again the set of coefficients a sub n is nothing but the inner products of the enth basis vector with a is equal to a let me call it m let me call this is supposed to be an M so the particular coefficient a is just given by the inner product of a with M that allows us to rewrite any vector in the interesting form sum of the N now let me write down the vector n first write in first and then after having written in let me plug in the coefficient n a this is a kind of expression that will occur over and over where you see an in standing next to an in like that summed over whenever you see that it's just a way of oops this should be closed like that a is a vector whenever you see an expression like n standing next to n with their noses pointed toward each other summed over in you can always say all of this sort of does nothing it just gives you back a so keep that in mind whenever you see a sum like this within who then you can it's there it's a real thing but it sort of just can be thought of as oh what is it thought of as the unit operator the unit operator on a gives you back a but if you didn't understand that that's not important this is a correct expression so now let's go back to the action of the linear operator K on an arbitrary vector let's take K and apply it to a what I want to do is I want to calculate or let lets let's come back over here one way of describing the vectors is just to describe them symbolically or abstractly as vectors but another way is just to give the coefficients a sub M if I know all the coefficients a sub M I know the vector a in particular if I have a specific basis picked out the set of coefficients is a representation of a vector it's a way of describing the vector and so there's an abstract notation for vectors and there's a more concrete way of describing them as a collection of components these are the components of the vector now here is an abstract operator being applied to an abstract vector I'm interested in working this out in terms of the components of the vector the actual numerical components of the vector so the first thing I might like to know K times a is a vector I would like to know its components its components are just gotten by taking the inner product with the basis vectors so this is the in component of the vector K times a or K acting on a let's work this out by writing a in terms of components here's a written in terms of components or like this so let's plug in for a m/m a summed over em that's the same expression that's written here the only difference is here I've used the index in for the summation variable here I've used the index M of the summation variable and now we can rewrite this as K n M this is the matrix element of K between the vector between the basis vector M and the basis vector n times a sub M summed over m so we can reduce the problem we can reduce the problem of calculating the components of k times a to a set of operations all on components and matrix elements this is another way of writing the matrix k acting on a column vector a right I assume that everybody here has done a little bit of homework and learned a little bit of linear algebra this is simply another way of writing the matrix K times the column vector a to get another column vector this is a column vector because it's a vector whose components are labeled by n so here's an example of the application of matrix or notation or matrix or construction to the idea of a linear operator acting on a vector now we can multiply linear operators together what does it mean to multiply two linear operators multiplying two linear operators let's call one of them whatever call it L let's K times L K times L if K and L are operators is another operator which means that it can operate on a vector a well it means is you first act with L on a to get a new vector l times AE let's say we can write it this way it's K acting or operating on the vector L multiplied by a so L times a gives you a new vector and then you'll hit it you'll hit the whole thing with K so whatever L on a does then hit it with K that defines the operation K times L but now there's an interesting question we can ask if operators are characterized by matrix elements what are the matrix elements of a product like K times L incidentally as most of you probably know K times L is not necessarily the same thing as L times K these are operators they're not numbers one has to check whether you can change the order of them and in general you can't linear operators are not commutative you can't change the order necessarily in which K in which they are act all right so let's consider the matrix elements of K times L what does that mean that means yeah that means sandwich the operator K times L between the basis vectors and in M alright now here is L times M let's write that in another way let's write that by sticky by writing L times M is the sum over yet another index I need a summation index now so let's call that summation index R we take L times M we take its inner product with R and then we multiply it by the vector R again this construction where you see ours and ours nose-to-nose like that it simply means well it simply gives you back the left hand side oops there's a missing bar and here there's a missing vertical line in there okay so that's L that's L times M let's plug it into here and then we can write that this is equal to n K with a little hat on top of it and then let's substitute this in for L times M it's our our summed over our okay now we have a nice construction we have everything written in terms of components let's get rid of this and we can now write that the components are the matrix elements in or let's K times L the nm matrix element of it this is the in emp'd matrix element of K times L what is that given by it's given by n K R that's K n R times l RM summed over R all right again this is simply matrix multiplication written out in longhand the matrix representing KL is simply the product of the matrices of K and L that's what this reads this is longhand notation for the matrix product of the matrix K and L so multiplying linear operators is an abstract idea you simply follow the action of one operator by another operator or it's a concrete idea that you can represent by by matrix multiplication now I'm not going to go very heavily into this because I'm going to assume that everybody knows it among other things it was discussed elaborately in the last class on quantum mechanics which I gave on quantum information and entanglement so I will assume everybody knows how to relate this the matrix multiplication next idea the idea of a hermitian operator mission operators are special classes of operators they play the role in our in in operator theory of things which are real real in the sense of real versus imaginary or real versus complex let's let's define hermitian operator let's spell hermitian operators that's not right what's that where her might yeah her Mike her married it's not a bug it's the name of a mathematician and hermitian hermite is a proper noun hermitian is an adjective hermitian operators what is a hermitian operator well it stands for something which in classical mechanics would be a real quantity a real thing that you can measure such as location of a particle the x-coordinate of a particle the momentum of a particle the angular momentum all of those are real quantities real as opposed to imaginary all right so a hermitian operator and I'm going to call hermitian operator by the generic term H now later on H is going to stand for Hamiltonian for now it just stands for hermitian so let's not worry about hamiltonians yet it just stands for hermitian hermitian operators as all operators have little hats on top of them hermitian operators are defined in the following way their matrix elements are such that if you interchange the bra vector and the ket vector keep the operator the same then all that happens is it becomes complex conjugated if you think about it for a moment if complex conjugating a the complex conjugated the ket vector a gives the bra vector a and complex conjugating B gives you the bra vector B then in order to get a complex conjugate of the left hand side on the right hand side you should also have two complex conjugate H somehow but H the hermitian operators are exactly the ones that you don't have to do anything to in order to get this work to work out let me give you another example supposing we sandwich a hermitian operator not between two different vectors but between the same vector in other words take a H a where a could be any vector whatever well according to this equation its equal to a H a complex conjugate interchanging the bra vector in the ket vector will do nothing if the bra vector in the ket vector are the same but then this equation just says that the left hand side is the same as the complex conjugate of the right hand side and things which are about their own complex conjugates are real all right so sandwich in a hermitian operator between the same state on either side necessarily gives you something real incidentally this can be taken to be the full definition of a hermitian operator all right you can derive the top from the bottom you can derive the top from the bottom I won't bother doing it but it does follow so you can take the definition to be any matrix or any operator which when you sandwich it between the same state or the same vector on both sides gives you something real it follows from that that if you interchange the bra and the ket in a matrix element of H that you get complex conjugate another way to write it H a B is the complex conjugate of HB a in particular it's also true the same thing is true for basis vectors so H m n if m and n stands for basis vectors is equal to H complex conjugate and M where I've interchanged N and M that is the cat out those are called hermitian operators sitting in say it again be hermitian eh no it's a complex conjugate of a hermitian be a hermitian a is always real but the top equation well they either obviously the second equation follows from the first just by setting B equal to AE but it's interesting that the first equation also follows from the second I will leave it to you to to try to prove that it's a it's a couple of lines of proof and if if nobody can prove it I'll do it another time all right so those those are the definitions of hermitian operators in order to understand the significance of permission operators obviously they have a kind of reality property reality again I emphasize reality in the sense of complex numbers they have a kind of reality property but in order to really appreciate their the significance of them we have to go to another concept the concept of eigenvalues so let's review very quickly I'm regarding this as quick review a concept of eigenvalues and eigenvectors of operators now in general when an operator acts on a vector here is a vector an operator acts on it it will do something to that vector one kind of operator might rotate the vector another might reflected about some axis there are all kinds of linear operators but in general it will change the direction of the operator it will take you from here for example to here there may be special directions particular directions which when you apply the operator for each operator given an operator there may be specific directions and there will be for hermitian operators if you have a hermitian operator there will be directions that if you apply h to the upper two vectors and those that it doesn't change their direction okay for example a simple operator are that you can think about would be an operator which takes every vector in any direction and stretches it along an axis stretches it out along one axis doubling its length in one direction keeping its length in the other direction fixed so we might have an operator which takes any vector and doubles the X component of the vector for example leaving the Y component fixed then if I take an arbitrary vector its directional change exactly as you see here but if I take a vector along the x axis then its direction doesn't change the direction stays the same those vectors whose Direction do not change when you apply a certain operator are called the eigenvectors of that operator eigen is the German word for proper I think proper hmm is that what I can means doesn't change I I was once told him improper yeah very very little literal oh okay good all right so the eigenvectors of an operator are the vectors whose Direction don't change but what does happen to them then let's suppose we have it we have a operator I'm taking it to be a hermitian operator now and I find some eigen vector let me call the eigen vector lambda what can happen to it if its direction doesn't change the only thing that can happen to it is it gets multiplied by a number let's call that number lambda now the first lambda here is just a notation for indicating a particular vector if we have if we find a vector that when H applies to it when H attacks it just multiplies it by the number lambda we call that vector the eigenvector of H with eigenvalue lambda eigen vectors I'll call them e vectors and evaluate vector lambda corresponds to the eigen vector lambda if H hitting that eigen vector simply multiplies it by the number of lambda that's it yes well say it again all operators yeah yeah all linear operators yes that is that is true not all operators have eigen vectors all hermitian operators do all hermitian operators and we're going to stay the theorem now now I'm not going to prove the theorem the theorem can be found on the internet version of my last quantum mechanics class on an entanglement and that sort of thing the theorem is an elementary theorem or has various pieces theorem number one all of the eigenvalues of a hermitian operator are what real okay there are these these are all easy to prove statements very easy from the definition basically this is this alone that the matrix element of a hermitian operator sandwiched between the same state is always real that's sufficient to prove it I'll let you figure it out so number one eval use of hermitian operators h or real real numbers okay second part of the theorem or second theorem - very easy to prove as I said the eigenvectors of hermitian operators for different eigenvalues if I find more than one eigenvalue if there is more more than one eigenvalue let's call it lambda 1 and lambda 2 and they're not equal to each other different eigenvalues the eigenvectors are what orthogonal for lambda 1 not equal to lambda 2 in other words for two different eigenvalues the eigenvectors are orthogonal this is not true of general operators it's true of the hermitian operators number three basically number three says there are a lot of eigenvectors of hermitian operators how many is the maximum they can be well they're all mutually perpendicular use they're all mutually perpendicular to each other how many mutually perpendicular up vectors can you find in a d dimensional vector space well maximum is d the theorem says that you can there are always d there exist the mutually orthogonal no for her missions not only at most but also at least commissioned hermitian matrices have D mutually orthogonal eigenvectors if two of the eigenvalues for two different eigenvectors happen to be the same number still you can find orthogonal eigenvectors associated with them in that case the two eigenvalues if they happen to be the same number a called degenerate but let's ignore the possibility of degenerate eigen values for the moment what does this mean this means that the eigenvectors of hermitian operators form basis in other words is enough of them they're all perpendicular to each other you can always choose them to be unit vectors why is it that you can always choose them to be unit vectors how do I know that I can always take an eigen vector to be of unit length unit norm unit in a product of itself well if I have an eigenvector and I multiply it by any number it's still an eigen vector with the same eigen value in other words if I take twice lambda here that 2 just goes straight through H on twice lambda is lambda times twice lambda so if I multiply an eigen vector by any number any are just ordinary complex number could be complex any complex number it still stays an eigen vector with the same eigenvalue so I can always multiply the eigen vectors by numerical numbers until I get them to be of unit length once I have them of unit length and they're all mutually orthogonal they form a basis this is the basic fact two basic facts about hermitian operators their eigenvectors form a basis and their eigenvalues are all real any questions up till now yes whether they exist say it again no rotating rotating axis is not our mission actually I'll tell you it's anti-hermitian will come to the definition out of that in a minute which means if you multiply it by I it becomes hermitian now you can ask well if you multiply it by I and it becomes hermitian then let's mean it has a bunch of imaginary eigenvalues yes rotation operators do have imaginary eigenvalues remember we're dealing with vector spaces over the complex numbers just an ordinary rotation happens to have a complex eigen value but if I were thinking about real vector spaces real vector spaces rotation of axes does not have is not hermitian so that's exactly right what does hermitian become what's the concept of hermitian do we have it here yeah what's the proposing we're talking about a real vector space where all the numbers matrix elements everything else are real numbers what is the mean what is the analog of hermitian the analog have emission we just ignore a complex conjugation no such operation is complex contour everything is its own complex conjugate and symmetric H a B is equal to H be a or H MN is equal to H n m so a special case of hermitian operators would be symmetric real things which are symmetric and real okay a rotation of coordinates is anti-symmetric there's an anti-symmetric matrix so in that sense it's not it is not hermitian so you a basis the vector space the complex vector space given an H yes but let's focus on a particular vector space and now take a certain particular hermitian operator in that vector space okay the collection of vectors hermitian operators act on that vector space give new vectors take that now if you want a concrete representation of it then the concrete representation is in terms of matrices basically this says that every hermitian matrix has a complete orthonormal family of eigenvectors and it's also simple as well yeah in that basis yes in that basis the form of H is very simple but there are it may be a difficult job to figure out what that basis is so I didn't want to I didn't want to belabor that point right now but you're right alright so if you like thinking concretely really concretely and not abstractly as I do then it's easy to think of operators simply as matrices matrices h-11 h-12 blah blah blah blah blah h2 1 dot dot vectors as column vectors a1 a2 a3 and then acting with a matrix on a column vector I will assume you know how to do that gives you a new vector if a is an eigen vector that means that the matrix when it acts on a just gives you a number lambda times the same times the same vector if it is an eigen vector not all like not all vectors will be eigen vectors most won't most vectors won't be eigenvectors but there will be a complete basis of them if H is hermitian so if you're like thinking concretely just think in terms of matrix elements are now we're ready to state the postulates of quantum mechanics the first postulate we've already stated the first postulate is that the states of a system the configurations the conditions describing the system at a given instant of time are described by vectors and the vector space so States equal the collection of ket vectors you could use the bra vectors they're in one-to-one correspondence but let's focus on the ket vectors that's the first postulate for every state of a quantum mechanical system there exist a vector in a in a vector space how many dimensions does the vector space have that depends on the particular system if the system is a coin and that's all and it can be heads or tails then it's a two dimensional vector space there are only two orthogonal states to mutually orthogonal space States if it's a die with six independent states then the dimensionality of the space is six real systems typically have a lot more States than that then for example a particle moving in space could could be located if we look for it in any position so we would have an infinite number of states but for each state there is a a a ket vector so that can be taken to be postulate number one now the number of postulates I'm going to write down is a little bit more than is absolutely needed to minimally xima ties quantum mechanics but it's it's nice to write them all down in a way which exposes the ideas and not try to be absolutely minimal okay yes say it again you don't observe the states of systems you observe the observables the observables correspond to the hermitian operators but we'll come to that you don't nobody has ever directly observed the state of a system are you observe the things which can be observed which are things like position momentum electric field magnetic field things which record measurements on an apparatus and they never never never give you enough information to be able to record the whole state of a system so you would never call the state a state of a system and observable we'll talk about later on at some point we'll talk about how you prepare state I don't know if we will that was talked about a previous quarter but the moment states of systems are described by vectors that's postulate number one postulate number two there's the things that you observe now observe means you make measurements of them you could call them observables you could call them measurables you could call them the things that you measure in an experiment the things that you measure an experimental call an observables it's the usual term for them the observables correspond we should really write this not equals but corresponds to corresponds to the collection of hermitian operators the collection of hermitian operators every hermitian operator corresponds to something that you can measure in principle not any old operators but the collection of hermitian operators number three third postulate the values you're going to do an experiment now and you're going to observe a particular observable that corresponds to a hermitian operator H what are the possible results that you can get of the measurement you make measurement of H whatever H is and you get a value the possible values that you can get a collection of values that you could get the values of the observable H are the eigenvalues of oops of H that's the meaning in the significance for us of eigen values that the values of the quantity that you can get when you measure it so they have their important is the point not only are they important as I said there are the things which you get in an experiment what else oh and remember any time you do an experiment of a quantity you get a number and it's always a real number if I measure the position of something I don't get a complex number I get a real number if I measure the x-coordinate or if I measure the angular momentum or I measure you know whatever electric field I get a real number so it's important hmm no they usually if you measure the angular momentum there's no error barque in the imaginary direction you may have error bars in the real directions it's a real number okay therefore it's important that hermitian operators have real eigenvalues after all those are the things you can measure furthermore the states or the states which if you find the system in that state or create a system with a particular state the states for which the observable H has a definite predictable value a certain value a value which is not subject to statistical fluctuations those states for which the observable H is let's say is definite or or certain as opposed to uncertain are the eigenvectors of H the eigenvectors of age in other words if by one means or another you created an electron in an eigenstate of some observables such as its position and then you measure the position the measurement will always yield every time the eigenvalue of the appropriate operator the position operator so the eigenvectors are the states in which the observable has a definite value and the value that it has is the corresponding eigenvalue any questions yes say it again well um I think it's more proper to say that you observe the observable and get a number and the number corresponds to the eigenvalue I would say that's more proper terminology and a more proper way to think of course once you do that then you know that the system was prepared in a particular state and in that sense you could say you observe it but I think it's a sort of linguistic misuse to say that you observe eigenvectors it's better to say you observe the observable and the result that you get is one of the eigenvalues of the observable and I think that's a more precise language say for yes okay all right so let me give you an example let's take an example such as the spin of an electron some particular component of the spin of an electron if you measure if you prepare an electron in an arbitrary state how do you do that you put the electron in a magnetic field go back the course on quantum mechanics number one and read about how you prepare an electron with its spin along a particular axis supposing you did that and then you measure the spin along some other axis then you may get plus or minus you may get with a certain statistical probability you will get different answers but if you measure the spin along the axis that you prepared it along then you will get a definite answer you'll get a definite answer corresponding to the spin that you're the way that you prepared it so there are certain states which if you measure certain quantities you will always get the same answer you do a certain experiment on this run experiment I mean you place it in a field you do something to it and at the end of doing something to it you measure something sometimes the measurements will give you statistically ambiguous answers with a probability distribution depending on what you measure and depending on what state it's been put in other times if the thing you're measuring corresponds correctly to the state that you've created the system with then you may get definite answers we'll we'll see some examples ah definite means there's no statistical uncertainty and the value that you get so there are certain states for each observable there are certain states where you will definitely get a particular answer and you'll always get the same answer namely if the system has been prepared in an eigen vector of a particular variable and you measure that variable you'll always just get the eigen value all right so that's another that's the fourth postulate and the fifth postulate is what happens if you start with a state which is not an eigen vector of the quantity of interest so in other words supposing instead of preparing a system with a state which happens to be an eigenvector of H you prepare some other state somehow and I'll give you examples we'll come to it but let's state the postulates for number-5 take an arbitrary state of a system made God knows how however it was however the system was originally created an electron may be created created by shooting it out of an accelerator or something in some particular way we make the electron in some particular state and we now measure something or other about the electron okay we measure the H Ness of the electron what can we get we can get answers which are the various eigen values let's call them lambda 1 dot dot dot dot lambda n these are all the possible answers we could get the various eigen value eigen values of lambda for each of these eigen values there is an eigen vector we can call it I don't know let's just call it lambda n that's the eigenvector that corresponds to mental definite value of H with a particular eigen value then the fifth postulate is about the probability of getting different answers lambda 1 through lambda n the answer is that the provid postulate is that if you calculate the component of the vector a along the axis lambda n okay think of these lambda ends as a basis set or a basis set which goes together with the particular observable made up out of the eigenvectors of that observable take the component of the vector a along the direction lambda n that's a complex number in general in general this is a complex number could not possibly be a probability but multiply it by its complex conjugate so multiply this by its complex conjugate one way of writing it is take the absolute value of it and square it another way of writing it is to just multiply it by its complex conjugate sorry lambda n multiply it by its complex conjugate its complex conjugate is lambda n times a you just invert bra and ket and that gives you the complex conjugate so take a no summation here a projected onto lambda N squared or complex conjugated that gives you the probability that's the probability that you measure lambda n if the system has been prepared in the state a so whatever went in the constructing a particular configuration of an electron the particular way that it came out of the excel a particular run of magnetic field electric field various conditions under which the electron was created to find the state a having defined having created the electron or whatever the system is you can then ask what's the probability that if I measure a particular observable H that I get any one of its possible eigenvalues the answer is the component of a along the eigenvector lambda n times its complex conjugate this is always a real positive number okay it's always a real positive number and so it can be a probability that those are the five postulates of quantum mechanics there are ways of diminishing the number of independent postulates but I think it's better to expose them all and and to say them all and not try to be clever and reduce the number of independent postulates okay well I will give you some examples soon enough I was going to talk about incompatible observables but let's not at this point let's let's do some examples the simplest examples correspond to systems with finite number of states like the coin the coin toss or the dice or so forth but I don't want to begin with that we've done that we did that repeatedly in the previous class in quantum mechanics and quantum entanglement and so forth which is basically about these simple systems with finite numbers of states let's jump right now to the motion of a particle on a line supposing we have our system consists of a particle in one dimension the particle can be anywhere as on a line it can move on the line classically we would just describe this by a particle with a coordinate X which could depend on time quantum mechanically we describe it completely differently very differently we describe the states of the particle by a vector space what vector space well I'll tell you right now what vector space the space of functions of X remember when we started and I gave you some examples of vector spaces one of the examples of vector spaces was functions of a coordinate complex functions of a coordinate sy of x sy of X corresponds to a vector in the vector space of complex functions we can think of it as a vector in a vector space because we can add functions and we can multiply them by numbers okay we can take inner product of these vectors let me remind you of the rule if I have two functions Phi of X and sy of X then the inner product between them is just the integral over the line the X of Phi star of X Phi of X Y Phi star of X because Phi is the bra vector sy is the ket vector so whenever you have a bra vector it always corresponds to some complex conjugation that's the definition of the vector space for a particle on a line the vector space can be thought of as as functions on the axis well actually it can be a little more abstract than that we can think of these functions differently we we can well let's not let's not be more abstract we can come back and be more abstract later all right let's think about operators now yes which no no inner products are not real in general what are real is the inner part in a product of a vector with itself all right so if I were to put sigh sigh that would be the inner product of size star with sigh that would be real okay all right good question keep the questions coming because what's that say it again here here and here yeah absolutely just common garden-variety complex multiplication right so that's the inner product now let me give you an example of some operators we can check that they are really operators check that they're hermitian find their eigenvalues interpret them as observables and see what their probabilities correspond to the first operator is the observable that corresponds to the location of the particle along the axis and I'm going to tell you what that operator is and how it acts on sigh of X we'll call that operator just X X with a hat on it and it operates on a vector psy in terms of functions what it does is it takes a function of X Phi of X and does something to it it does the simplest possible thing it multiplies it by X that's its definition the operator X hat acts on the ket vector sigh well I should say corresponds to not equals in such a way that the new vector if the old vector was represented by psy of X the new vector the result of multiplying it by the operator X hat is just to multiply the function by X very simple well what would it mean to have an eigen vector of the operator X hat in terms of ordinary functions it would mean that or what does it mean first of all in terms of the abstract notation the abstract notation X hat on psy of X well no you can't multiply a ket by X you can only multiply it by the operator X hmm oh oh oh good good good good yes you could do that thank you yes that that would be legitimate in other words it's the ket vector whose representation is x times I of X and just instead of sigh of X yes that's fine that's good okay are there eigen vectors well what would that mean that would mean that X hat on psy of X is just equal to some number Oh before that let's check that it's hermitian let's check that it's hermitian what does it mean that it's hermitian are the necessary and sufficient condition is that a hermitian a is real for all a that's necessary and sufficient for a hermitian operator for any for any vector a ok let's just check that all that means is that psy of X X hat sai of X is real but what is that x times I of X just corresponds to the vector X I of X just corresponds to the function X I of X taking its inner product with the bra vector psy of X means multiplying it by size star of X and integrating this is surely real so I of X x sized star of X is real X is real DX is real this is a real number all right whatever sigh is this is always real so it follows that the inner profit the that the matrix element of X between equal vectors is always real that's necessary and sufficient for X to be a hermitian operator so X is hermitian that must mean has a lot of eigenvectors so let's see if we can find the eigenvectors the eigenvectors abstractly are defined by x hat on psy of x on psy is equal to a number I'm sorry if we can find functions which satisfy that those are the eigen functions or the eigenvectors the eigenvectors of x hat what does it mean in terms of ordinary functions all it means we erase this all it means is find the function phi of x so that when you multiply it by x you get a multiple lambda of the original function now that looks a little absurd doesn't it a function such that when you multiply it by x just gives you a number times the original function if you think about it for a minute that's quite impossible well it is impossible strictly speaking but you can get very close to this so let me show you what's involved let's write this as X minus lambda times I of x equals 0 all I've done is transpose the right hand side over to the left X minus the number lambda on psy of X is equal to 0 this should be thought of as a an equation for a function psy of X and it must be true for all X when you write a thing like this you mean that it should be true for all X well here's product of two things which has to be zero according to the equation what do I know if a product is equal to zero I know that at least that at least one of the factors has to be equal to zero yes we will assume that we will assume that the integral of size star psy always converges we'll find out why in a little while converges in particular that means psy goes to zero at large distances whatever psy is it should go to zero far away do whatever it wants from here and then go to zero your body the interpretation of that will become clear in a little while so it doesn't leak off to very very large distances and get bigger and bigger it goes to zero assumption fast enough to make this integral converge okay so let's take X minus lambda psy of X equal to zero what does this equation tell us it tells us that anywhere is where X is not equal to lambda is lambda right over here x equals lambda right over here any place where X is not equal to lambda psy has to be equal to zero that means the only place where psy is not zero must be where X is equal to lambda at X equal to lambda you can have sine not equal to zero because at that point X minus lambda is equal to zero anywheres else if this equation is to be true psy has to be zero so let's plot what psy has to look like so I is a function which is zero everywhere except that X equals lambda as x equals lambda right there so it's zero everywhere except that there's one point where it can be nonzero now function like this is not a nice continuous sensible function in particular you can't square it and integrate it a function which is only nonzero at a single point and so what directed was to invent a kind of idealized arm general generalization of a function the generalization of the function is probably well known to most of you and it's called the Dirac Delta function at the moment I don't want to get deeply into the mathematics I'm more interested now in illustrating some of the concepts that we've gone through so let's imagine that what this really means is that it's a function which is nonzero over a tiny tiny interval in the end we'll allow that interval to shrink to zero let's call the size of that interval epsilon epsilon for small number epsilon and the height of the function will take to be 1 over Epsilon that's not in itself very important at the moment but just to put some area under that function let's make the function as high as it is narrow that's a function which is zero everywhere except in this tiny interval near x equals lambda near x equals lambda it's a high narrow function which is zero everywhere else except in that tiny interval that function is what Dirac called the Delta function Delta of X minus lambda it's zero except when the argument is equal to zero except when X minus lambda is equal to zero then the Delta function is a high narrow spike but when X is not equal to 0 or when X minus ly there is not equal to zero the function is zero that's called the Dirac Delta function now to give a precise mathematical definition you get uncomfortable with functions which are ism as loosely defined as this and you should there is a precise definition of the Delta function we may omit yeah that is yeah when when yes it's not orthonormal it's ortho but not normal the reason it's ortho let's let's see if it's ortho first of all ortho means or well first of all let's prove that this is an eigen well of course it's an eigenvector of X it satisfies this equation X minus lambda times I of X is zero either because sy is zero or because X is equal to lambda so yes it is an eigenvector it's an eigenvector with eigenvalue lambda but let's first ask about ortho before normal ortho means that if I have two different eigenvectors with two different values of lambda that they must be orthogonal to each other so let's draw another eigenvector corresponding to x equals lambda prime that's a function which looks like this now as I said we will have to imagine in the end that this epsilon shrinks to a smaller and smaller size but these two functions are orthogonal to each other because when you multiply them together let's call Phi and sorry when you multiply them together one or the other is zero everywhere when you multiply this by this one or the other or both a zero everywhere is wherever this one is nonzero this one is zero wherever this one is nonzero this one is zero so when you multiply them together you always get zero and therefore the inner product of them is certainly zero on the other hand if you take the eigenvector and take its inner with itself you'll be squaring this high narrow function and the answer won't be zero so they're orthogonal to one another normalized worried about another day they're not the amount of unit norm okay this is an example of an orthonormal orthogonal family of functions which are the eigen vectors of the operator x 1 eigen vector for each value of lambda for each possible position along the x axis now in fact we've even found out what the eigen values are the eigen values are simply all the possible values of X along the real axis we could erect one of these Delta functions anywheres any place we erect it it will be an eigenvalue or sorry an eigen sometimes I use the word eigen function eigen function is another word for eigen vector it's an eigen vector of the operator x with eigenvalue lambda and lambda can be anything on the real axis so that's our first example of a hermitian operator a spectrum of eigenvalues spectrum just means the collection of eigenvalues orthogonal 'ti of the different eigenvectors and now let's discuss what a wave function really means sigh of x is called the wave function but let's let's discuss the following for simplicity I'm going to take the height of this function to be 1 over the thickness of it and that means the area under it is 1 the area under a Delta function is 1 that's a definition of the Delta function well let's take sigh of x and take its projection it's in a product with the eigen value with the eigen vector lambda eigenvector lambda corresponds to a particle located at position lambda in other words it's the eigenvector it's the eigenvector of the position and it's the state in which the position has a definite value what is this inner product this is the inner product gotten by taking sine of X and integrating it with the wave function corresponding to lambda but what is the wave function corresponding to lambda it's Delta of X minus lambda DX this is the inner product of psy with the eigenvalue lambda with the eigenvector lambda so I of X integrated with the eigen vector this function is equal to something special it so u equal to something very simple remember that sine of X is zero everywhere is that I could accept that x equals lambda if it's zero everywhere is it except x equals lambda except x equals lambda it's only sensitive to the value of psy at x equals lambda in other words over this whole tiny tiny range here the only values of psy which are important are the values of psy at x equals lambda since the height of the function since the area under the function is 1 it's easy to prove okay here's that here's the argument between this point and this point psy doesn't vary very much assuming that psy is nice and continuous and if we take this interval sufficiently small psy will not vary very much across the Delta function all right what is it right at the Delta function it's just equal to sigh of lambda so to a good approximation and a better and better approximation as I shrink this interval here I can simply replace sine of X here by psy at that one point psy of lambda times Delta of X minus the integral sigh of lambda now can be brought outside the integral this doesn't depend on X anymore so we bring it outside the integral and we just have to integrate Delta of X minus lambda the integral of Delta of X minus lambda is 1 because I've defined the area under the function to be 1 so we just read off from this then that this is just equal to sigh of lambda this is equal to sigh of lambda in other words the wave function itself sine of lambda or we could call its I of X we could replace X by X here we can read this as saying the inner product of psy with a state which is localized at position X is just the wave functions I of X that's what's I of X is it's just this inner product where this inner product now where I've changed notation instead of calling the eigenvalues lambda i I've called them x themselves now this corresponds to a wave function localized right at X ok so where are we now we now know mathematically that sigh of X is sort of just a component of the vector sigh along the basis vectors X but now let's use the postulate what have we done we have erased all the postulates oh my goodness we really believe what was the fifth postulate not Euclid's fifth postulate but the quantum fifth postulate was about probabilities it said that the probability for getting a particular lambda is the square of the vector projected onto the eigenvector lambda in this context it says the price R squared in this context it says the probability for detecting a particle at position X the probability to detect the particle at position X is just equal to sy X quantity squared or absolute value squared but what is this IX it's just sy of X times size star of X in other words we've now found out what the meaning of sy of X is that it's the thing that you score out it's not the full meaning of it but a partial meaning of it is it's the thing whose absolute value squared is the probability to detect the particle at X so we've used the postulates of quantum mechanics to determine in terms of the wave function what the what the probability to locate a particle at X is ya know I mean so I could be any old function but for any old function there will be a probability distribution whatever sy is whatever sy is and so I can be complex so I need not be real it can be negative in places so sy is some complicated complex function that can be negative positive and imaginary and all those things if you take it and multiply it by its complex conjugate you'll get something real and positive that real positive thing is the probability to find the particle at different locations on the x axis that's the implication of the postulates of quantum mechanics in particular it says that probabilities are given by the squares of certain complex functions now if all you get out of it was the probability for for finding particles in different places you might say why the hell don't I just define the probability as a function of X why do I go through this complicated operation of defining a complex function sigh and then squaring it the reason is because there are other things that you could measure besides the of the particle there are other hermitian operators which correspond to other observables in particular let's think about other possible hermitian operators I'm just going to give you another simple one the simple one corresponds to a very basic thing in quantum mechanics I'll name it as we go along but before I name it let's just define it in abstract the operator sense not abstract a concrete operator sense again we're still doing the particle on the line its states are described by functions Phi of X in other words it's the vector space is again the functions of X same exact set up as before but now I'm going to think about a different observable a different observable which is characterized by a different operator the operator now is not multiplication by X multiplication by X is what we did over here now a different operation differentiation just differentiate sigh of X that's an operation in fact it's a linear operator ah if I differentiate I can differentiate the sum of two functions I can multiply the function by a number if I multiply the function by a number its derivative is just the numerical number times the times the derivative the original function I can add functions and differentiate them D by DX is a linear operator acting on the space of functions I of X so this is another operator does it correspond to an observable not quite not quite because it's not hermitian it's what's called anti hermitian anti hermitian means all you have to do is multiply it by I to get something hermitian but let's first prove that it's not hermitian in fact I'll prove that it's anti hermitian let me tell you what anti hermitian means I'll just define it right now it's a class of operators that were discovered by her mites ant anti-hermitian he our anti-hermitian Ibrahim mission means our mission means that a Sai B is equal to the complex conjugate of B OMS inside H excuse me h h a complex conjugated that's the meaning of a mission in particular it says that a hey I don't and why not don't know I think I do yeah all right anti-hermitian says that a HB is equal to minus B H a that's called anti-hermitian if you spend five minutes you realize that for every nth oh this is anti-hermitian let's call it h what about the Hat you mean this notation here yeah we're splitting hairs but properly so I don't know where to do it I don't know what they call it let's just call it a anti-hermitian yeah yeah sorry star it's exactly the same as permission with an extra minus sign it's very easy to prove that if you have an anti hermitian operator if you multiply it by I it becomes hermitian and the reason is because the complex conjugate of I is minus I if you have an anti hermitian operator and you multiply it by plus or minus I let's say - I oh boy this looks like I doesn't it okay anti hermitian anti hermitian it's not a times h it's anti hermitian if I take an anti hermitian operator and multiply it by I I get a hermitian operator that's a little theorem to prove do it yourselves but it just follows from the fact that the complex conjugate of I is minus I so no H bar please not now you're confusing me okay but in fact D by DX is anti hermitian but let's just see why it is that ah that D by DX is not hermitian and why you discover an extra minus sign there I'll tell you what let me bit let me cut it short let me cut it short by simplifying you pay right now that the thing which is hermitian is I times D by DX in other words or - I doesn't matter I on - I - I D by DX on psy of X is or minus ID by the X's and operation it's a linear operator on the space of functions and it's a hermitian linear operation let's see if we can prove that let's see if we can prove that it's hermitian the simplest way to prove it is to prove that let's call let's give it a name I'm going to give it the name K hat I'm not going to call it H I'm going to call it k but its hermitian K hat on psy of X the simplest way to prove it is to prove that for any psy psy k psy is real now that sounds a little odd because K has an I in it it looks awfully much like it's going to be imaginary but it's not let's check it we have to calculate size star that's the left that's the bra vector K hat is - I D by DX times sy or just decide the X now it doesn't look real but it is real the prove that it's real we want to prove that it's its own complex conjugate so let's prove that this thing is its own complex conjugate and the way we prove it is by integrating by parts does everybody know how to integrate by parts integrate by parts is a very simple thing if you have the product of two functions F of G F times V G by D X and you integrate the product of a function with the derivative of another function the answer is minus G times the derivative of F you simply interchange which of them is differentiated instead of differentiating G we differentiate F and you throw in an extra minus sign that's called integrating by parts it's a standard elementary calculus theorem what am i missing out of this the endpoints of the integration but the argument for the endpoints is that I've chosen functions I which go to zero far away if they go to zero far away I don't have to worry about the endpoints of the integration so let's integrate this by parts to integrate it by parts I simply throw in another minus sign this must be equal to plus we have to change the sign plus I times the integral and now I interchange which of the which of the things gets the gets the complex car or gets the derivative it becomes the size Staller by DX times I that's this all right so I have this is equal to this integral psystar times - I decide by the X is plus I times integral psi star by DX now I assert that this the second term the second expression the right hand side is simply the complex conjugate of the top let's check that let's take the complex conjugate of the right hand side of the equation the complex conjugate of the right hand side first of all has the complex conjugate of ie so let's say well I'm going to rewrite the complex I'm going to rig to do that I'm going to write the complex conjugate of this not all right equals just complex conjugate here it is Cici complex conjugate what's the complex conjugate of ie - I what's the complex conjugate of sigh sighs star right what's the complex conjugate of the sized star by the X beside the X but look the complex conjugate of the right hand side is just the left hand side - I sighs star decide by the X so what have I proved I've proved that this is its own complex conjugate by integration by parts I prove that the left side of the equation is equal to the right side but then I prove that the right side here is just the complex conjugate of the left side though yeah so I prove that this is its own complex conjugate if its its own complex conjugate it means that it's real and it means that minus ID by DX is a hermitian operator the extra sign or this it's a little surprising you might have thought that just D by DX is hermitian but it's not okay so minus I K which is just minus ID by DX is a hermitian operator let's find out what its eigenvectors are if it's a hermitian operator it must have a lot of eigenvectors let's see if we can find out what the eigenvectors are and then see if we can interpret them and figure out what observable what quantity that you can measure this operator K corresponds to so here's the eigenvalue equation let's write the eigenvalue equation - i decide by DX should equal this is the action of k on side this is K on sorry K is minus ID by DX if it's to be if sy is to be an eigen vector what does the right-hand what does the right side have to be lambda times I where lambda is the eigen value let's call the eigen value K just to call it by its traditional name let's call it little K this corresponds to an eigen value x I of X so we're looking for functions the eigenvector functions are those whose derivatives so - I derivatives are equal to K times I of X itself everybody know how to solve an equation like this anybody know how to solve an equation like this Exponential's Exponential's the solution of this equation is that psy of X is proportional to e to the I KX okay let's check that rather than solve it let's just check that this is the solution of the equations e to the ikx of course means cosine KX plus I sine KX let's check that let's check that minus ID by the air decide by the X is K times I so to differentiate this every time you differentiate it it pulls down the coefficient I K so decide by the X is equal to I K times e to the I KX which is just I K times I now I want minus I times this so let's multiply by minus I what's minus I times plus I is 1 so we find out that this function here is an eigen vector of the operator minus ID by DX so in fact we have found the eigen vectors of the operator minus ID by the X and they are simply the functions cosine X plus I sine X let's think about those functions for a minute those functions are very very different than the eigenvectors of X the eigen vectors of X are these highly peak Delta functions these functions here are spread all over the map the map meaning X they're spread all over the map there are oscillations which endlessly go on and on cosine looks like this sign looks like let me draw a sign in blue yeah I can't I can't draw a sign no but I can draw a sign and that's it don't add don't add the blue to the black multiply the blue by I before you before you add it to the black but I can't draw that let's say it starts here and and it has the same wavelength I don't know has the same wavelength but it's 90 degrees out of phase we know what sine and cosine look like I'm not going to try to draw them all right this is the eigenfunctions and it's got a plus I cosine KX plus I sine KX all right what is the magnitude supposing I multiply sine of X by its own complex conjugate what do I get one the number one psy times I star is just 1 e to the ikx times e to the minus ikx e to the minus ikx is the complex conjugate of psy psy times I star is 1 remember yes this is an ambiguous constant in here all right but whatever that constant is size star psy is constant that means that the probability to find the particle anywheres in space is uniform complete uniform probability distribution there's no information in this where the particle is but the function oscillates and those oscillations must mean something ok let's see if we can figure out what they mean by using first of all let me say this Frizzle according to the postulates K corresponds to some kind of observable its eigenvalues are little k here just numbers any number from minus infinity to plus infinity will do will be an eigenvector will be an eigenvalue what are we call this operator well we've got to give it a name to give it a name I'm going to relate this to some very primitive observations about quantum theory before real quantum mechanics was discovered going back to Einstein and de Broglie this is for the purpose of giving it a name yeah yes yes yes you're entirely right and in fact there are no true eigenvectors of sigh of x which go to zero and infinity but you can find you can find functions which are arbitrarily close to this arbitrarily close to solving this by taking functions which oscillate and then faraway gradually diminishing in magnitude you are right I don't want to get into that now are the normalization of the functions but you all right and one has to do a little bit of explanation here about why I'm allowed to get away with functions which don't go to 0 at infinity I didn't want to do that now I want mostly I wanted to illustrate the principles by working out some examples of eigen vectors eigen values and so forth so allow me the freedom to ignore that issue of whether the functions go to 0 for the moment at least they don't blow up at least they don't get big and that is important at the moment I want to find out I want to name this observable if I name it I mean I want some intuitive connection with something that I already know all right so the eigen functions the eigen functions after all are the wave functions which correspond to a definite value of this observable what shall I call that observable well first of all let's think about the wavelength of these oscillations these are waves all right the wavelength is this distance the wavelength is the distance that you have to move before cosine KX comes back to itself cosine K cosine zero starts at one how far do you have to move before cosine comes back to warn well the answer is 2pi but it's not x equals 2pi its KX equals 2pi right so the wavelength corresponds to K times the wavelength L is equal to 2pi or the wavelength here l is equal to 2pi divided by K so we can either characterize these waves as having a certain value of K a certain eigen value or we could equivalently characterize them by their wavelength and their wavelength is L right now next we make use of a intuition which goes back to the Broglie and also to Einstein that if we have particles corresponding to waves of a given wavelength those particles have a certain momentum now we're only using this to name a certain thing I cannot legitimately claim at this point to explain to you why we're going to call something momentum we will do that later in order to see that something is really behaving like momentum we have to do a lot more but for the moment I just want to put you know sort of a what should we call it just throwing a bunch of stuff together that that we've seen earlier in order to give this thing a name a familiar name all right what the Broglie said is that if we have a beam of particles corresponding to a wave of wave length L that those particles have a momentum and the momentum let's call the momentum P it's the same term that we used in classical physics anybody remember what the Broglie said P is equal to H over L this is also the thing that Heisenberg used the shorter the wavelength the higher the momentum of the corresponding particles and it's plunks constant H which goes in there so if you have a set of photons for example a set of photons which are described by a class cool wave of wave length L then once tine new in which you didn't say but with the Broglie reinterpreted and generalized to any old particles not just photons is that the momentum of those particles described by a given wave is given by H divided by L well L is 2pi over k so let's plug that in that says that this is H and then we have to turn L over so that's K divided by 2 pi H over 2 pi has another name it's called h-bar so this is h-bar plunks other constant times ke momentum is h-bar times K so we now have an interpretation it's an interpretation that we're going to have to check later when we understand the connection between quantum mechanics and classical mechanics momentum is a classical concept we're now using sort of seat-of-the-pants old-style quantum mechanics the intuitive confused ideas of that were before Heisenberg and Schrodinger but let's use them and justify them later that wavelength and momentum are connected in a certain way where is it wavelength and momentum are connected in a certain way and if I then plug in I find that momentum is connected to K momentum is h-bar times K do I have that right yes yes momentum is h-bar times K well that now interprets for us what the physical meaning of the observable K is it's just the momentum of a particle accepting units of h-bar K is just the momentum of a particle in units of h-bar if you don't like calling it momentum don't call it momentum just call it the observable K all right you can just call it the observable K and you see that it's eigen values and it's eigen functions are just the numbers k and the wave functions e to the ikx the interesting point here is that whatever K is it is of course it is of course the momentum but whatever K is the eigenvectors of it are completely different than the eigenvectors of the position x the eigenvectors of the position x are these high narrow functions which are also slightly ill-defined and the eigenfunctions of momentum are these functions which oscillate on and on forever and ever and have a completely uniform probability distribution in each case is something mathematically a little wrong in one case this is clearly a thing which needs a better mathematical definition and in the other case we are allowing functions now which don't go to 0 and infinity these two things are related but without being too precise and without being too mathematically rigorous we now see that whatever these object are whatever these wave functions are they're completely different for position and momentum input in particular no eigenfunction of position is also an eigenfunction of momentum well that sounds familiar it sounds something like the uncertainty principle if the eigen functions of position are the states in which if you measure the position you would know exactly what it was and if the eigenfunctions of momentum are the wave functions which corresponds to a definite value of momentum it's quite clear that they sort of clash with each other if you know the position of a particle that's equivalent to saying that you know that its wavefunction is peaked like that if you know the momentum of a particle it means that the wave function is spread out all over the place it can't be both and that of course is the source of the uncertainty principle which we don't have we're not going to do tonight but it's the source of it and it's an example of incompatible quantities quantities which correspond another way to say it is if you think of the eigenfunctions of position is forming a basis of vectors in a very high dimensional vector space but I'm going to draw it is just three dimensional then the eigenfunctions of momentum are simply different eigen vectors pointing in different directions so that no eigenfunction of position is also an eigenfunction of momentum and no eigenfunction of the momentum is an eigenfunction of position it corresponds to different basis sets which are at angles relative to each other we've now kind of gone through one example of a space of a vector space the space of functions of X and the interpretation of a couple of different linear operators the linear operator corresponding to position and the linear operator corresponding to momentum what would we really have to do to see that it's more or less clear intuitively that a wave function which is peaked at a location in space corresponds to a particle at that location of space what would we really want to do in order to see that that this other observable really does correspond to the momentum of a particle well momentum is a classical concept and in order to understand in what sense this thing corresponds to momentum we have to be able to understand the limit of quantum mechanics in which it behaves like classical mechanics it's quite obvious that there must be such a limit if quantum mechanics governs everything it governs electrons that also governs bowling balls but bowling balls are very heavy and they correspond to a limit of quantum mechanics a limit of quantum mechanics of heavy objects what we will have to do in order to see that this object that I've called momentum really behaves like classical momentum is to understand that limit better and we will we'll come back to that limit for the moment calling K momentum or calling h-bar times K momentum is just a name it's just a name for an object name for an observable okay any questions yes I agree so let me let me say um good so I mean let me just restate what you just said more completely a complex number e to the ikx is a number whose magnitude is 1 e to the ikx times e to the minus ikx is 1 which means that it's a number in the complex plane which lies on the unit circle okay it'll cross it's equal to cosine of something cosine KX plus i sine KX now if I imagine the complex plane being oriented this way and the x-axis oh this is different X sorry this is a sy Society sy is a point on the complex plane and if it has if it's of the form e to the ikx that means it's on the unit circle here sorry on the unit circle this would be the real part of sy real part of sy in this direction and the imaginary part of sy in that direction let's not confuse x and y with the position of the particle now supposing I draw the complex plane this way remember the wave function is a complex function and the x-axis the position of the particle that way that means that at every point of space I could draw the complex plane and plots I as a point on the complex plane at every X so at every X sy is a point on the complex plane and in fact for this kind of function it's a point on the unit circle so as was pointed out what it corresponds to is a point which as you move along wines around like a helix wines around on the units go like a helix that may help some people helps me small K means very long wavelength means it winds slowly small K means large wavelength right large K is small wavelength so large K means that it varies varies very quickly right so large K means it whines very quickly as we move along small K means that it whines very slowly as we move along right yeah that's it that's a good point any other questions or comments some of the comments are helpful so phosphorous didn't say anything about like why laughs no not at this point not yet not yet not yet um one of the point one of the point that we haven't talked about which we will talk about next time we have not talked about the probability for different Momentum's we talked about the probability for different positions it was just sighs star of X I of X we have not talked about the probability for different momenta right so we'll talk about that next time a little bit and right we have not talked about basically if not talking about the measurement process itself and what it does to the system the actual act of measurement we only said the probability for a given measurement is given by this or that yeah BK is small L Big L and vice versa that's what we just said no II didn't we said small K corresponds to slow variation which means Big L large K corresponds to rapid oscillation which means small right well okay momentum we usually think of as having to do with the X by DT right what we do I so let me then let me state it for you then there's an approximation to quantum mechanics which is good for heavy particles and heavy objects and in that approximation the wavefunction moves around tends to form a lump and this lump moves around what we're going to want to prove is that this lump moves around in more or less the accord with the classical equations of motion and we're going to want to prove that the velocity that it moves around with is related to how fast the wave function is varying so if we have a wave function which varies quickly like that so that it corresponds to a large K we're going to want to discover that it moves across the blackboard faster as if it was a high momentum particle but to do that we have to understand how things change with time we haven't even brought time into it yet so the correspondence between momentum and what we normally call classical momentum is through the question of how wave packets move around we're going to want to see that wave packets move around in accord with pretty much the classical equations of motion good any other questions about one more minute before I collapse before my wavefunction collapses you know how many hours I've been teaching today for for my father used to work 18 hours a day you know but you're supposed to feel sorry for me the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Channel: Stanford

Views: 270,166

Rating: 4.8010659 out of 5

Keywords: relativity, Physics, math, calculus, geometry, algebra, statistics, quantum, mechanics, linear, operators, vector, space, hermite, hermitian, states, observable, values, eigan, complex, numbers

Id: epzh76hNl8I

Length: 116min 50sec (7010 seconds)

Published: Thu Apr 10 2008