Lecture 4 | Modern Physics: Quantum Mechanics (Stanford)

this program is brought to you by stanford university please visit us at stanford.edu let's just review what happens if we have some quantum mechanical problem involving some discrete variable that could take on some values uh finite number of values from 1 to n or something like that little n little n could be a discrete variable variable going from 1 2 3 4 and so forth up to some number uh something that we can measure and if we measure it we'll get a number and that number will have a probability the probability let's call p of n standard rule the sum over the probabilities should add up to one total probability should add up to one and so always sum of probabilities should add up to one that's one rule another rule related to it not completely disconnected is that probabilities are governed by amplitudes amplitudes are related to the inner product of state vectors with other state vectors let's just write down one or two simple rules first of all the basis vectors that are associated with n let's call them m and n can be taken to form an orthonormal basis that means each of the vectors has length one and they're mutually orthogonal and that we can summarize by writing that that's equal to delta n m the chronic or delta symbol which is 1 if n is equal to m and 0 if it's not now we can always think of delta nm which looks like it's a function of two variables two discrete variables we can always think of it as really a function of only one discrete variable namely n minus m we could call it delta of n minus m with the rule that when n minus m is anything but zero delta of n minus m is equal to zero if n minus m is not equal to zero and delta of n minus m is equal to one if n minus m is equal to zero so in fact delta of n minus m is a function of two variables but it has a particular form and it's really only a function of the difference between them zero if there's a difference non-zero if there's not a difference i just point that out as something to uh to keep in mind if we if we plot the probability as a function of n then it has its values on the integers and so it one integer maybe one value another integer maybe another value another integer and so forth and so on and it's these heights which add up to one that's the picture for a discrete variable if like the position of a particle if the variable that we're measuring is continuous if it can take on any real number for example or take on some value over a range of real numbers perhaps from zero to one or more simply any real number altogether then we have to change the rules a little bit we have to think a little bit what we're doing it doesn't make sense to ask what's the probability that the coordinate of a particle is three the probability that it's exactly three exactly three oh boy we're getting oh this is a fancy uh please come and come and join me i'll i will have some the probability that a continuous variable is equal to any exact particular number is infinitely small a particular number is a set of measures zero and so the probability for any particular value is zero but what we speak about is the probability that the position for example is somewhere in some range of values and we just we define a probability density a probability density let's call it let's still continue to call it p it's a probability density as a function of x and it's a continue typically it's a continuous function for some reason i don't seem to have any decent pins tonight all right so it's typically some some continuous function could be something like that because it doesn't have to be a bell-shaped curve like that it could have bumps and so forth but a bell-shaped curve will illustrate the point and we don't ask what's the probability that if we look for the particle we would find it exactly at some value of x we ask what's the probability that we would find it on an interval on some interval between x and x plus delta x let's say an interval of width delta x we define the probability density so that the probability itself is given by the integral of the probability density over the interval from x to x plus delta x and that integral that sum of infinitely small probabilities but there's an infinite number of them in there that's the probability let's call it the total probability to find the particle between x and x plus delta x i'm making up notation as i go along i may not even use it again but but that's the idea in that case the first equation up there the sum over n of p of n gets replaced and it gets replaced by let's uh let's put it up there it gets replaced by the integral dx of the probability density is equal to one all right so that's the first replacement we have to make now um what about the inner products between basis vectors ah before we talk about that we should talk about the delta function the delta function is a useful tool for studying analogs of these equations for continuous variables basically it's the continuous version of the kronecker delta symbol that's what it is it's the continuous version of the kronecker delta symbol for variables for a variable for single variable x which is continuous again it can be thought of as a function of the difference of the variables so let's the dirac delta function delta function let's think of it as a function of two variables x and y but it's really a function only of the difference between them delta of x minus y and it's defined so that it's zero if x is not equal to y just like the kronecker delta and what is it when x is equal to y the answer i'm not going to give because the answer is undefined but if you want to think about it um qualitatively we draw the x-axis here's the point y and think of y as a parameter now just think of y is fixed this is a this is a function of x which varies as you vary x and as long as x is not equal to y it's zero but when x hits y it gets very large not one but very large it's defined in such a way that the area under the delta function under the spike is equal to one the area under the spike is equal to one so it's defined so that delta that the integral of delta of x minus y dx doesn't depend at all on y it's just equal to one now is there any real function which does that no no honest to goodness mathematically well-defined function can be zero everywheres except at one point that's okay it can be zero everywhere except at one point but have an area which is equal to one if it has an area which equal to one well it must be infinitely high but we can approximate it or we can think of it as as the limit of a sequence of approximations of functions which have a width let's call it epsilon and a height one over epsilon now a mathematician would never accept this as the actual definition you would say the limit doesn't exist the limit just just doesn't exist okay what a mathematician would say is a little bit different he would say the delta function is defined in the following way it's not really a function it's an operation which you do to functions what do you do to functions with the delta function you integrate it delta of x minus y with functions of x in other words whenever you see the delta function it's something to be integrated with a function of x again think of y as just a parameter y could be zero in which case it would just be the integral of delta of x f of x or y could be one in which case it would be delta of x minus one times f of x and so forth the delta function is defined in a particular way so that this is equal to something particular but in order to let's let's use this um intuitive picture of the delta function to motivate what the answer to this integral is this is integral dx right first of all the answer can only depend on f at x equals y it cannot depend on f anywheres except x equal y because the function that it's being folded in with vanishes everywhere except that x equals y so here's a way to think about it supposing here's this is this point is the point y we have some function f of x and the value of the function away from this point is quite irrelevant to the integral because when we multiply f of x by the delta function the delta function is zero on both sides here and only the value of the function in this narrow interval can be important now assume that the function in this narrow interval is continuous and so it doesn't vary very much in fact it varies negligibly over the tiny range epsilon and as epsilon goes to zero the function will vary less and less over the range epsilon and so if that's the case we might as well just evaluate f of x at the point y we won't be making a mistake uh in the integral because the integral is only sensitive to f at the point y so this must be equal then to the integral well let's just do it in steps delta of x minus y f of y dx but an integral over x we can take f of y on the outside let's just take it on the outside since it doesn't it doesn't depend on x at all it just becomes f of y and now we use the fact or the assumption that the integral of delta of x minus y dx is just equal to one and the result is of course just the rule this is the rule now which defines the delta function it is not really a function it's an operation that you do on a function but whatever that operation is it always gives just f at the point y that's basically its definition from a mathematical point of view an operation that you do on a function that gives the value of the function at the point y i don't find that terribly satisfying i find it much more satisfying to intuitively think of it as a very high narrow function with area one with total area one uh anyway that's the delta that's the delta function now let's come back to the particle on a line and ask what's the analog of the inner product of the nth state vector with the nth state vector now n and m are replaced by the position of a particle and so the natural question is if the state vectors of the system are spanned that means if there's a basis of state vectors labeled with the coordinate position x and x is a continuous variable then we can ask what is the inner product between x and y the answer is they're orthogonal if x is not equal to y that's the basic rule that if two configurations are measurably different then the inner product should be zero it's the analog of delta m n equals zero if n is not equal to m but if x is equal to y it shouldn't be zero the natural analog is to set it equal to delta of x minus y now notice that means that the normal that the norm of the vector x x itself is not really defined it's really the delta function at the origin it's the delta function at the point x equals y and so it's infinitely high in fact we never actually wind up using this quantity we always wind up using it in this form as you'll see i'll give you some examples but that's what replaces the inner product structure uh that's there for discrete spaces for spaces with a discrete basis of vectors all right now let's go back and discuss the rules of probability we don't need this anymore let's go back and discuss the rules of probability in quantum mechanics and how they're related to these abstract definitions first in the discrete space in the discrete space the probability in a particular state if we have a particular state that's been prepared the probability p r o b probability the probability that a measurement will yield the value in is given by if psi is the state of the system and we're going to talk about preparing states making them and so forth but for the moment psi is the state of the system it's a vector in the vector space it's in a product with the discrete state n is called the probability amplitude for the state n but it is not the probability we have to take its absolute value and square it another way of writing it is that it's psi n times its own complex conjugate which can be written as psi n n psi not summed no sum over n that's the probability to detect if you like the value n for the observable that's labeled by n what's the natural analog of that the natural analog for discrete for discrete sorry for continuous systems is that the probability density let's call it p of x now or p let's call it p of y y does not represent a different coordinate from x they just represent two values x and y represent two values of the same variable here's the line that the particle is moving on let's call this point x and let's call this point y it's not the x and y axis it's all the horizontal axis and this corresponds to two distinct points the probability that we find the particle at y in a given state now how is a state described it's described in terms of a function of x but let's just uh let's just write it it's given by the inner product of the state vector of the system with the state in which the particle is known to be at y sorry that's not this probability that's the thing which you square to get the probability this thing squared which is also equal to psi y y psi but that now raises the question how do we calculate the inner product of psi with y now the vector space that we're dealing with is the vector space of functions of x each vector is described or labeled in terms of a function psi of x a wave function wave functions like this which are complex functions which depend on a single variable form a vector space so they form a candidate for the description of a particle and we've already discussed the inner product in this space the inner product in this space let me just write it down the inner product between two vectors phi and psi is just equal to the integral of the x phi star of x sine of x the definition of the inner product so now we can apply it to calculating y psi let's just do it it's equal to the integral dx psi of x and now the wave function associated with a particle known to be at y that wave function is just delta of x minus y a particle known to be at y its wave function is zero unless x is equal to y and the delta function effect sorry it's zero unless x is yeah i said it right unless x is equal to y so the phi vector here is just delta of x minus y and it says the amplitude well it's it says what it says just just the wave function where did i write it over there so this is the wave function of a particle known to be at y remember think of y as a parameter and x is a variable and this is the wave function of a particle known to be at y zero unless x is equal to y okay we know the rule for calculating this integral the rule for calculating this integral is just to substitute for x the value y that's the definition of the delta function that's just equal to psi of y so here's what we found we found that the probability amplitude for detecting the particle at y is nothing but the wave function psi of y we've already said that previously but i'm just describing it in a more formal mathematical rigorous uh way that the amplitude to discover the particle at point y is given by the inner product of psi with y and it just is the wave function at point y from that we immediately follow follows that the probability density is just equal to size star of y psi of y all right so that's applying the axioms to a particle moving on an axis on a continuous axis and the only really new thing is that we have to uh that we have to deal with the continuum of eigenvalues or the continuum of possibilities by substituting for the kronecker delta the dirac delta any questions yeah yes could you go back to the first statement on the top of that board there and go through it again oh this is this is uh this is one of the postulates of quantum mechanics the probability if we have a discrete set of possibilities all right a discrete set of possibilities could mean heads or tails if it were a die it could mean one through six if it were some other variable which could take on any of a discrete number of variables then for each value of that variable for each possibility for each independent possibility there's a state vector a bra vector and a ket vector labeled n one of the part one of the one of the postulates of quantum one of the four or five postulates that i gave you the last time for quantum mechanics is that the probability to detect eigenvalue n i'll use the word eigenvalue n or that the observable has value n is the inner product of the state vector whatever the state vector happens to be with the basis vector n squared okay that's just this times its complex conjugate psi n that's this and here's the complex conjugate okay okay all right now it's just a matter of substitution a matter of formal substitution wherever you see n stick y instead of thinking about discrete probabilities think about probability densities things that we will integrate in order to find probabilities this is now the probability density replace probability by probability density replace n by y up to event no difference and then in a moment we will discuss that okay added ingredient that the inner product structure that we postulated for this for the vector space of functions particular complex functions was just to integrate over x phi star with psi okay that's a that's another postulate if you like another um assumption of quantum mechanics point is it works of course and it leads to a nice uh set of ideas okay so then we can calculate the inner product of psi with the vector representing the particle localized at y it's just given by the integral of the wave function of the particle psi of x times the wave function describing a particle known to be at y known to be localized exactly at y that's this delta function this high narrow function which is zero everywhere except that x equals y and has area under it equal to one that's the delta function that integral is just psi of y definition of the delta function so if we know that y of psi is sorry y inner product with psi is just the wave function psi of y we can use that in here and we see that the probability is just psi star psi we could have we could have begun with this it would have been fine to begin with this and work in some other direction but this is the basic um set of rules a set of postulates for quantum mechanics that iraq laid out uh i don't know 1930 somewhere somewhere around that it's never been replaced by anything it still stands today exactly the way he uh the way he expressed it i don't understand the inner product well do you understand do you understand the i can you say the divided functions for essentially let's check and prove that delta of x minus y is the eigen function of the operator x with eigenvalue y is that what you're asking no i don't understand i don't know that's not what i mean here no the inner product the inner product is a number but it depends on which x and which are come back to here it's the dirac delta function it's the direct delta on the right-hand function of the equal sign is a function then so is the x is it is a variable on the left side so it's no but the thing on the right side is y it's not one it's infinite if x equals y sorry by the vectors themselves having an infinite length by the vectors having an infinite length now if you really want to do this right and we it would take us another two hours to do it right what you really do is you replace the continuous line by a dense set of points a dense set of discrete points okay and instead separated by a tiny distance a then instead of labeling the particle by a position on the line we might label it by which site it's at which of these n sites it's at okay we could take we could then use the standard definition delta n m the implication of this is that the inner product of one of these with itself is just one okay what we would do to get to the vectors x is to multiply the vectors in by a numerical value which would be one over the square root of a you just multiply the vectors by a large number when a gets small the one over the square root of a gets big and you simply multiply all of the vectors then what i'm this x here means x can be identified with n it's simply the coordinate position of a point located n units down the line okay the vector x is one over the square root of a times n that's the official definition the way things uh the way things are set up but then if you take the inner product of x with y what you'll get is delta of n minus m right that's n minus m y here corresponds to the m site x corresponds to the nth side divided by a okay well when a gets to be a very very small number this inner product becomes infinite or just very large if you like just very large that's the limit in which this becomes delta of x minus y you can work everywheres if you like with with a discrete space like this but it's useful this is a useful definition the point is this is a useful definition let me show you something else when you integrate when you integrate p of x dx that's not really the sum that's not equal to the sum of p of x it's equal to the integral what's the relationship between the integral of a function of a smooth variable and the corresponding discrete sum well the answer is that the integral is the sum times delta x p of x times delta x where each little delta x here that gives you an area the height times the width and the integral is the limit of the sum of p of x times delta x or just times a so you see this little a interval gets into equations it gets into equations uh it gets into this equation in this form and it gets into this equation in this form this is what we're doing this is what we're doing let's see can we do a little bit better than that um well i think if you trace it through you'll find out that the definitions are completely consistent with this and this a here is the same as this a here so we're just taking all of the vectors and stretching them out by a distance one over the square root of a if we stretch them out by distance one over square root of a then all the inner products get to be very big now remember this delta function well here look here's another way to think about it what is this delta function it's a high narrow function it's a high narrow function whose width can just be taken to be a and whose height is one over a e right let's uh let's take that definition of the delta function its width is a and its height is one over a okay now here i've told you that i've stretched every every vector out by one over the square root of a so we can now calculate what the inner product of x with y is it's just delta of n minus m divided by a right but this is just a delta function it's a function which is 0 everywhere except when x is y or when n is equal to m and there it's of height 1 over a so we've just stretched the vectors out for convenience mostly for convenience we've stretched them out by this factor one over square root of a and then the inner product goes from being the kronecker delta to being the dirac delta the dirac delta differs from the kronecker delta if we you know if we approximate the continuous situation by a very very dense collection of points then the kronecker delta and the dirac delta simply differ by a factor of one over the spacing and if we put that factor of one over the spacing into the definition of the vectors here then we come out to the nice formal um uh convention that the inner product of x with y is delta of x minus y you don't ever really have to do this it's just a nice tool for for um eliminating lots of dependence on this small variable a okay so that's the particle on a line or the wave function for a particle on a line the wave function is a probability amplitude and it plays the same role as n psi in discrete quantum mechanics all right let's study another system let's study a system of a particle moving on a circular line on a circular line meaning a circle circular line means a circle here's our circle and there's a particle on it somewheres now i'm going to lay out the circle i'm going to cut the circle up here or someplace now i guess maybe i'll cut the circle uh down here cut it down here and take the entire circumference of it and sort of lay it out on a line just to think of it as a line interval instead of a circle what's the difference between a line well let's let's let's come to a moment instead of labeling the circle with an angle i'm going to label it with a coordinate x let's take the radius of the circle to be r and the total circumference of the circle is 2 pi r that means if i lay the circle out on a line it goes from 0 to 2 pi r the particle can be found anywheres on this line interval or equivalently anywheres on the circle so how do we represent vectors how do we represent the space of states the space of states is again labeled by position but now x only goes from zero to two pi r precisely the same setup eventually we come to the idea of a wave function psi of x the square of which is the probability for finding the particle at different positions we come again to the idea of a probability amplitude to find the particle at different locations and the square of this when i say the square i mean the absolute value of the square the absolute value of the square of this is the thing which represents the probability on this axis but there's one thing we have to take into account if we really want to think about it as something on a circle if something is on a circle if a function is defined on a circle then it has the property that when you go all the ways around the function comes back to itself if a function is defined on a line interval that is not necessarily so so in order to be studying a particle on a circle we have to restrict ourselves to wave functions which are periodic which means they come back to themselves after going from 0 to 2 pi r another way to say that is that the space the vector space that we're talking about is the space of functions psi of x which have the property that they're equal to psi of x plus 2 pi r consider the class of functions the special class of functions which when they start at 0 and get to 2 pi r they come back to the same value they're called periodic functions okay special class of functions first question is are they a vector space the first thing you should think of to yourself is am i following the rules of quantum mechanics is this a vector space well if we take a function which comes back to itself and we multiply it by a constant any complex constant it will still be a function which comes back to itself a periodic function when you multiply it by a constant is still periodic what happens if you add two periodic functions each of which comes back to itself after going around the circle the sum of two periodic functions is still periodic so the periodic functions are a vector space an allowable vector space for studying a quantum mechanical system that's that would be the basic setup for quantum mechanics on a periodic interval or equivalently quantum mechanics on a circle okay the last time we began to study momentum so let's come back to momentum thus far we've been studying positions let's come to momentum the position operator as i explained to you last time is equivalent to multiplication of a function of x by x itself it takes any function of x and multiplies it by x so it's an operation that you do on the vector space that multiplies it by x i'm interested now in the momentum operator the operator which represents momentum we discussed it last time let's call it p and we discussed it last time it's i h bar minus times d by dx now what this really means is that the abstract operator when it acts on abstract vectors has the same action as minus i d by dx or minus i h bar d by dx when it acts on functions of x there's a one-to-one correspondence between vectors and functions and the things that you can do to a function are to differentiate it multiply it by something and so forth so every operator corresponds to an operation that you do on a function and this is the definition or this is the operation that you would do to define momentum well we worked out the eigenvectors of this operator the eigenvectors were just defined by minus i h bar d by dx psi of x is equal to k k is the eigenvalue psi of x uh no p psi of x momentum not probability here momentum let's call let's sorry it's p for momentum what's the solution of this equation the solution of this equation is psi of x is equal to any numerical multiple incidentally times e to the i p over h bar times x let's check that if i differentiate with respect to x it pulls down a factor of i p over h bar i multiply by h bar that gets rid of the h bar and then i multiply by minus i that gets rid of the i so this operation when it when it hits psi of x if this is psi of x let's call it psi p of x it means the wave function associated with a particle of momentum p the wave function associated with a particle of momentum p is e to the i p over h bar x all right to check that what we do is we check that the psi of x is an eigenvector of the appropriate operator now why this is called momentum is another question as i emphasized last time to check that this makes any sense to call it momentum we want to understand how classical how wave packets move we want to understand how wave packets for heavy objects move and we want to check that the motion of the wave packet is consistent with calling this object momentum but for the time being this is just a name momentum is just a name for the observable that goes with this operator here are its eigenvalues all possible real numbers and here are the eigen functions it's traditional to put a square root okay let's uh let's be careful now we're doing particle on a circle let's consider the inner product of this wave function with itself let's normalize it let's normalize it that means let's multiply it by a constant so that the integral of psi star psi dx is equal to one this is for momentum eigenfunctions psi sub p for the same eigenvalue in other words equivalently let's require that the total probability for finding the particle anywheres adds up to one anywhere's on this circle it's just a statement the total probability of finding it somewhere is on the circle is one all right what is psi star psi if psi is given by this what is psi star psi it's just one star psi is just one so this says the integral of one but what is the integral over the integral goes from zero to two pi r from zero to two pi r of 1 is certainly not equal to 1. it's equal to 2 pi r so in order to get this to come out to be 1 we have to multiply the wave function by the square root 1 over the square root of 2 pi r it's just a number it's just a numerical number that happens to depend on r and if we do that then psi star psi is just 1 over 2 pi r this becomes integral of 1 over 2 pi r from 0 to 2 pi r and now it is equal to 1. this is just something to keep track of that it's uh convenient to normalize these wave functions so that their norm so their inner product with themselves is equal to one it doesn't play any great role in what i'm going to say but it's nice to get the formulas right okay what about the inner product between wave functions with different values of p say p and q if p is not equal to q well then the answer is zero and you can check that just by knowing the properties of exponentials like this but the reason that we know it is because the eigenvectors of hermitian operators with different eigenvalues are always orthogonal so this is a this is something that you can check that the integral of e to the i p h x e to the minus i q h x or whatever that that's equal to zero but that follows from the hermitian property of the operator p which we checked last time we checked it last time by working out uh the property of being hermitian okay so now we have the wave functions for these particles but the something we haven't we haven't guaranteed yet we haven't guaranteed this periodicity here thus far a general wave function like this is not necessarily periodic the space of states that we're interested in is the space of periodic wave functions so let's ask what is the constraint what do we have to do to restrict ourselves only to those wave functions which come back to themselves well the answer is that if we take e to the i p over h bar x and we add to x two pi r that it had better just be e to the i p over h bar x in other words after x makes an excursion of 2 pi r it must come back to itself that's the condition of periodicity it's the condition that the wave function is really on a circle and not on a line all right well this is equal to of course e to the i p over h bar x times e to the i p over h bar times 2 pi r just the property of exponentials and that has to be equal to e to the i p over h bar x we can cancel this out and our rule is that e to the i p over h bar times two pi r must be equal to one that's a restriction on the possible values of momentum that's a restriction on the possible values of the eigenvalue so that we actually are in the appropriate space of functions if we choose p to be anything other than something which sets this equal to one it's not an allowable wave function because it's not periodic it's not really on a circle the condition that this is equal to one is the same as saying that p over h bar times two pi r has to be an integer multiple of two pi e to the two pi n where n is any integer times i is equal to one and only if n is an integer is this equal to one e to the two pi i is equal to one e to the 2 pi i to the nth power is equal to 1. and so the general solution to requiring the wave functions to be periodic is that p over h bar times 2 pi r should be an integer multiple of 2 pi we can cancel out the 2 pi we can divide by r r is just the radius of the circle and what it tell and we can multiply by h bar it tells us that the allowable values that the momentum can take on are integer multiples of h bar over r let me write let me clean that up the allowable values of the mo of the momentum are integer multiples now i forgot what i said integer multiples of what h bar over r if we choose any other values of p in this wave function it won't be periodic and it will not correspond to motion on a circle so what we learn among other things is although the position on the circle is a continuous variable the momentum is a discrete variable the momentum is a discrete variable and it can only take on integer multiples of a certain quantity h bar over r now as the circle gets bigger and bigger as r gets what's the let's uh let's plot the possible values of p here they are they're discrete n equals 0 could be over here n equals 1 n equals 2 but the values of p are n times h bar over r so that means the separation between different values is not a separation in space it's a separation of the um possible values of momentum they're separated by amount h bar over r as r gets bigger and bigger if we well if we imagine a bigger and bigger circle which means that this line goes from zero to some very very large size then the spacing between the different values of momentum gets smaller and smaller and eventually any value of momentum or very dense values or a very dense collection of momentum become possible so by making the circle very very big and tending toward the situation where it becomes an infinite line in that limit the momentum the possible momentum become anything but if we're actually on a circle if we're actually talking about a particle moving on a circle then the momentum are quantized quantize means that they come in integer multiples of some particular value of momentum that's the r that's always the origin of discreteness in quantum mechanics uh discreteness of various things which in classical mechanics can take on any value and in quantum mechanics only take on discrete values energies and things like that always has to do with some periodicity and the wave function having to be periodic let's uh go one step further let me just remind you from classical mechanics if we have a particle moving on a circle it has an angular momentum the angular momentum is called l we worked it out last quarter and it can be written in a number of ways but one way that it can be written is the tangential component of momentum let's just call it p it's the component of momentum along the circle which we've called p times r angular momentum is momentum times distance or the moment arm of the momentum what does that say the possible values of angular momentum are the possible values of angular momenta are r that's this r here times the possible values of p which is n h bar over r so we see before our very eyes that angular momentum is quantized in units of h bar incidentally angular momentum and h-bar have the same units units of length times uh times momentum action it's called angular momentum comes in integer multiples and it doesn't matter how big the circle is however big the circle is the size of the circle cancelled out of this and we found out quite independent of the size of the circle the angular momentum is quantized angular momentum always comes in discrete integer multiples of uh except when it comes in half integer multiples but that's another story for another day okay angular momentum is quantized in quantum mechanics what happens what should we do when the circle does get very big the circle is a convenient uh starting point for studying particle on an infinite line just by making the circle bigger and bigger and bigger eventually it approximates a particle on an infinite line but as i said in that limit the spectrum the collection of eigenvalues of momentum becomes denser and denser in other words it approaches a continuum the discrete spectrum of momentum becomes continuous what was the spacing between different momenta it was h-bar over r h-bar over r was the spacing between the different momenta here and ozil gets bigger they get denser and denser and in that limit we start to use the continuum notation instead of describing uh instead of describing vectors p which have inner products which are kronecker deltas we stretch them out we stretch them out and p and q represent two different momenta and use the dirac notation p minus q in the limit where the where the line gets so big that basically every value of the momentum is possible in that limit we want to think of the momentum as a continuous variable and just use the notation of dirac delta functions we don't really do anything new we just stretch out the vectors by 1 over the square root of this distance here stretch them out and replace the kronecker delta by the dirac delta there's not much physics in this replacement of the chronic delta by the dirac delta it's just a convenience and a trick to be able to pretend that this is a continuous function which it is not of course as markets uh tends toward infinity would you then expect the regression back towards class of campus no not well in one respect only in the respect in the respect that the possible values of momentum become anything but that does not mean we go back to classical mechanics necessarily in fact we'll discuss that right now in the angular momentum if i multiply p times r p times r is angular momentum so the spectrum of angular momentum the r cancels out and the spectrum of momentum it doesn't yeah good right okay angular momentum momentum just differ by this factor of r and it's just definition of angular momentum but it's a good definition of angular momentum so the quantization of angular momentum ultimately has to do with the periodic nature of the angle variable the angle around the origin here is clearly a periodic variable wave functions have to be periodic even if you're not on a la even if you're not restricting yourself to motion on a circle even if you can move anywheres in space still when you go around the closed loop here the wave function has to come back to itself and that's the content that's the ultimate content of the quantization of angular momentum that if you rotate around by two pi you come back to yourself you come back to the same thing all right now all right so now i was asked the question are we essentially doing classical mechanics when we uh when we let the interval go to infinity no not at all um in classical mechanics a particle can have both a known position and a known momentum in other words it can be in a state in fact every state of a particle has both a position and a momentum with infinite precision position and momentum are simultaneously specifiable not so for a quantum mechanical particle specifying the position of a particle means choosing an eigenvector of its position operator the position the eigenvectors of the position operator correspond to the states in which the particle has a very very definite position the eigenvectors of the momentum operator correspond to the states where the particle has a definite momentum the eigenstates of position are these high narrow functions that look like that what about the eigenstates of momentum they have the form e to the ipx which means let's uh this the thing in the denominator here is not so important they have the form of cosine p over h bar x plus i sine p over h bar x each one of these functions is more or less uniformly spread around the circle if we tend to let the circle get an infinitely big radius it means they're infinitely spread over the entire axis why do i say they're infinitely spread well the easiest way to see it is just to take psi star times psi the magnitude of the function it's just one it doesn't vary on the circle so each momentum state has equal probability of being anywheres on the circle each one of the momentum states separately has equal probability of being anywheres on the circle okay each position state is extremely well localized in space so no wave function is simultaneously an eigenvector of position and momentum just can't have any for which the position and the momentum are both specified so we're not doing classical mechanics we're quite far from classical mechanics but let's discuss a little bit the idea of simultaneously specifiable two variables are called compatible if you can specify both of them uh simultaneously position and momentum are not compatible in that in that sense what is the condition that let's say we have two observables a and b i think i called operators with little hats on top of them supposing i have two operators a and b which are simultaneously specifiable what does that mean that means that they have common eigenvectors that means that there are eigenvectors which are eigenvectors of both of the operators simultaneously in fact the best possible situation for two operators being completely compatible is that if all of the or if there is a complete basis of vectors let's label it i'm going back to the discrete case but it doesn't matter that there's a complete basis of vectors which are simultaneously eigenvectors of both a and b then each one of those eigenvectors both a and b can be specified and can be uh can be definite so if we have a complete set of eigenvectors for which both a and b we have a complete set of eigenvectors which are simultaneously eigenvectors of a and of b then a and b are compatible and they can both be known by simply putting the system into one of these combined eigenstates well let's see what that means that means we have a complete basis of vectors labeled from 1 to capital n such that for every one of them a on n is some eigenvalue let's call it alpha sub n times n this just says that these vectors are all eigenvectors of a with eigenvalue alpha n the first vector has eigenvalue alpha one the second one has eigenvalue alpha two but simultaneously the same set of vectors exactly the same set of vectors are also eigenvectors of b with eigenvalue let's say beta n times n the necessary the question is what is the necessary conditions necessary in fact in sufficient conditions that two operators a and b have the same set of eigenvectors so in order to motivate the conditions let's let's study what happens if you multiply a times b and then act on n what does that do well we know what b does when it acts on n it just gives us the eigenvalue beta n times n so we can rewrite this as a times the number beta n now this is a number beta n is not an operator anymore times n beta n is just a number we can take it out put it on the left hand side of the of the operator a and so that becomes beta n a times n numbers ordinary numbers you can push them back and forth at will they just multiply the vector n by two and using the linearity of a we can easily just move beta to the other side now what is a on n that's alpha n times n so what do we get we just get beta n alpha n times n in other words the same set of vectors are eigenvectors of the product a times b with eigenvalues which are just the products but we can also see from this that it doesn't matter which order we multiply a and b if they have a simultaneous set of eigenvectors in other words if there's a basis which are simultaneous eigenvectors of both operators then it doesn't matter which order we multiply them because in any order we're just going to get beta n times alpha n times n the order that we multiply two ordinary numbers is immaterial alphas and betas are just numbers and it doesn't matter if we write this as beta n alpha n or alpha and beta n and from that we see that the order that we multiply a's and b's won't matter if they have a complete basis of common eigenvectors all right the necessary and sufficient conditions i won't prove this but the necessary it's clearly a necessary condition it's also sufficient the necessary insufficient conditions that there exists a basis of vectors which are common eigenvectors of both simultaneous eigenvectors of both are that a times b is equal to b times a that's necessary and sufficient another way to write it is a times b minus b times a is equal to zero now remember a's and b's are operators they're not numbers in general but if they're compatible if they're compatible then it doesn't matter which way you multiply them and the symbol for a b minus b a is called the commutator a b the bracket this is not a bra and a ket the square bracket with a and b sandwiched inside it with a comma that's by definition a b minus ba so with all of this little bit of notation the necessary and sufficient condition for two operators to be compatible is that they commute that their commutator is zero this of course is not always true for every pair of operators but when it's not true they don't have a complete set of common eigenvectors and they're not compatible you can't in general measure both of them or you can't know both of them simultaneously what about position and momentum are they compatible well from what i told you it's clear that it did not there are no common eigenvectors none whatever right so they can't be compatible but we could check this by checking whether position and momentum commute in other words it should follow that position and momentum don't commute if there are no common eigenvectors we can check that we can just do the calculation of multiplying p times x and x times p and see if we get the same thing so let's do that and in the process we'll find out exactly what xp minus px is is not zero so let's work it out here's what we want to calculate x xp minus px now if we operators now remember what x does to a wave function it just multiplies it by x what does p do minus i h bar d by dx so we can write this as x times minus i d by dx minus minus i d by dx times x all i've done is substitute for p minus i d by dx but what on earth does this mean you're not allowed to just write d by dx you have to d by dx has to do something to a function so let's take this and see what it does to a general arbitrary function let's try to figure out what this operator is by seeing what it does when it operates on a general function so let's take this thing x minus i d by dx minus now we just minus times minus is plus plus i d by dx times x and apply it to a function of x psi of x now it makes sense now it makes sense the first term here is just x oh what did i leave out h bar h bar h bar multiplies the whole thing okay h bar times the whole thing x times minus i d by dx times psi of x remember when you multiply operators a b and you apply them to vectors you first apply b and then you take what you get and you apply a to it okay so here it is we first apply minus i d by dx to psi let's take these two terms and work them out separately first of all we get h bar and then this term gives us x minus i d psi by dx all i've done here is apply this d by dx to psi the x is to the left so it doesn't get differentiated only the psi gets differentiated and then we multiply it by x in the other term we have plus i h bar and now first we hit psi with x that means we have x times psi of x and then we differentiate the whole thing what's the derivative of x times psi of x we have to work that out the derivative of x times psi of x has two terms in one of them the derivative hits x and leaves psi and the other one the derivative hits psi and leaves x so just this piece of it here i h bar gives us the x by dx that's 1 times psi of x plus x d psi by dx this is just differentiating the product x the x psi so there's an i h bar and another i h bar here let's see what we have we have minus i h bar x d psi dx and then we have plus i h bar x d psi by dx they cancel in other words the term that you get when the derivative hits psi cancels this one here so that cancels we can just forget this and forget this all that's left over when you do this operation is i h bar psi of x so here's what we learned we learned that x p or just uh let's just yeah x p minus p x when you think of it as an operator hitting any wave function just multiplies the wave function by i h bar in other words this thing does exactly the same thing as multiplication by a simple number a simple complex number i h bar one writes this by saying that the commutator of x and p x p minus p x is equivalent to or is equal to multiplication by the number i h bar x p minus p x is i h bar now as a classical equation that's nonsense x times p is equal to p times x but as an equation for operators operators don't necessarily commute the order in which you multiply them counts just like matrix multiplication what this is telling us is that x and p are operators which don't commute and the worst possible situation incidentally is when their commutator on the right hand side is just a number that's as bad as any commutator can be and it is the reason that there are no common eigenvectors at all there are no common eigenvectors if there was a common eigenvector then at least if you were to put that common eigenvector here then at least when it acts on that common eigenvector you would get zero but there are no common eigenvectors and uh that's an indication of the fact that the commutator is just a number it's as bad as it can be okay let's stop for a minute and yeah every vector is an eigenvector of the commutator this is true that is true with its same eigenvalue yep true well that just tells you there's no information in the commutator i mean there's no the commutator is not an interesting observable every time you observe it you get the same number it's okay so it's not that doesn't tell you anything about the system i'm talking about thinking of this as an observable it doesn't tell you anything about the system when you measure it xp minus px is always equal to ihbar in classical physics it's always equal to zero this was heisenberg's crazy equation xp minus px is equal to i h-bar that people thought he had lost his marbles when he wrote it down maybe he didn't lose his marbles but in any case uh it's a correct equation it seems like x times x times any vector is just x times that vector so why isn't there a vector a vector no because x is not a number yeah here's what it would mean to be an eigenvector you have some function arbitrary function and if you have an operator which operates on that function and gives you a multiple of that function let's say two times that function then it's just two times the same function okay if the eigenvalue is i then it gives you i times the same function if you multiply this function by x it doesn't give you the same function back even a multiple of that function it gives you a function which will look like this because you're multiplying it by something bigger over here than you are over here okay so for example let's suppose this function we're just e to the minus x squared e to the minus x squared has a bell shape like that that is not a numerical multiple of x times e to the minus x squared okay the only eigenfunctions of x are delta functions because when you multiply them by x only one value of x gets picked out so it is multiplication by a number for uh for a delta function right so it's not every function is a multiple of every other function if you if you allow variation in in the uh in the coefficient when we say it's an eigenvector we mean with a numerical constant coefficient here not an x but a three or three plus two i or whatever okay so that's uh i knew that was going to come up i was waiting for somebody to ask it thank you okay so here's the setup again we have two slits now let's suppose we closed up one slit closed up one slit and took a beam of particles coming through here described by a wave function each the particles are described by a wave function some psi of x and the simplest thing would be to have a psi of x which was just for example an e to the ipx this would describe a particle with momentum p coming in from the left yeah p over h bar right the particle passes through the hole and this and the probability wave psi spreads out like that in fact i could even write down what it is but i don't want to write down what it is we'll just uh just draw a picture of it right now it hits a screen and when it hits the screen it's varying from here to here from here to here it's a wave then has a trough and a high point low point high point low point and so forth when it hits the screen in the vertical direction here we also see variation we just see the variation of the wave radially imprinted on the screen as a variation along the screen now if you think about it even if this were even if these waves were uniformly separated like so they would not look quite uniformly separated on the vertical screen they would look more closely spaced up here than they do down here just try it out take a compass and a piece of paper draw some concentric circles like that and you'll see that they get more closely spaced up here than they are here if we were to approximate the wave vertically by an e to the i let's call it p over h bar y that wouldn't be such a good approximation because the wavelength varies as we move up and down the wavelength is shorter up here than it is here right all right but let's just say over some some interval so over some interval it looks pretty much like an e to the i p y like that some vertical variation variation along the y axis here this is the y axis now let's open a second hole put the second hole over here oh before we open the second hole what is the probability of finding the particle at different values of y now this is only a good approximation for a small interval but still for that small interval what's the probability for finding the particle at different y it's uniform because if i multiply this by its complex conjugate i just get one all right so the probability vertically over some small interval where the function can be approximated like this is uniform the magnitude of e to the i p over h bar y is constant now supposing we open a second hole a second hole will also create a wave similar kind of wave which will add to the wave that comes out of here so the wave that comes out of here will get added to by a second wave which comes out of here what we're doing is we're taking some state described by e to the i p y and we're adding to it another state something we never do in classical physics we never add states in classical physics but we're adding a second vector this is a vector describing what happens if the particle goes through the first pinhole if the particle goes to the second pinhole it creates a different wave vertically here and in fact the different wave up in here has a slightly different variation we could represent that by saying the second wave that comes out of here might look like e to the i q over h bar y over the same interval over the same interval they have slightly different variation slightly different variation because they're at different heights vertically relative to the whole so they have slightly different variation now supposing you took either one of these either one of these has a magnitude equal to one so if we only had one hole opened vertically here over some range of distances we would have a uniform probability distribution a probability distribution which would not vary vertically no variation vertically in the probability distribution if we only opened one hole it might vary slowly but the oscillations implied in this function here would not cause any kind of oscillation in the probability because this thing squared the oscillations go away right okay what if we take the sum of these two which is what happens if we open both holes another way to say it is if we open whole number one we get out a wave function psi of x we get a state psi let's call it psi one if we open open whole number two we get outside two if we open both holes the state of the electron or whatever it is becomes psi one plus psi two in other words it becomes the vector sum of the two states that it started with the two states from the individual wholes okay now let's ask what the probability to find different y is all we have to do is take this and multiply by by its complex conjugate so let's do it let me call this let's get rid of the h bar let's work in units where h bar is equal to one we get tired of writing it so it's e to the i p y plus e to the i q y but then we have to multiply it by e to the minus i p y plus e to the minus i q y what have i done i've taken the wave function and multiplied it by its complex conjugate okay if i only had one of these then then the probability would be uniform along the y-axis either one but now i have both of them let's calculate this e to the i p y times e to the minus i p y what's that one we also have e to the iq y times e to the minus iq y that's also one so the terms e to the i p times e to the minus i p and e to the i q times e to the minus i q all together give us two then we get plus e to the i p y e to the minus i q y that's coming from this term times the far term over here and then we have plus e to the i q y times e to the minus i p y or to write it slightly differently this is twice e to the i p minus q y 2 plus excuse me 2 plus plus e to the minus i the rate to the i q minus p y what is this e to the i times something plus e to the i times the negative of that twice cosine twice cosine so this becomes twice 2 plus 2 cosine of p minus q y 2 plus 2 cosine p minus q y or even simpler just twice one plus cosine of p minus q y what does this do is this constant no it's not at all constant it has a constant piece and it has an oscillating piece the oscillating piece is just a cosine the oscillating piece the piece there's one piece which is just one i'm forgetting the two here there's one piece which is just one and there's another piece which is a cosine here's a cosine of something we add them together where they enhance each other they're twice as big where they cancel each other where the 1 cancels the cosine the answer is zero in other words there are places in this function where the answer is zero in fact what the sum of the two functions looks like is it looks like this there are places where it goes to zero what does that mean that means if we open only one hole the probability distribution is uniform if we open the other hole the probability distribution is uniform if we open both holes there are places where the probability is zero despite the fact that uh that particles could get here quite easily if we opened only one hole so opening both holes makes it harder to get to this point here than opening either one of the holes this is the puzzle or the puzzling not the puzzle but the puzzling phenomenon of interference and it's a particle phenomenon in other words it's a phenomena that happens even if we only let one particle through at a time one particle through at a time if we open one hole we get a constant probability distribution if we open up the other hole we get a constant probability distribution if we open both holes we suddenly find that there are places where you can't get to that's the significance of adding states psi plus phi psi plus phi can have a complicated dependence even though the probability for psi and phi might be perfectly smooth so interference oh notice notice how important complex numbers were here well i we've already we've already discussed the importance of complex numbers in uh in momentum and having a momentum e to the i p dot x if we didn't have complex numbers we could never have an eigenvector of minus id by dx obviously okay um questions let's start for questions now yeah yeah the bigger the the bigger the distance between the slits the closer the spacing and the uh and the um smaller distance between the splits the bigger the spacing that that's not yes that is correct that is not what i wanted to point out here though i just wanted to point out that the phenomena of interference has to do with the ability to add state vectors which is something you don't do in classical physics but the answer is the closer the slits the closer the slits the further the interference pattern the bigger the interference pattern the further the slits the shorter the distance between the interference maxima and we can work that out we'll work that out another day yeah if that's right that's right that's right yeah okay so um if you let a wave go through a hole and you project it onto a screen over here what you'll see oscillates it oscillates pretty much like one of these e to the ipy's but it oscillates with a changing oscillation so for example over here let's just take i'll draw it as an oscillation of course it has a real and imaginary part but the oscillation will have a longer wavelength over here than it does as you move out the wavelength will get shorter as you move out you know why okay i'll tell you why anybody have an idea i mean you can just do it by calculus you can take a wave going out like this and calculate how how close the the maxima are when they intersect but there's a physics point of view also how does the particle get over here it gets over here by getting a kick in its momentum so in order for it to get over here it must have a larger vertical momentum than over here right at the center it required no vertical momentum to get to here to get to here down here it required a somewhat bigger vertical momentum remember now the bigger the momentum the faster these functions oscillate the bigger p the faster the functions oscillate so when you actually plot the function that you get from a wave going through a hole you find out that pretty much anywheres it looks like an ordinary wave except with a slowly varying slowly modulated frequency where the frequency gets a little bit yes a little bit faster as you move away from the hole now now you take the second hole and you open it over here all right at this point in this region over here the vertical frequency due to one hole is a little bit different than the vertical frequency due to the other hole because they're at different height here okay so it's just like super posing taking this wave and then superposing a replica of it shifted you'll be adding together two waves of slightly different frequency just because from a single hole the frequency varies as you move vertically and that variation has to do with the fact that to bend for the trajectory to bend it has to pick up a vertical component of momentum that extra vertical component the momentum makes it vary a little more rapidly down here but you can work it out i can tell you what the formula is if you if a wave goes to a hole over here and let's say that hole is at the the vertical origin then the wave that comes out looks something like e to the i p r r is radial distance divided by r r is radial distance here okay now let's put the screen in here what is the radial distance to a point that distance y what's the radial distance from here to a point at y square let's call this l r is the square root of l squared plus y squared and what's down here is not so important square root of l squared plus y squared here you can see that what goes on here is not a simple plane wave e to the i p y but e to the i p times a function which has a more complicated variation if you take this function here you'll find out that pretty much anywhere along here it looks like an ordinary oscillation but the oscillation gets a little bit faster as you go out here gets a little bit faster but very large y you can ignore l and then it just looks like e to the ipy but in closer the variation is a little bit different and so what you're doing is you're putting together two functions with slightly different oscillation wavelengths well of course what we're doing here is calculating beats these are nothing but beats two slightly different frequencies are and now we're not talking about time we're talking about vertical height to slightly different vertical frequencies are getting interphase out of phase interphase outer phase and creating this uh this extra oscillation with a cosine here that's all this is it's beats but beats in the probability distribution and it's coming because the wave that comes out or the state of the electron that emerges from one hole here is some psi which is characterized by one wave function if you open the other hole you superpose a second wave function which is characterized by a shifted wave function and the sum of the two of them has this oscillation that neither one of them has neither one of them has an oscillation in the probability together they have an oscillation is that the frown over diffraction pattern what's that front loafer diffraction plastic i don't know where from halfway ahead of there is what is farming halfway remind me i forget what fraunhofer is it's just a diffraction power pattern yeah it's just a different it's a diffraction pattern it's the diffraction pattern in this case of two holes right okay now somebody asked me about um the distance between the uh the maxima if the two holes are very close together then you're adding functions which are almost identical in that case the p and the q here are almost the same okay if the p and the q here are almost the same it's like beating together two almost equal frequencies all right you get a very very long wavelength so if the two holes are close together then this cosine here has a very long wavelength if the two if the um if the two holes are far apart then you're beating together two waves with somewhat different frequencies and so you see a shorter wavelength for the uh or a shorter interval between the maxima any other questions yes but one uh we certainly have not gotten to it yet because in order to get to it we have to talk about a composite system the composite system now being the electron plus the detector and talk about the concept of entanglement between the detector and the uh and the electron right so what's being discussed here which i'm not going to talk i will talk about it but i won't uh i won't derive it now because we have to understand how to combine systems together at the moment we're talking about a single electron or we're talking about a sequence of experiments each one involving a single electron um supposing there was a object over here which detected which of the holes the electron went through now by detecting which hole it goes through it means the electron leaves an imprint on on something it leaves an imprint on some other system telling that other system i went through this hole or that hole when that occurs the interference pattern is destroyed but we're not in a position yet to discuss that because we have to discuss entanglement before we do that incidentally was the subject of the previous lectures on quantum mechanics and it is in there but we'll try to discuss it a little bit again okay so as i said that's part of the problem of combining systems together the systems being whatever it is that detects the electron and the electron more complicated story well in order for it to deviate from the horizontal it must have picked up some component of y momentum all right if it picked up a component some significant amount of y momentum or another way to say it is if it doesn't deviate from the vertical it must have meant that there was no y momentum and in fact that must mean in this location over here the function is varying particularly slowly which it is all right the slowest variation over here in order for it to get out here it must have gotten a kick to get a kick it now has a y component of momentum and the y component the momentum means that it has a certain oscillation over here so where does the energy start oh there's no energy to change the momentum the energy is proportional to the square of the momentum and the square of the momentum hasn't changed it's uh roughly speaking like bouncing well you have a you have a a target over here a stationary target which doesn't move and you bounce a ball off it the ball changes direction but there's no loss of energy so that means the x momentum yeah yeah yeah yeah exactly that's right the magnitude of the momentum the magnitude of the momentum is determined by the distance between these uh maxima and minima in the radial direction that is just the original same spacing as the spacing that came in spacing that goes out is the same as the spacing that came in so the wavelength coming in is the same as the wavelength going out but it's converted into a spherical wave but when projected when projected onto the vertical axis you see a variation in the in the frequency vertically and that corresponds to having picked up some some vertical momentum and as you say that would also imply that the horizontal component of momentum must be a little bit less up here than it is down here that's a good point no other question yes yeah so a question on on the so we have a brain function that's now hitting both holes that's right the wave function right so is it meaningless to talk about a postulate particle going well we're sending through one particle at a time out of the electron gun one particle at a time is coming through one particle at a time doesn't make a wave-like pattern like that it just makes a spot on the detector but we do it many many many times and we discover that there's a probability distribution on the detector here so each event consists of a spot on the detector it's only the accumulated effect of many spots on the detector that will create a pattern on here and the pattern will be the pattern that looks like this notice one thing this is always positive positive or zero is negative nowheres cannot be negative that's because we multiply the thing by its complex conjugate so the probability distribution is never negative but it does vary and it varies uh in some places is equal to zero yes how is this compared to like the interference of life well the interference of light is a special case light consists of a large number of photons photons are particles they satisfy the same rules as electrons if we attenuate the beam of light for example by sending it through a series of filters which filter out almost all the light then we'll find out that very few photons are coming through and what we'll see is a flash a flash a flash a flash eventually building up to the pattern of uh of the way of the wave-like interference if on the other hand we send a lot of light through all at once well then we'll just see the uh the probability distribution uh as a uh as a pattern sort of created all at once but that's just because a lot of particles came through all at the same time so uh we see the particle aspects if we slow not we don't slow the particles but we um uh yeah we decrease the number so that they're coming through one at a time let's say 10 years between photons and we see a blip every 10 years well they come through randomly but let's say on the average once every 10 years then we see a blip but if we wait a thousand years then those 10 years per blip blips add up to a uh add up to a an interference pattern so as i said we're not going to explain strange quantum phenomena but this is the description yes i'm still having trouble with the one whole experiment and trying to understand where the oscillation comes from the limited oscillation you see somehow well you don't see an oscillation from just one chord um what do you see on the screen or what's the problem the distribution of probabilities on the screen from one hole just the blob yeah uh right in the all right what you the question is what does the wave function look like so let me let me uh roughly write down what the wave function looks like it looks like an e oh boy i must remember to tell them to put fresh pens can that be seen not very well i'm afraid all right probably red is better than blue i imagine let's try it we're talking about one hole right one hole so basically for one whole you see some kind of e to the i uh some some function of x which is not exactly x p x but it's something like p x uh times a another function let's call it rho of x where this is a rather smoothly varying thing this is a smoothly slowly varying thing oh i should call this y let's call this y y right so there's some oscillation multiplying something which is not oscillating very fast roughly speaking it looks like e to the i p x p over h bar x times some smooth function row of x incidentally p over h bar is likely to be a rather large number which means that this oscillation is generally very fast this row of x is a smooth function which doesn't vary very much over the ray over the range here now what happens if we multiply this by its complex conjugate if we multiply and this is a real function this is real this is here's where the imaginary parts come if we multiply this by its complex conjugate then this goes away and we just get row of x squared okay and rho of x squared is a nice smooth function and we see no oscillations from one hole being opened up row of y row of y i keep writing x but i keep meaning y row of y now if we and of course as i said this isn't exactly e to the ipy you can think of it as a slowly varying p as you vary up and down here so there's a p which very slowly as you vary as you move up and down here times a nice smooth function but then you open a second hole and the second hole is some e to the i let's call it p prime or q over h bar y times another smooth function a second smooth function row of y where the second smooth function might just be the first smooth function slightly displaced okay slightly displaced because you lowered the hole either one of these if you were to square them you would not see the oscillation but when you add them together and square it then you see the beats between the between these two things so one by itself you see no no interference pattern yeah okay right all right it depends well you see something that depends on the details of the hole okay now what's it yeah it's a good point and let me just spell it out if the hole has rather sharp edges right then what you see is a wave coming out of here but you also see a wave that got produced right at the edge the sharp edge created a little wave around the sharp edge and the other sharp edge creates another wave around the sharp edge and it's almost as if you had two holes if the edge is not so sharp if it's a kind of fuzzy hole then it doesn't it doesn't create an interference pattern then the interference pattern of the single hole is not there it's true a single hole can make a an interference pattern by scattering off the two uh by scattering off the different parts of the hole uh and that's correct yeah yeah but it's quite distinct from this different uh right right yep you see some circles but the circles are different than the interference pattern that you see because of the two holes right so in the two-hole situation shooting one electron through at a time there are places on the screen that will never be hit that's right the two holes are over zero last forever there'll never be any clouds now i mean nothing is ever quite that good i mean uh there's a bit of an idealization here but this is essentially correct yeah there's dark spots which are can be very very dark if you uh if you have good tiny tiny holes and everything is very very ideal no cosmic rays no no no noise in the room everything is as you know as ideal as possible then you can see some pretty dark interference spots in there and those dark interference spots are for uh in an idealized situation there would be no particles hitting there yeah as long as the two holes are small if the two holes are small then it really doesn't matter whether they're uh identical or not if one hole is big and the other is small then of course it'll make a difference but yeah small compared to the separation right and the wavelength it's also important that everything yeah it's also important that everything be small compared or small or not too much bigger than the than the wavelength of the light yeah i i say electrons but if you were to do this experiment in the laboratory you'd want to use photons yeah uh photons have much longer wavelength i mean light ordinary light has wavelength many many times longer than the typical wavelengths that you can make for electrons it's hard to make long wavelengths for electrons but in principle if you could if you could create electrons with long wavelengths you could do pretty much the same thing long wavelengths what does long wavelength mean it means very low momentum very low momentum means very slow right how do you make very very slow electrons it's not so easy because where do you get electrons from you get electrons out of hot wires for example the hot wires eject the electron with some velocity how do you get very very very slow electrons not so easy to get them so if you want to make long wavelengths you want very slow and it's not so easy to get very slow what's that for electrons yes oh uh you need to make the slits very very small and very very close together to see this that is true so even even optically even optically you uh you need to make the slits pretty close together uh they can they're visibly close together they're not so close together that you can't see them but but you need to make them close independence of the sauce you change the sauce and you still get the same curve yeah yeah pretty yeah you get the same sorts you see yeah you get the same effect um once the electron comes out of the hole here it pretty much forgets what it was doing over here and it just becomes an electron which was produced over here it becomes an electron which was shot out from that point that's the idealization that you can approximate the electron as an electron ejected from exactly that point or exactly that point an electron ejected from this point or from this point uh add up to a wave which has this kind of interference pattern yeah there's a way to bring the same electron back say it again if there's a way to bring the same electron back it's still in the same type if there's a way to send the electron back once it has gone out yeah the same electron somehow cycled back through the hole through the hole here you want to cycle the electron through one hole and back to the other no the first issue of reversibility oh oh oh oh you want to know what would happen if the electron passed through and then you reversed everything is that what you're asking if you recorded it on the screen oh well um you could imagine that the electron bounces off the screen leaves a little spot there and bounces off and then you could uh of course you could then try to focus that electron back through the hole you could do that and you would get the same kind of pattern on the side no you want a cyclotron no i don't think you want a cyclotron you can use the same electron over and over you can use the same electron over and over you can collect the electron from here bring it around bring it through the hole whatever you want and do the experiment over again with the same electron oh yeah you can definitely you can use another electron you can use the same electron the last time you told us to if you if you detect it then that changes it if you don't detect it if you detect it over here that changes it yeah oh no if you detect it at the hole ah okay um let's come back that you would you now you're asking the question about reversibility that's a different question than whether you can use the same electron over and over for the same experiment okay that's all right let me come to that next time let me come to that next time it involve in fact let me come to that after we talk about hamiltonians we have to understand hamiltonians and evolution before we can answer that question the preceding program is copyrighted by stanford university please visit us at stanford.edu

Channel: Stanford

Views: 264,595

Rating: 4.7333331 out of 5

Keywords: relativity, Physics, math, calculus, geometry, algebra, statistics, quantum, mechanics, discrete, variable, amplitudes, state, vectors, vector, space, variables, eigan, permission, operators, angular, momentum, wave, function, particles, probability, complex, numbers

Id: oWe9brUwO0Q

Length: 119min 33sec (7173 seconds)

Published: Thu Apr 10 2008