HC Verma Lecture on Dirac Delta Function

so this short torque is on Dirac Delta function which is very widely used in quantum mechanics and the Dirac Delta function at such it's something which is zero everywhere and infinity at one point so if you try to visualize it in a geometrical form it will be something like so it's called Dirac Delta function and Dirac is in fact greater physicists PA M Dirac who has contributed a lot in development of quantum mechanics and so that name is Dirac Delta function it will look like something like this it goes to infinity and then comes back and becomes 0 it remains zero everywhere so if this is X and this is that function which we generally write like this Delta X then this is X equal to zero so everywhere everywhere on this side that side that means for X less than 0 this function is 0 X greater than 0 again this function is 0 and at X is equal to 0 one single point Delta X is infinity so this is how it looks like and the in addition to this in addition to this there's another restriction another equation which defines this Dirac Delta function and there is area under this curve that means integration of this function from let's say minus infinity to plus infinity ok this area should be 1 so the height is infinity the width is 0 and if you calculate the area under that that should come out to be 1 so that is the restriction that is the definition these four condition one two three four these four define a Dirac Delta function a function which follows all these four is defined as direct Delta function so you can realize this Dirac Delta function as a limiting case of various functions that you can create which are more friendly for example you can have a function you take a rectangle like figure here rectangle like figure here this is X is equal to zero this is let us say X is equal to a by 2 this is X is equal to minus a by 2 this point so that this width is a and this height let us take 1 by a so the function is defined as function FX is defined as this is equal to or for X less than minus a by 2 FX is 0 then from minus a by 2 to plus a by 2 X between minus a by 2 and plus a by 2 this is 1 by a and then for X greater than a by 2 it is 0 once again and at minus a by 2 at a by 2 you can take any value you can take this value or this value or any value or you can define your own values so let's say X is equal to minus a by 2 FX can be 1 by 2 a and X is equal to plus a by 2 again you can take it 1 by 2 a and so on so this is a function the area will be 1 this is a and this is 1 by a so area will be 1 and this side is 0 this side is 0 in between it is 1 by a now you take the limit you take the limit of this FX not value limit of a function itself how does this function transform itself when you take the limit as a tends to 0 if a tends to 0 1 by a tends to infinity this gap becomes 0 so everything is at X equal to 0 collapses at X equal to 0 so before X equal to 0 it remains zero function it means zero after this X equal to zero it remains 0 at X equal to 0 the functional value is 1 by a which is infinity and then the area is 1 so this becomes Dirac Delta function if you are worried about the discontinuities and all that is a discontinuous function of course you can still get in the area under the curve you can start with a triangle also you can take this kind of picture and then you can write this is X is equal to 0 and you can write equation of this straight line equation of this straight line up to here and then it is 0 here then it is 0 here and once again this this width and this height they are to be properly chosen so that the area is 1 and when you take that weight going to 0 this becomes a Dirac Delta function other variety is Gaussian functions which has certain width and certain height you can always connect that width and height so that the area remains 1 and then take the limit so that the widths become 0 height becomes infinity and it is Dirac Delta function so that's for the definition point now one beautiful representation which is actually very widely used in quantum mechanics and which has a very interesting interpretation also that I will write that functional form of direct Delta function I will write and that is integration from minus infinity to plus infinity e to the power 2 pi X and then some variable P and integrated over P DP since we are integrating over P so this is the variable and this is constant during this integration this is constant but you can use another constant another constant another constant P finally you will be putting values so this whole thing will be just a function of X now this function happens to be Delta function right it's not obvious but it happens to be Delta function and we'll just see what it is why this is Delta function so let's define a function FX which is integration from minus L to plus L and then this e to the power 2 pi X P DP and then we will take limit as L goes to infinity to get this function and this is a very integrate very simple integration so this will be e to the power 2 pi X P divided by we are integrating over P so divided by 2 pi I X and the limits are minus L 2 plus L and you put the limits so this is to 5i X here and P remember we are integrating over P so it is e to the power 2 pi X L and minus e to the power minus 2 pi and I Excel okay you put P equal to L here you get this term then P equal to minus L here you get this term and then you have to subtract and this is anyway common factor here now you know what this is you know this P to the power IX minus e to the power minus I X divided by 2 I this is sine X this is sine X and if you put plus here and don't put I here it will be cos x so this is sine of 2 pi X L it is 2 pi X and divided by PI X sine of 2 pi Excel ago I will be there and then - I will be on the other side so PI X here so this is the this is this function FX and we want to see how this function behaves as L goes to infinity alright so let's try to plot this let's try to plot this so I draw the x-axis and then I draw the y-axis and let's first see at X equal to 0 what happens you know X is equal to 0 sine X by X sine X by X that limit goes to 1 as X tends to 0 so let me write here PI X to L so 2 L here into 2 L and into 2 L so that it is of this type sine X by X as X goes to 0 sine of this argument divided by this thing itself and then as X tends to 0 this whole thing tends to 0 and therefore it will be 1 and therefore as X tends to 0 this will go to 2 well so at X equal to 0 this function is somewhere here this is to it so from this side from this side they are all converging here now what happens if this argument is PI that means 2 pi X L 2 pie XL you are increasing X now this is X equal to 0 and now we are increasing X slowly so when you are increasing X slowly this thing at some stage will become PI so X is equal to 1 by 2 L at X equal to 1 by 2 well let's say this is X equal X is equal to 1 by 2 L and this is 2 by 2 L and this is 3 by 2 L and so on similarly on this side minus 1 by 2 L minus 1 by 2 L then you have minus 2 by 2 L and so on at 1 by 2 L this is PI sine of PI and 0 so your function has to be 0 here again at X equal to 2 by 2 L this function will be 2 PI sine of 2 pi is also 0 and remember now your denominator is not going to be 0 and therefore 0 divided by non zero will be 0 so at all these points this function should be 0 there the first thing the second thing is as you go away from origin you are dividing by a larger and larger number we are increasing X so that denominator is becoming larger and if you divide by a larger quantity that value decreases so because of this denominator if the denominator is not there it would have just a sine function with equal amplitude but now this amplitude will keep on decreasing because you are dividing by larger and larger value so it will be something like this it will become 0 here and then it will be maximum here don't it's not to the scale this looks larger than this but doesn't matter so so it should have been maximum here here it will be PI by 2 then Phi this will be 3 PI by 2 negative and maximum but not maximum is not going to be 2 L it will be less than that and therefore again here it will be 0 so it will be something like this this is larger this is shorter once again here it will be maximum here but that maximum will be shorter than this so it will be like this and then it comes down and then it goes like this and it comes down so it's this type and this is an even function symmetric if you put minus X in place of X nothing changes the sign X also gets its negatives negative and this will be its negative so if you put minus X in place of X here and minus X in place of here both those minus will cancel out and you get the same thing so the same thing will be on this side symmetric like this there's a kind of function now what happens if L goes to infinity okay we we are working with this function and we are targeting this function so L going to infinity if L goes to infinity if L increased beyond limits this 1 by 2 L will come here 2 by 2 L will come here 3 by 2 L will come here all these Maxima which you are seeing here here here here here all these Maxima they will all come at the origin as L tends to infinity and what is left what is left 0 everywhere 0 is becoming shorter and shorter this is becoming shorter and shorter ok and all these Maximas are coming here so what is left it's 0 everywhere and what is this height it is 2l so as L goes to infinity its shape is going to be just it comes here it goes to infinity and then comes back and become 0 so that is a kind of Delta function is the area under the curve 1 will the area under the curve 1 it looks like a delta function fine but our conditions on area should also be satisfied so in this situation if you calculate the area you can get the answer there's a wonderful integration and that is integration of sine X by X DX minus infinity to plus infinity now it looks simple sine X by X but then when you go to integrate it say is not that easy there are various ways of integrating it there are research papers on how to integrate this and people have devised varieties of techniques to get this and research papers compare which one is easier which one is harder how many steps and so on so and so and so on one way is using complex plane and then integrating some function e to the power Z by Z and blah blah blah blah blah blah but the result is very simple and the end result is that this integration is equal to PI so we will be using this result so the function we had obtained was sine of two pi X L and divided by PI X right this was the function that we have obtained after writing that minus L 2 plus L and integrating we got this so let us take this function and integrate it over minus infinity to plus infinity so if I integrate it to minus infinity to plus infinity over X a very simple change of variable I can make 2 pi X L this I can write some X Prime let us say and then DX will be DX prime divided by this 2 pi L it is this and this PI X will be PI X will be X prime divided by 2l so let's write these things here and this equation becomes minus infinity to plus infinity sine of X Prime that's what X prime is 2 pie XL so 2 pie XL and this DX you can write here DX is DX prime divided by 2 pi L and then denominator PI X you can write as X Prime and then divided by 2l so multiplied by 2 L here you can write to L here or divided by L here so this is it now this 2l cancels with this 2l and it is reduced to this pi you can write here 1 by PI and then it is minus infinity to plus infinity sine of X Prime and divided by X Prime and then we explain this as I told you this is PI and this is 1 by PI so this is equal to 1 the interesting thing is it is independent of L it is independent of L and therefore as L tends to infinity this integration still remains 1 ok this function this function itself when L tends to infinity becomes Delta function Dirac Delta function and the integration of this function from minus infinity to plus infinity since it is independent of L even if L goes to infinity this integration remains 1 that means this Delta function which we are constructing the area under that Delta function is also 1 this function this function which is on the board with some L the total total area is 1 right the total area under this curve is 1 whatever L is and for just like our triangle and rectangle the total area we fixed that one and then we took the limit so take the limit and the area remains one and this is Dirac Delta function so we have established this now very interesting things come out of this one is you can consider it just take this part just take this part what is this okay now let me it is this what is this e to the power 2 pi I XP and consider this as a function of X consider this as a function of X now I am NOT integrating and I am NOT doing this I am just taking this part and considering it as a function of X for some particular value of P this is a periodic function the important thing that this is a periodic function because this is what this is once again you know e to the power I x is cos x plus I sine X and so this is cos of 2 pi XP or 2 pi px plus I times sine of 2 pi px each of them each of them is a periodic function and what is the period at what value of x it repeats so that is given by this is 2 pi P X so 1 by P right Delta X equal to 1 by 3 if you put if you increase X by 1 by P 1 by P cancels with P so you are increasing this by 2 pi increasing this by 2 pi and therefore it repeats sine cos they have periods 2 pi so 4 X 1 by P you are repeating it is a periodic function and that period or the frequency in this much of length those oscillations are there in the graph so that depends on P so essentially if any one of this or this if I if I draw it will be like this is a periodic function and then this is this depends on P now if I take a different P I get again a periodic function but this will be changing so I make it like this or I make it like this so the periods are changing and what I am doing here I am taking such functions for different P that means I am taking this I am taking this I am taking this I am taking this and all varieties of P going from minus infinity to plus infinity continuously vary so continuously varying all varieties of this periodic functions we are adding them together integration is just adding I have I am getting I am getting a delta function so when you are adding this function to this function to this function to this function to this function to this function with various these frequencies with equal amplitude don't increase or decrease amplitude what you get is this Plus this Plus this Plus this Plus this Plus this you are getting this very interesting so this is one beautiful thing about this representation now I'll touch upon one more thing which is known as or to normality orthonormal functions if you have two functions say f1 X and f2 X we define some kind of a dot product scalar product it's called inner product and people also write it this way this is called inner product of what of the two functions but the product is a number these are two functions but the inner product is a number and how is that defined dran is defined as integration from minus infinity to plus infinity the first functions complex conjugated the second function and then DX this is the definition of inner product and then if if this inner product is zero two functions are there and if the inner product is zero then the functions are called orthogonal to each other remember if you have two vectors a and B and if a dot B is zero you say that a is perpendicular to be orthogonal to each other in that same spirit two functions are called orthogonal to each other if this inner product is 0 and then normalization not to normal for normalization if they are same you are taking inner product of a function with itself then what should its value that will be infinity but then it's 0 here it's infinity here so it should be some kind of a delta function so for orthonormality it should be some kind of a delta function the beautiful thing is that these functions these functions these functions they are orthonormal that means if you take different values of P how are you getting two different functions f1 X and f2 x by taking two different values of P so if you take two different values of P and write two functions and take the inner product that is going to be 0 and not only 0 so let's take two values of this and construct two functions so f1 X is e to the power 2 pi X P 1 and F 2 X is e to the power 2 pi X P 2 what happens if i take the inner product f1 f2 this will be integration from minus infinity to plus infinity the first function complex conjugated so e to the power my two pi X P 1 the second function remains at such eetu the power 2 pi I XP 2 and DX that summer definition two functions of X are or inner product of two functions of X are defined in this way first function complex conjugated second function as such integrated over X in the full range minus infinity to plus infinity now write this this is minus infinity to plus infinity and then e to the power 2 pi I X P 2 minus P 1 and DX let us write X on this side okay so that's the inner product now compare this and this compare this and this here integration is over P is a function of X and that is Delta X but the same identical structure is here now it's a function of X I'm treating this as a function of X and this p2 minus p1 is fixed here I was integrating over P this was a function of P and this X was a parameter but that parameter is changing for different integration different values of X you keep taking different values of X evaluate it for each X and you get a function of X the same story is here now it's a function of X and this p2 minus p1 is a parameter take different values of p2 minus p1 and keep on integrating and you will be getting a function of p2 minus p1 mathematical structure it's safe and therefore this is nothing but delta p2 minus p1 this is a delta function in p1 p2 so if you if you plot on fee scale if P 1 P 2 are same it goes to infinity if P 1 P 2 are different then it's 0 so again it'll difunctional so if I fix let's say I fixed this P 2 and P 1 is here at P 1 is here T 1 is here P 1 is here P 1 is here so it's 0 that inner product is 0 0 0 0 0 P 1 P 2 are different P 1 P 2 are different so it is 0 and then here P 1 P 2 are same and it goes to infinity and then again it is 0 so these functions are also or to normal these properties are used in quantum mechanics to talk about position talk about momentum and how a definite position can have too many momenta together a definite momentum state can have too many positions together all these things are described in terms of these Dirac Delta functions

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Length: 31min 52sec (1912 seconds)

Published: Mon Sep 02 2019