 the more accurately we can measure the position of a particle the less accurately we can measure the momentum of the particle and the more accurately we can measure the momentum of the particle the less accurately we can measure the position of the particle and I've gone a little bit further in the third case scenario where the wave packet is even more localized then the Fourier transform leads to a much more wider spread in the k values. We cannot know the future for sure because we cannot know the present for sure. This kind of introduces some very deeply philosophical ideas about how nature functions in the microscopic world. In quantum mechanics we regard a particle as a localized wave group or a wave packet. The moment we introduce this picture then it automatically leads to certain fundamental limitations in measuring the particle's position and momentum. You see we have talked extensively about this concept of a wave packet in my previous videos however we have not talked about the consequence of a highly localized wave packet versus a wave packet that is spread out in space. You see this wave packet is constructed by a wave or a wave function that we call as psi. Now what is the physical interpretation behind psi? Well it basically gives us an idea about the likelihood of where the particle is going to be found. That means wherever the wave has a greater amplitude there is a greater likelihood that the particle is going to be found there. To be precise psi mod square gives us the probability density of a particle being found at a given location. That means wherever the wave packet is spread out wherever its amplitude is non-zero there is a greater probability that the particle would be found in that region and lesser probability away from that particular region. So essentially the localization of the wave packet gives us an idea about the region in which the particle has a greater likelihood of being found which leads to a very simple interesting conclusion. The greater the localized wave packet the greater is the accuracy of the particle's position. So if we have a highly localized wave packet then it means that we are certain about the particle's position to a greater degree. We are more certain about the particle's position in this wave packet as opposed to the lower wave packet which is spread out over a larger region. Now let me ask you a question what about the wavelength? In which of these wave packets are you more sure about the wavelength of the particle? Of course in the lower case because wavelength has to do with the periodicity of the particular wave. So the greater the spread out nature of the wave packet the greater certainty we have regarding the wavelength and the lesser certainty of wavelength we have in this case here. So we know from the De Broglie hypothesis the wavelength is related to the momentum. So if we have a wave packet where we are more sure regarding the wavelength that means we are more sure regarding the particle's momentum and if we have a wave packet where we are not that sure about the particle's wavelength where we have a greater amount of uncertainty regarding the particle's wavelength that means we have a greater uncertainty regarding the particle's momentum. So that means for a highly localized wave packet we are more sure about its position but less sure about its wavelength or momentum but for a spread out wave packet we are more sure about the wavelength or momentum and less sure about its position. This inverse correlation that exists between the uncertainties of the position and the momentum is what leads to the Heisenberg's uncertainty principle. It simply states that it is impossible to measure the exact position and exact momentum of a particle at the same time with absolute accuracy. This principle was given by Werner Heisenberg in 1927 and represents one of the most significant laws in physics today. So in this video we are going to talk in detail about the Heisenberg's uncertainty principle and I'm going to give you a somewhat proof of this particular principle based on our understanding of the wave packet in position space and momentum space. I'm going to divide this entire video in essentially two parts. In the first part I'm going to give you a qualitative idea regarding this kind of an inverse correlation and in the second part of the video I'm going to derive the exact expression for the uncertainty principle for a Gaussian wave packet. But to do that first I must introduce to you the concept of Fourier transforms. Fourier transform is a mathematical technique or a mathematical operation in which we can convert a signal or a function from one domain into another. If you are familiar with what Fourier transforms are usually Fourier transforms are done in signal processing where we convert a signal from our time domain to a frequency domain but in our case we are going to make transformations between position domain and momentum domain. So let me elaborate on this a little bit further but first of all let us talk about the concept of the wave packet. We have talked extensively about this so I'm just going to introduce to you the function that describes the wave packet properly. So we are going to use psi here which is a function of x and t which is equal to 1 upon root 2 pi integration from minus infinity to plus infinity phi k a to the power iota k x minus omega t d k. What this simply means is that you have a large number of plane wave solutions that you superimpose over each other so that all those plane wave solutions or sinusoidal variations constructively interfere in a very tiny region and you get the peak maxima and destructively interfere everywhere else. This is essentially the concept of a wave packet. Now this wave packet is not just a function of x it is also a function of time so right now we are not really interested in the evolution of the wave packet we are interested in looking at the wave packet at a given instant in time. So I'm going to say okay at time t is equal to 0 what is the nature of this wave packet function I'm going to say okay psi x 0 is equal to 1 upon root over 2 pi from minus infinity to plus infinity phi k a to the power iota k x d k. So essentially the time component has been removed because for a time t is equal to 0 this is essentially the wave packet. What this wave packet simply represents is that you have a plane wave solution you know plane waves are essentially complex functions that has sine and cosine components. So you must be familiar with e to the power iota theta which is equal to cos theta plus iota sine theta. So essentially this complex function has its representation in both the real plane as well as in the complex plane and in both the planes it's simply a sinusoidal variation. When we add up a large number of these sinusoidal variations then let's suppose these sinusoidal variations all of them are at phase at x is equal to 0. So as x is equal to 0 then e to the power iota k x tends to what 1 which I can write as e to the power iota k x tends to 1 for all the plane wave solutions and they all constructively interfere to give you a large peak but as you go away from x is equal to 0 as you go from let's suppose x far away from 0. In that situations e to the power iota k x simply gives you a large number of random or arbitrary oscillations which ultimately cancel each other out and we end up getting destructively interference away from the center of the wave packet. Now in this particular expression phi k essentially represents the amplitude of each of these sinusoidal waves but the amplitude is not constant it is a function of k which is essentially related to the wave number and this is where Fourier transform comes into picture. The Fourier transform of this particular wave essentially gives us a technique to find out phi k as a function in terms of psi x. So if I find out phi k then phi k comes out to be 1 upon root 2 pi for minus infinity to plus infinity this particular function psi x comma 0 e to the power minus iota k x dx. What this function does is that it basically gives us an idea about the nature of the wave packet in what is known as k space. You see Fourier transform is a mathematical operation which converts a signal from one domain into another domain. Now if you look at the wave concept here the wave packet here this wave packet is represented in terms of position. So if I draw the wave packet the wave packet is represented in the position space or x domain. So here I can say this is the real part of psi x. This is the real part of psi x which is represented in the x domain but the moment I do a Fourier transform it transforms the x domain into k domain. So essentially I should get from in a Fourier transform the same wave packet but in k space instead of in position space. So what we are trying to do is we are trying to go from position domain or position space to k space or k domain also known as momentum space. Now why is it called momentum space? You see k here is not just some arbitrary letter here. k is related to momentum. You see what is k by the way k is what we call as wave number. So when we say wave number we essentially mean that k is a quantity is equal to 2 pi upon lambda but what is lambda? Lambda is equal to h upon p. So if I write this here this becomes 2 pi h upon p but what is h upon 2 pi h upon 2 pi is a constant which we call h cut. So essentially this becomes k is equal to p upon h cut. h cut is a constant and because h cut is a constant k basically gives us an idea about the momentum of the particle. Since we know from De Broglie hypothesis the wavelength of a wave is somewhat related to the momentum of the particle. Therefore if we know the wave number of that particular wave we are essentially talking about the momentum of the particle in quantum mechanics. So here when I am going from a function of x to a function of k essentially what we are getting an idea of is the momentum of the particle. So we are going from the position domain to k space or momentum space from position space to momentum space and that is what we are going to look here. We are going to take a look at the wave packet in position space versus the same wave packet but in terms of the variable of k or in momentum space. So let me give you a very simple visual example of what it means to do a Fourier transform between signals and I think then it would become much more clearer for you. If we take a singular wave a single sinusoidal variation in position space so if we have some kind of a wave where the wave is represented by psi x and then it is in the domain of x and what I do is I essentially say that this is a singular sinusoidal variation that means it's just some sort of a wave cosine wave or sine wave whatever you want to draw it is a simple wave like this having a fixed wavelength. Alright so if I do this then what do we know about this waves what do you call a wavelength? So a very simple sinusoidal or cosine wave like this has one wavelength and because it has one wavelength it has one wave number. So in the k space or in the momentum space so I am going to call this as k and this I am going to call as phi k what we are going to get is we are just going to get one singular line here something like this. Essentially there is only one let's suppose I call it k0 corresponding to some lambda0 which is corresponding to this particular signal or sinusoidal wave function and they represent each other's Fourier transform so essentially I am going from a sinusoidal wave in x space to that same sinusoidal wave but in k space so they are each other's Fourier transform but I am not just interested in a simple sinusoidal variation I am interested in a wave packet. So let's take this argument forward what I am going to do is I am going to take the linear superposition of multiple waves you see at the end of the day the wave packet is a linear superposition of an infinitely large number of sinusoidal waves in both the real plane and the complex plane. So to go there let's first look at finite number of waves so if I take let's suppose three distinct waves with slightly different amplitudes let's suppose I have a wave which has a wave number k0 I have another wave which has a wave number k0 plus del k and I have another wave which has a wave number of k0 minus del k then what kind of a wave should I find here what we will end up getting is we will end up getting a combination of three distinct sinusoidal waves if these wavelengths or wave numbers are very near to each other we will end up seeing recurring beats something that looks like this it is always better to get these kind of conclusions from some sort of an example so let me demonstrate to you an example of these kinds of linear superpositions of waves so I have a couple of examples to show you and let us just take a moment and sort of get a feel for what we are discussing here kindly note that I'm trying to give you an idea a qualitative idea of how the position and momentum are correlated to one another and once I am able to convey to you this qualitative idea then I will move forward to giving you a quantifiable expression or mathematical equation so to give you a qualitative idea let's first look at the first example so I have taken a sinusoidal variation so I'm taking a cosine function here cosine function sine function same thing now the reason I'm taking a cosine function is because here in this particular software I can only plot real functions so even in this particular case I have taken a plane wave the plane wave is epsilon iota kx but that epsilon iota kx can be decomposed into a real function and an imaginary function so I'm only looking at the real function here but the same can be concluded for the imaginary function as well so here is a quite simply put a very simple sinusoidal variation or cosine variation of a function now what happens if I add three functions together so here I have three functions three cosine functions having a different wave number you see here we are talking about cos kx so I'm taking a k value of 1920 and 21 so it is centered around the k value of 20 and I'm also tweaking the amplitudes a little bit so that here the amplitude is larger compared to the other waves so if I am going to add these three waves what do I get I should get some sort of a recurring beats or something like that so let me shift this particular signal a little bit above here and then here I'm going to plot this and let me shift it a little bit further so now if we look at these functions and we should see it here you see these this is the answer we get this is essentially the sum of three sinusoidal waves and because these waves are similar to one another in terms of its wavelengths which are slightly different from one another what we end up getting are periodic beats you see these these are periodic beats or repetitions or recurrence of these wave groups so essentially we are interested in these wave groups but I don't want multiple wave groups right I basically want some kind of a singular wave group something that looks like this so how can we go from these multiple recurring wave groups to a singular wave group okay so for that I have a little bit of a qualitative example here let's do one more thing let's add up more number waves but keeping the range of k values to be the same so we are keeping the range of k values from this is my example that I've taken you can try on your own I've taken the example of range of k values from 19 to 21 units so I'm adding five waves in this particular range what is going to happen or what change we are going to see let's take a look and let's plot another wave here so this basically gives us an idea about now the superposition of five waves now kindly note here these are the superposition of sinusoidal waves and they are all leading to certain kind of periodic recurrence of these wave groups the central pulses you can say but the moment I add more number of waves with slightly varying wave number in that same range of wave number I end up getting wave groups that are far apart from one another or we can say in a region the number of wave groups have decreased let's take this argument a little bit further if I take now in the same range of k values from 19 to 21 all right I take even more number of waves from 19 19.2 19.4 up to 21 basically wave number units and I try to plot this then what do I get then let's let's see if we can get something similar yeah we get something similar in the sense that we can see what's happening here you see you see this purple so this is essentially the the sum of multiple waves but because I've taken more number of waves you can see what's happening here okay let me just create a little bit of distance here between these waves you see what's happening and what's so interesting here at this particular moment that I am taking a larger number of waves to add them up together but I'm keeping the range of wavelength in the same range okay I'm just taking multiple wavelengths in that particular range and when I do that the number of wave groups decrease in a particular region or the distance between individual wave groups or pulses increase so essentially what does this lead to this leads to the argument that if I want a singular wave packet if I want a singular wave group what I need to do is I need to add infinite number of sinusoidal waves in a given range of wave number or wavelength if I add infinite number of sinusoidal waves with infinite similarly small variation in its wave number or wavelength then I can theoretically construct a singular wave group a pulse or a wave packet that is constructively interfering only once and never again so essentially what I want is I want a large number of waves which are slightly different from one another or infinitesimally different from one another in terms of its wave number or wavelength so I want a large number of waves infinite waves in fact that are different from one another by an infinite symbol change in its wavelength or wavelength and they will constructively interfere to create a singular wave packet this is beautifully illustrated in this particular textbook by Hecht a textbook called optics I will give you the source for this particular textbook where they have given a very similar sort of a representation on the left hand side you can see the momentum space or the k space and on the right hand side you can see in the position space so initially you can see that for a singular wave you have only one wavelength or one wave number but then there are three waves having different amplitudes and then you end up getting beats but then you have multiple wavelengths or wave numbers then you get beats but the wave groups are further apart or there are lesser wave groups in that particular region when you increase that the number of wave groups decrease and if you reach the final conclusion here where you have an infinite number of waves having wavelengths which are infinitesimally different from one another you can theoretically construct a singular wave packet in this particular manner now we could have easily constructed some sort of a different function here I could have taken some kind of a rectangular uniform band of k values there is a very specific reason why I have taken sort of a this sort of a representation so what this distribution does is it gives us an idea about the amplitude corresponding to those particular wave numbers you see if we take a Fourier transform of different kinds of bands of wave numbers then essentially we end up getting different kinds of wave packets so for a uniform band of frequencies I may get some kind of a sync function for a Gaussian distribution we will get a Gaussian wave packet for a Lorentzian momentum space distribution we will get a double exponential distribution of wave packet in position space but I am interested more or less in the Gaussian distribution because you see in quantum mechanics whenever we try to describe a particle most of the time we are interested in a Gaussian distribution this is the most common sort of an approach and also in the context of the uncertainty principle as we will discuss later on the Gaussian distribution also represents a minimum uncertainty condition therefore I have taken a more or less a Gaussian distribution of this wave number in which I am including not just an infinitesimally large number of waves in a given range of k values but I am also taking the k values to go from minus infinity to plus infinity something like this from minus infinity to plus infinity now I will just mention it here that the wave packet that we get here in this kind of a situation and in position space and momentum space are both Gaussian in terms of the mathematical functions so a Gaussian wave packet in a position space gives us a Gaussian wave packet in a momentum space but now let us take this sort of a we are having a little bit of a play with these functions this argument a little bit further I want to show you something more interesting so coming back to our sort of a you know we are playing around with these functions what happens if I increase the band of the k value so initially I took a large number of waves in a range from 19 to 21 so let me take that wave again okay so let me take that wave in the range from 19 to 21 and let me plot it here okay I end up getting these recurring bands fine no issues now what if I sort of do something else so let me shift it vertically a little bit I'm just shifting it vertically okay to distinguish between the different functions so what if we go for a greater range of k values you see earlier it was 19 to 21 but now I want to take from 18 to 22 and the reason I'm doing this is because I just want you to see for yourself what happens when we add a large number of waves okay I just don't want to give you the answer I want you to see also through visualization what we get so if we look at another sort of a wave here but having a k values which are more spread out if you compare these two pulses you see that in the earlier case where k values were less spread out the wave group was more spread out but in this situation where the wave k values are more spread out the wave group is less spread out okay so let me take this argument a little further if I take the k values from 17 to 23 okay so I'm increasing the range of the k values and I'm adding a large number of waves in that particular region of course I'm modulating the amplitude also to a certain degree and I'm going to plot this here what you see is the width decreases even further the width of this central wave packet decreases even further you see that if I compare three distinct beats see I'm unable to infinitely add a large number of infinite waves in some sort of a program so I'm just adding a finite number of waves since I'm adding a finite number of waves I'm getting these recurring wave groups fine I just want you to notice that the moment I increase the k values initially the k value was from the range of 19 to 21 then I took the k values from 18 to 22 and then I took the k values from 17 to 23 so I'm increasing the k values and the range then this automatically leads to a more localized wave packet a much much more localized wave packet you see this the first one is the most spread out wave packet the second one is the most localized kind of wave packet now I want to take this argument a little bit further and to do that I have actually created a program to sort of do a Fourier transform of a Gaussian wave packet and I have tried to tweak the spread or the standard deviation of the Gaussian wave packet so in the initial case if you look at the first correlation the Gaussian wave packet is kind of spread out in space but if you look at the k values then the k values are slightly localized in the momentum space or in the k space but in the second case scenario I've tried to localize the wave packet by bringing the points closer and in that sense if we do a Fourier transform we end up getting the distribution of the k values to be a little bit more wider in its spread and I've gone a little bit further in the third case scenario where the wave packet is even more localized then the Fourier transform leads to a much more wider spread in the k values see the k values are essentially the values of the wavelength right these are the values of the wavelength of all the waves that we are adding up together and if we take a greater range or greater contribution of these wavelengths or these wave numbers in the linear superposition we get a much much more localized wave packet in the position space so essentially what happens is that if I take a much more spread out sort of a distribution in k space so again these are all this is phi k right in terms of k this is psi x alright and again this is psi x and this is phi k so if I sort of take a greater spread in the k values here so if I take a distribution of k values which has a greater spread or standard deviation then what that essentially means is that we end up localizing the wave packet even further so we might end up seeing something like this kind of a wave packet this is by the way a very very interesting feature of how waves behave when we represent a linear superposition of sinusoidal waves then essentially we require an infinite number of waves having slightly varying wavelength or wave number and they can create a singular wave packet but if we try to increase this distribution or if we try to increase the standard deviation of this kind of a distribution function in this particular manner then what we end up getting is we end up getting a wave packet which is much more localized in space what does this mean it simply means that we have increased the accuracy of finding the particle in a given region which means that we have decreased the uncertainty in the particle's position but we have increased the uncertainty in the particles k value or momentum this in a sense gives you an idea about the Heisenberg's uncertainty principle so to sum it up if we sort of represent the spread of a wave packet in position space by some kind of a quantity called I'm going to call it as del x so del x represents the uncertainty associated with measuring the position of the particle and similarly I'm going to say that okay there is a certain spread in the distribution in k space I'm going to call this as del k so del k gives us an idea about the uncertainty associated in measuring the wavelength or the wave number or the momentum of the particle then based on this argument we get a qualitative idea about the nature or the correlation between del x and del k so del x is inversely correlated with del k or del p so this is a qualitative argument behind the Heisenberg's uncertainty principle then the moment we take the concept of a wave packet to describe a particle and the wave packet can only be constructed by a linear superposition of a large number of sinusoidal waves or plane waves having slightly varying wavelength or wave number then the distribution of that wave number and its spread is inversely correlated to the distribution of the wave packet in position space and this inverse correlation introduces an uncertainty that we cannot go beyond it this is a fundamental limit that is introduced by the Heisenberg uncertainty principle in terms of measuring position and momentum of a particle the more accurately we can measure the position of a particle the less accurately we can measure the momentum of the particle and the more accurately we can measure the momentum of the particle the less accurately we can measure the position of the particle now this simply gives you a qualitative idea I also want to give you a quantifiable mathematical expression and for that let me take the example of a Gaussian function. A Gaussian function is also known as a minimum uncertainty wave packet because it is one of those functions which corresponds to minimum uncertainty relation. So let me obtain the actual Heisenberg's uncertainty equation for the case of a Gaussian wave packet. So let me rub some of this board first. All right, so here I am taking the example of a Gaussian wave packet, but instead of the wave packet I am taking the mod square of the wave packet because essentially at the end of the day the mod square of the wave function gives us an idea about the probability of the particle's position and the mod square of the wave packet in case space gives us an idea about the probability of the particle's momentum being a certain value. So therefore, because the wave packet is Gaussian and its Fourier transform in the momentum space is also Gaussian, we have two distinct Gaussian wave functions of psi square and phi square. All right, because we are just interested in the spread of the wave packet or the spread of the probability density associated with the wave packet in both momentum space and position space in case space and x space. So that is what we are interested in. I have taken two examples. These are two unique functions. Let me describe what they are. You do not have to get scared. They look complicated, but they are not really. So in the position space I have the wave function which wave function simply represents this plane wave. This is a plane wave and with the plane wave we have this Gaussian function distribution. Now I told you that we end up creating a wave packet by a linear superposition of an infinite number of sinusoidal waves, but if we are successfully able to do that we can simply replicate with some kind of a Gaussian function in this particular manner. So this here represents the plane wave. This represents some sort of a Gaussian function and that arises from the infinite series sum of all the plane waves and this is just some sort of a normalizing function or a normalization constant. And if we do a Fourier transform from here to here I simply end up getting this form. Now some of the students may be good at mathematics or interested in mathematics. So I give you this problem to find out the Fourier transform of this particular function. If you find out the Fourier transform of this function you should get this and if you find out the Fourier transform of this function you should get something like this. If you are interested you can try that particular calculation on your own in your notebook. Now since I am interested in the spread of the probabilities in the position and momentum space so what I am going to do is I am going to say or I am going to define the half width. So the half width here I am going to call this as del x and the half width here I am going to call as del k. So to find out the half width we need to find the mod square of psi and phi. So let us first find out the mod square of psi and phi. So mod square of psi because mod square simply means that you have a complex function. If you take the complex function mod square you multiply the complex function with its complex conjugate. So the moment you take a complex conjugate of e to the power iota k naught x it will become e to the power minus iota k naught x both of them will cancel each other out and all we will have left is 2 upon pi a square root over and e to the power minus 2x square upon a square. This is psi mod square and if we are interested in the phi mod square so this basically gives us again a square upon 2 pi root over e to the power minus a square k minus k naught whole square upon 2. Now you have to keep in mind that the wave packet and as a result its probability density is centered around x is equal to 0 and the wave packet in momentum space and its corresponding probability in momentum space is centered around some k is equal to k naught value that is why I have k minus k naught here okay. So now we are going to define this del x and del k so this is basically the half width or the standard deviation of this distribution function where the function with respect to its central value takes this particular form. So at this particular width so you have psi del x plus minus and times 0 mod square upon the central value which is psi 0 0 mod square this has a fixed definition of e to the power minus half and similarly to find out the half width in this particular case we take phi k naught plus minus del k mod square upon phi the central value which is at 0 or no so it is at k naught square is equal to e to the power minus half. So we maintain a consistency in defining the half width at a particular range. So let us just solve this and see what we get okay. So if I solve this particular case I of course the constants will get cancelled each other out and all we are going to be left with because x is equal to 0 so this is going to e to the power minus 2x square upon a square is equal to e to the power minus half this gives us minus 2x square upon a square is equal to minus half which simply gives us that half minus minus gets cancelled 2 becomes 4 so x square is equal to a square upon 4 so x is equal to or this should be del x by the way I am sorry this is del x okay this is del x okay this is del x minus 2 del x square so this is minus 2 that del x square okay because we are defining del x at this particular location here so therefore del x is equal to a upon 2. Now of course I am taking the positive value the reason I am taking positive value is ultimately because del x is what del x is the uncertainty associated in the particles position and that is always positive so it comes out to be a by 2. Now if I do the same thing here what do I get here in this particular the constant gets cancelled and for k is equal to k naught plus minus del k k naught k naught gets cancelled and we are left with e to the power minus a square and k is equal to k naught plus minus del k so this is basically del k square upon 2 is equal to e to the power minus half again what do we get from here this basically is equal to minus a square del k square upon 2 is equal to minus half minus minus gets cancelled and 2 2 also gets cancelled and finally we are left with del k is equal to 1 upon a del k is equal to 1 upon a again I take the positive values because we are interested in the uncertainty which is a positive number so let us now multiply del x and del k if I multiply del x and del k this turns out to be a by 2 into 1 upon a so this is equal to half so we have finally here del x del k is equal to half so this is the uncertainty relation in terms of position and k value however we are interested in the correlation with momentum I already said that in the last slide that k is related to momentum as k is equal to 2 pi upon lambda lambda equals h upon p so k is equal to p upon h cut right so the uncertainty in k is equal to the uncertainty in p upon h cut so if I substitute this particular expression above here then what do I end up getting so del k is equal to del p upon h cut so this simply ends up becoming del p equals h cut upon 2 this my dear friends ladies and gentlemen is the Heisenberg's uncertainty relation for a Gaussian wave packet keep in mind that I have taken a very particular example of a Gaussian wave packet because usually in quantum mechanics when we are describing the motion of particles we tend to deal with Gaussian wave packets they are a best representation of the particles motion however that is not the only kind of wave packet that is possible the Gaussian wave packet is also known as a minimum uncertainty wave packet because it is one of those special cases where we end up getting the theoretically minimum value of uncertainty is involved for other kinds of wave packets we do not get equality for other kinds of wave packets we get an inequality the uncertainties are only going to increase so we end up getting an inequality so the del x del p is always greater than h cut by 2 the minimum possible value is only for Gaussian wave packets but we only get a greater amount of uncertainty for other kinds of wave packets keep in mind that we have not yet talked about the uncertainties that gets associated due to measurement due to human errors due to instrumentation errors due to errors of you know calculations statistical errors all these things we have not really talked about we are only talking about the fundamental limit to the accuracy of position and momentum that is introduced in the theory of quantum mechanics as a result of the wave packet picture of a particle right and that wave packet picture of a particle introduces this fundamental limit to the accuracy of the measurements of position and momentum and this is the Heisenberg uncertainty principle that the product of the uncertainty in the measurement of a particle's position del x and the uncertainty in the measurement of a particle's momentum del p is always greater than this particular value h cut by 2 now let me clear the board once more before i make the last final statements so if you want to just take a snapshot or take a note of it you can do that and let me rub this so this product in the uncertainty of the particle's position and the uncertainty in the particle's momentum is always greater than this particular constant is known as the Heisenberg's uncertainty relation we have proved that this is equal to h cut by 2 for a particular Gaussian wave packet but keep in mind that the uncertainty relation does not take into account other things for example the uncertainty relation does not take into account statistical errors the uncertainty principle does not take into account instrumentation errors we are not really talking about building a measuring device which is more and more precise you know we can always try to develop instruments that are making more and more precise measurements that are trying to get more and more precise values we can always do that but what this expression is talking is not about the precision of instruments it is talking about a fundamental limit to the accuracy of these measurements that automatically arises in the theory of quantum mechanics all right so the moment we consider instrumentation errors the moment we consider statistical errors the uncertainty of obviously increases right this is only the limit the uncertainty increases because of all these factors right so if we try to make the measurement of the particles position very accurate so if I try to create a highly localized wave packet so we try to measure the particles position with as much accuracy as possible that means the uncertainty in the particles position tends to zero then this expression tells us that the uncertainty in the particles momentum will tend to infinity and if we try to measure the momentum very accurately so the uncertainty associated with it tends to zero the uncertainty in the position tends to infinity we can never have both the measurements accurately at the same time the more accurate we know the position the less accurate we know about its momentum the more accurately we know about the momentum the less accurately we know about the position you see the position here theoretically is spread out from minus infinity to plus infinity and the momentum is just one singular value something that replicates a delta function and lastly it's the opposite so increasing the localization of the wave packet in the position space so increasing the accuracy of position decreases the accuracy of momentum and vice versa so before we leave let me also mention that this kind of a correlation does not exist just for position and momentum it exists for other physical quantities also as we will see later on this Heisenberg uncertainty principle is actually true for many pairs of physical quantities and quantum mechanics so for example this is just for a particle moving in one dimension along the x-axis so what we have discovered is essentially the position uncertainty of a particle in the x-axis and the momentum uncertainty for its component along the x-axis all right but there are other kinds of similar uncertainty relations for example if we look at the y-axis and the momentum component along the y-axis then also we end up getting a similar uncertainty relation if we look at the z-axis and the momentum uncertainty in the z-axis then also we end up getting a similar uncertainty relation. In fact beyond momentum and position if we measure the energy of a system repeatedly then the time period between those measurements and the uncertainty associated with the energy of the measurements are also inversely correlated in this particular manner. Further if we want to measure the angular momentum of a particle then there is an uncertainty associated with its angular displacement which are also correlated by the Heisenberg's uncertainty relation. So there are effectively a large number of such pairs of physical quantities whose accuracies or uncertainties are inversely correlated and it all arises from this very interesting picture of quantum mechanics in which the particle is best described by a wave packet and the wave packet has a certain kind of spread in the position space which is inversely correlated to the spread of the same wave packet in the momentum space. And this principle therefore brings into our minds something very interesting about the nature of the subject of quantum mechanics that we do not really see in other branches of physics. You see we can never really measure the position and the momentum of a particle in absolute accuracy that is what this principle says. That means we can never really know exactly where the particle is and what its velocity is at a given point in time right now and because we cannot know where the particle is and what its velocity is at this very moment exactly we can never really know where the particle will be what the particle's velocity will be in the future. It introduces a certain kind of indeterminacy in the subject of quantum mechanics. It tells us that nature functions in such a manner that we can never really know the exact position and velocity of the particle right now and because of that we can never really know the exact position and velocity of the particle in the future. We cannot know the future for sure because we cannot know the present for sure. This kind of introduces some very deeply philosophical ideas about how nature functions in the microscopic world. Well that is all for today. I hope you have enjoyed this lecture you have understood what the uncertainty principle is and how it arises from. See usually in university lectures and college classes teachers do not really go so much in detail in depth. I myself face this issue so I hope that this video is solving the problem of this in-depth discussion of this particular topic that we do not really see in university classrooms most of the time and hopefully in that process illuminates your mind about how nature functions at a deeply fundamental level. So I hope you have enjoyed this lecture as much as I have enjoyed delivering it and preparing for it. I am Divya Jyothidas. This is for the love of physics and I will see you next time. That is all for today. Thank you very much. Have a nice day.