 Alright so I have here a couple of examples of some functions. Now all you have to do is you have to figure out which one of these wave functions are allowed and which one of them are not allowed. Let's do this exercise and you can test yourself. Hello everyone welcome back I am Div Vishwati Das and this is for the love of physics. In today's lecture we will try to answer the question what is a wave function? You see in quantum mechanics the most fundamental equation is that of the Schrodinger's equation. It's a second-order partial differential equation. So if we want to study any kind of a physical system we have to solve this equation and if we solve this equation what we end up getting is this psi. This is a Greek symbol pronounced as psi which is a function of x and t because we have taken the Schrodinger's equation for a 1D case. We could have taken it for 3D as well but for simplicity I am taking it for a 1D case. This solution of the Schrodinger's equation is known as the wave function. Wave function. Now what is interesting about this wave function is that it's a complex function. This wave function is a complex function so that means it represents what exactly? You see the Schrodinger's equation is a wave equation that is supposed to describe the matter waves of microscopic particles. We know by this point that microscopic particles demonstrate wave particle duality. Its motion has a certain kind of wave behavior associated with it. So whatever that wave behavior is the Schrodinger's equation tries to be the wave equation corresponding to those matter waves. So now the question arises that if this is the wave associated with the particles motion then what is waving? What is this wave of actually? A quick announcement for my students who are preparing for CSI and net physical sciences or gait physics. We at Elevate classes organize live classes for these examinations every six months. The next batch is starting January. So this is a full-fledged live batch with live interactive classes recorded access test series and everything that you need. It's a complete package. So if you're interested you can check out further details at elevateclasses.in. You can avail the early bird discount coupon which is available for the first 50 students and if you're not really very sure about these online classes then the first week of lectures are free to attend upon registration. So elevate your physics preparation with Elevate classes. You see when we talk about let's suppose ripples on the surface of a pond then we know that these ripples are essentially atoms which are displaced from their equilibrium position. When we talk about light waves we are talking about electric fields with respect to a particular propagation of a wave. So when we talk about the wave function of a moving quantum mechanical particle what is this wave made of? Now that is a very important question and that is probably a question that many students ask themselves whenever they first study quantum mechanical theory and the sad part is the answer is it is nothing. At least nothing that is physical. No physical quantity is associated with this particular wave function. It's a mathematical function. Its existence only makes sense in the context of the Schrodinger's equation. It's a computational device. It's a tool. It doesn't have any physical significance associated with it or as you say physical quantity associated with it but it does contain some information about the particle because if it didn't then the whole point of this process would be useless because at the end of the day I'm trying to solve the Schrodinger's equation because I want to obtain some information about the system or information about how the particle is behaving in the presence of some sort of a potential. So if this quantity doesn't contain that information what's the point right? So no it does contain some information it's just that this quantity itself doesn't resemble a physical quantity. So we have to perform some kind of a mathematical magic on it to extract that information and one of the very first magic that we can perform was given by the Born's Statistical Interpretation which we explained in the previous lecture. So the Born's Statistical Interpretation simply says that if there is a particle restricted to moving in the x-axis we solve the Schrodinger's equation we come up with the solution. Then the mod square of the wave function which is essentially the complex conjugate of the wave function times the wave function gives us the probability density of the particle being found in a given region which means that if I integrate this quantity with respect to dx at a given point in time between let's suppose locations x is equal to a to x is equal to b then this simply gives us the probability of the particle being found along the x-axis in the region x is equal to a to x is equal to b. Thus the wave function is a mathematical function which let's suppose if I solve for a given situation and then I plot this mathematical function mod square and then that mathematical function may look something like this and then that mathematical function simply gives us an idea of the probability of the particle being found between two locations if we made a measurement if we made a measurement what is the likelihood that the particle will be found somewhere that is the information contained in the wave function as given by the Born's interpretation. So clearly the idea of a particle being found somewhere has something to do with the wave function yes but the wave function isn't exactly a physical quantity. So in a sense if the particle has a some wave function then there is greater likelihood of the particle being some found wherever the amplitude of the wave function is greater right. So if I have a particle and it has a wave function then wherever the amplitude of the particle is greater the wave function is greater the particle has a greater likelihood of being found there it has a greater probability of being found there. So, this all makes sense in the context of the quantum mechanics being in deterministic theory. The quantum mechanics that we are studying does not predict where the particle exactly is going to be in the next moment. In fact, the Schrodinger's equation cannot predict where the particle is going to be exactly in the very next moment. The only thing that quantum mechanics provides us is an idea about where the particle is more likely to be. Okay, so this is a theoretical prediction of where the particle is most likely to be. But if I made a measurement, I could as well as find the particle anywhere possible. It's just that if I made a large number of measurements, most of the time the particles will be concentrated wherever the amplitude is the greatest. So, thus the wave function turns out not to be a physical quantity, but a Pandora's box. It contains a lot of information inside. It doesn't really mean anything, it contains a lot of information inside, but we have to extract that information out of it. And the entire quantum mechanical theory is how we can extract that information. And this is the first piece of method of how we can do it. Now, it is not enough for the wave function to just be a solution of the Schrodinger's equation. There are other things involved. You see, the Schrodinger's equation is a differential equation. It gives us a mathematical solution. But is that mathematical solution even relevant to us in the physical world? Because at the end of the day, we will find that there are many situations where the solutions of the Schrodinger's equations are not useful to us. They cannot replicate a physical scenario. What I'm talking about is that the wave function is a function which has a lot of restrictions associated with it. There are a lot of criteria that it has to satisfy for us to be able to use it as a proper wave function for a particle. So, there are certain criterias that the wave function has to satisfy for it to be useful for us. And these are the following criterias. The first point is that the wave function must be finite single valued continuous. Now, why must the wave function be finite single valued and continuous? Well, it comes back to our interpretation of the wave function. The wave function basically gives us an idea about the probability of the particle being found somewhere. If the wave function is not finite, if the wave function blows up to infinity at a certain location, it simply means that the probability of the particle being found at that location will also become infinite, which doesn't make any sense. Like, if you look at the idea of probability, the probability can either be 0 or it can be 1. If there is 100% probability that the particle is there, then it's 1. If there is 0 probability that you'll never find the particle there. So, the probability of a particle is always between 0 and 1. It can never go to infinity. So, we cannot allow the wave function to be anything other than finite. It must be single valued. It must be single valued because there is only one probability associated with the wave function at a given location. Let's suppose there were multiple values of the function. It simply means that for one particle in one system, there are multiple probabilities of the particle being found in the same location, which again doesn't make any sense. So, we can't have that kind of an absurd behavior. So, we must impose that the wave function is single valued and finite. And it must also be continuous. Why must it be continuous? It must be continuous because we want the derivatives of the wave function to also behave well. The derivatives must exist and the derivatives must also be single valued, finite and continuous. Why? Because you see we have this second order derivative with respect to x. This must also exist and this must also give us some meaningful information. So, this brings us to the second point that the derivatives of psi, derivatives of psi. So, for example, we have d psi upon dx for a 1d case. And if we are considering 3d case, we will have del psi upon del x, del psi upon del y and del psi upon del z for 3d situations. They must also be, they must first exist, they must be finite, they must be single valued and they must also be continuous. But the question is why? Well, the answer depends upon two things. First of all, you see as I just now said the wave function contains a lot of information that we have to extract via some sort of a mathematical operation. Then one of the things that we are interested in is the momentum of the particle and we will see later on that whenever we are trying to find the momentum of a particle from its wave function, then essentially what we do is we take the derivative d psi upon dx if we are interested in let us suppose the momentum component along the x axis. So, because the momentum is an actual physical quantity, it cannot be infinite and it must also have a certain smooth variation and all these different conditions that are usually associated with realistic physical quantities, that is one of the reasons why the derivative of psi must also be finite. If it is not finite, it blows up, it simply means that the particle's momentum is blowing up or the particle's momentum has become infinite. If it is not single valued, again absurd physical scenario. If it is not continuous, that means the second order derivative will become infinite. Now, we do not want the second order derivative of psi to be infinite either because the second order derivative is present in the Schrodinger's equation. If this blows up then the entire equation itself doesn't make sense. So, in a way not only we want the wave function to be finite single valued continuous, we want its first order derivatives to be finite single valued continuous and its second order derivative to exist. Fine. So, these are some of the criterias that we have to impose. Otherwise, the wave function could be any mathematical function that makes zero sense. It could give us some very absurd weird situations which I am going to come to in a moment. But there are further conditions attached to it. The third condition that is attached to the wave function is that the wave function must be normalizable. Now, what is the meaning of this term normalizable? It has to do with normalization. I am going to come to that but essentially what it means is that if I want to find out this probability, right, if I am given a wave function and I take its complex conjugate multiply with the wave function. So, I essentially I get wave function mod square. I take it as an integral with respect to x. Then this between minus infinity to plus infinity must be a finite number. It must be a finite number. We do not allow this particular expression to either go to zero or go to infinity. These two conditions are not allowed. Again, it comes back to the idea of probability. If the wave function mod square from minus infinity to plus infinity comes out to be zero, it simply means that the probability of the particle being found in the entire universe is zero, which does not make any sense. The particle has to exist somewhere, right. And if it comes out to be infinite, it simply means that the probability cannot be made to be less than one, which again does not make sense because the maximum probability of the particle being found is one. So, it must be some finite number because if it is a finite number, we can make it to be less than one by multiplying that number with some sort of a normalization constant. So, this is basically known as the concept of the wave function being square integrable. This is called the wave function must be, should be square integrable. So, we should be able to integrate the square of the wave function and that should exist. Now, I am going to come back to this point of normalization a moment later. Let me first also mention a few more points. Another point is that for bound systems, okay. So, whenever a particle is bound in some kind of a potential. So, let's suppose the particle's energy is less than whatever potential barrier it is stuck in around it, all right. So, you can imagine a particle being stuck within some sort of a bound sort of a system. In that situation, the wave function should go to zero at infinity, okay. The wave function should go to zero as x tends to either positive infinity or negative infinity. You know the wave function cannot go to infinity or it cannot even be constant at infinity because it would mean that the particle then is not bound, all right. So, this condition is necessary for a bound system. And lastly speaking, for any kind of a system, the wave function must satisfy whatever boundary conditions comes from that system. So, psi must satisfy the boundary conditions necessary for that particular system, okay. So, for example, if I talk about particle in an infinite square well potential, then at the walls the wave function should be zero because if the potential is infinite at the walls the particle cannot exist. So, the wave function must also be zero. So, it must maintain these conditions imposed by the boundary conditions. It must be continuous, single valued finite. Its derivatives must also be single valued continuous finite. It must be normalizable. These are some conditions that we have to impose on the wave function apart from the fact that it's a solution of the Schrodinger's equation. So, it's not just sufficient that a wave function is a solution of the Schrodinger's equation. No, there are going to be large number of mathematical functions which are solutions of the Schrodinger's equation that we have to reject unless they satisfy these criteria. Now, I'm going to give you a small test. I'm going to give you a list of different wave functions and you have to tell me which one of them are allowed or possible and which of them are not allowed, all right. So, let's do the simple sort of exercise, okay. All right. So, I have here a couple of examples of some functions. Now, all you have to do is you have to figure out which one of these wave functions are allowed and which one of them are not allowed. So, what you can do is you can pause the video right now for a couple of seconds or minutes, take a look at these functions and note down which one is allowed, which one is not allowed. So, if a wave function is not allowed, why is it not allowed? Just write it down. Let's do this exercise and you can test yourself. All right. So, now let's move on to the answers, okay. Let's look at these wave functions one by one. So, here I have in the first case scenario, okay. I'm not just giving you any mathematical expressions. I'm giving you a diagrammatic representation. So, in this first case scenario, what do I have? I have some sort of a wave function which is heading towards infinity. So, is this allowed? Is it allowed? It is not allowed. Why? Because wave functions must be finite. It cannot blow up to infinity because if it does, it means that the probability of the particle being found is going to be infinity, which is absurd, doesn't make any sense. Wave function has to be finite. So, that is not possible, not allowed and does not represent a physical particle. Let's move on to the next one. What about this wave function? If you look at it closely, this is not a single valued function because if you look at this region, any point in this region if you draw a line, the wave function has multiple values. See, this is not a one-to-one relationship between x and the function and we need a one-to-one relationship because the wave function corresponds to one probability value at a given location. It cannot have multiple probability values at a given location because then that is an absurd situation. What does it even mean? So, we want a one-to-one relationship. We want a finite and single valued function. This is not single valued. So, this is not allowed. Let's move on to the next one. What about this? This doesn't seem to blow up to infinity. This doesn't seem to not be single valued, but there is a problem with this. What is this? There is a sudden jump here at this location. That means there is a discontinuity here. There is a discontinuity in the wave function, but I said what? That the wave function must be continuous. So, this is not allowed. Why? Because if there is a discontinuity, that means the first order derivative of psi with respect to x blows up. We don't want that, right? Because of the second condition. So, therefore, this is also not allowed. What about the third condition? The third condition is that again it's going to infinity at as x tends to infinity. Again, it will it is not going to be finite after a certain interval of time and also it is not going to be normalizable if you notice because it is going towards infinity, right? So, this is also not allowed. What about the next one? The next one looks simple, but if you look closely, if I try to find out psi mod square for this particular condition, then what is psi mod square dx? It is essentially the area under this curve, right? So, what is this area under this curve? It tends to infinity, but we don't want that. We don't want that. So, therefore, this is also not allowed. What about the last one? Well, the last one is finite. It is single valued. It is continuous. It is smooth. Hopefully, its derivatives exist. Seems to be normalizable. This is a possible function. This can be allowed, all right? So, you see, out of all of these, only this one was possible and this one is allowed. So, now you can cross check with your answers and see how many of them matched with yours. So, the whole point of this exercise is to understand that the Schrodinger's equation could have many different solutions, but all those solutions are not necessarily allowed. They are not something that we want. We want the solutions which satisfy certain conditions and some of these conditions are written here. Now, let's move on to this next point of normalization. What is normalization and why is it so important? Many of the students who are studying this particular chapter in quantum mechanics is always stuck with this particular point. What is normalization? Even in examination, they might give you a problem of normalization. Normalize this function. What is normalization constant? So, let's try to understand what is this whole business about normalization of a wave function, all right? You see, there is no need to get scared of this process of normalization because it's actually quite simple. So, whenever we start a problem, we start with the Schrodinger's equation and we solve it, we obtain some kind of a wave function, all right? Now, we make the wave function pass all the tests that I mentioned, perfectly fine. If after all these tests, it turns out that this is actually a appropriate function, then what do we do? We try to find out the probability of the particle being found at two different locations A and B, let's suppose A and B, which essentially comes out to be the wave function mod square dx, it turns out to be some number. Now, if I ask you a question that if A is equal to minus infinity and if B is equal to plus infinity, okay? I am taking minus infinity and plus infinity along the x-axis because I am only interested in the particle's motion in the x-axis, all right? By definition. What is this probability going to be? What is the probability of the particle being found between minus infinity and plus infinity along the x-axis? Is equal to one, right? Because the particle exists somewhere in the entire universe along the x-axis, right? It exists somewhere, so the probability has to be equal to one. That's what it actually means. Now, it might happen that in certain situations, we might not actually end up getting one. Then, what do we do? Then, we simply multiply the wave function with some constant so as to make this condition true. Now, that constant is the normalization constant. To appreciate this argument a little better, let us take a classical example, all right? If I take a classical example, let's suppose in the classical world, we have a simple pendulum, all right? So, if a simple pendulum is present and we are interested in finding out the motion of a simple pendulum, then most of you may be familiar with the equation associated with a simple pendulum. It might be something like d2 theta upon dt2 plus omega square theta is equal to 0. This might be the equation corresponding to the motion of a simple pendulum. Of course, under the condition of small oscillations. Now, what is the solution of this kind of a second order differential equation? The solution of this kind of a second order differential equation is of course, theta t. Now, the most general solution is basically a sin omega t plus phi, where a is the amplitude, phi is the phase difference. These are constants that depends upon the initial conditions. Now, depending upon the initial conditions, I might have a different kind of particular solutions. For example, I might have a solution theta 1 is equal to sin omega t, all right? This is a possible solution, fine. I might also have a solution that theta 2 is equal to 2 sin omega t. This is also a possible solution. I might have a solution that theta 3 is equal to 3 sin omega t. Are you seeing what I am saying? These are all possible solutions for a simple pendulum and they all represent certain kind of motion where the amplitude is different. So, if I draw these sort of a graphical sort of a representation for this motion, maybe theta 1 representation could be like this, okay? This is theta and this is time. And maybe theta 2 could look something like this and maybe theta 3 could look something like this, all right? What I am trying to convey to you is that all of these particular solutions are perfectly fine. They represent a some slightly different motion for the simple pendulum. However, when we talk about the Schrodinger's equation which is also a differential equation and its solution, then we do not have this sort of a privilege. We do not have this kind of a window of a large number of particular solutions because it is not just enough for the wave function to be a solution of the Schrodinger's equation. We must also impose this additional condition. This is actually a very important condition because if this condition is not met, then the wave function is not going to make any kind of a physical sense. If the probability in the entire universe is greater than 1 or less than 1, then that wave function is absurd. So, out of all possible solutions which are different from one another by some multiplicative constant, only one of the solutions is allowed which satisfies this condition. Are you understanding what I am saying? In classical physics, we have a large number of possible particular solutions depending upon the initial conditions. But in quantum physics, the Schrodinger's equation might give us a lot of mathematical solutions which are different from one another by multiplicative constants. But only one of them will satisfy this and that is the answer we are looking for. So, what if you get a question in the exam that, okay, this is a wave function, normalize it. What is the process of normalization? The process of normalization is actually quite simple. It just involves us making sure that this condition is met. So, if I have a wave function, let's suppose psi, all right. Now, if I check this condition and somehow the probability between minus infinity plus infinity for that wave function is not equal to 1, then what I do is, I simply multiply the wave function with some constant. Let's suppose I call it n. This might be a complex constant. It's possible. It could be a complex constant. I simply multiply the wave function. So, let me write it like this. I have the original wave function. I simply substitute this, okay. I simply substitute this with n psi. That's it. Such that n star psi star n psi dx integration between minus infinity to plus infinity equals to 1, which gives us n mod square from minus infinity to plus infinity psi mod square dx is equal to 1. So, I solve this equation. I obtain n and then I substitute n here as a multiplicative constant for the wave function. So, then this is the normalized wave function and this process is called the process of normalization. So, this is a very simple step. I hope you have understood it. When we need to do the normalization, so as to make sure that the final function that we are dealing with is actually a function that can represent a physical particle. And for a physical particle, this condition is a must, all right. So, this is the sort of weird world of wave functions. Wave functions are not just mathematical functions of Schrodinger's equation. We must also impose a large number of conditions so that finally we can use that for some sort of an actual physical particle. So, in the next video, we will see something more. What other information does the wave function contain? What other magic can we perform to reveal further information about the physical system? Those things we will discuss in the coming lectures. I am Divya Jyothidas. This is for the love of physics. Thank you so much. Take care. Bye-bye.