 Hello everyone, welcome back. I am Divya Jyothidas and this is for the love of physics. Today is a very special day. It's a special day because finally we are discussing the Schrodinger's equation. After so many lectures on the origins of quantum mechanics, finally the time has come to actually study the quantum mechanical theory and we do that by introducing the most fundamental equation in quantum mechanics or more specifically non-relativistic quantum mechanics. You see the Schrodinger's equation plays the same role in quantum mechanics, what Newton's laws plays in classical mechanics or Maxwell's equations plays in electromagnetic theory. In a sense, if we want to study a quantum mechanical system, this is where we start. If we want to understand the behavior of a quantum mechanical particle, its position, its momentum, its energy, its trajectory, if we want any information about that quantum mechanical particle, this is where we start. So the Schrodinger's equation gives us the starting point of studying any kind of a quantum mechanical system. So what are we going to do in today's video? First of all, in today's video we are going to try to understand this equation. Second of all, I'm going to try to come up with a derivation, if possible, for this particular equation. Even though it's not really a derivation as such, it's a set of arguments that we can use to come up with this particular equation and I'm going to go through all that. And third of all, I'm going to introduce to you the time-independent Schrodinger's equation. You see what we have right here is a time-dependent Schrodinger's equation. We can reduce it further to its time-independent form. So let us start. A quick announcement for my students who are preparing for CSI-Net Physical Sciences or GATE 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. As you can see, the Schrodinger's equation is a second-order partial differential equation, which contains a large number of terms. Let me first introduce to you what those terms are. So here we have h-cut. Okay, h-cut simply means h upon 2 pi where h is nothing but the Planck's constant. What is m here? m is the mass of the particle that we are studying. We are trying to study the motion of a particle, right? Even if a quantum mechanical particle, the mass of the particle is m. What is x here? x, of course, we are interested in the most simplistic version of the equation. So we are kind of considering a 1-D motion. So let's suppose if the particle's motion is restricted to only the x-axis. In that situation, the equation looks something like this. So x basically represents the x-axis along which the particle is in motion. And t, of course, represents time because the motion not only happens along the x-axis. We can also figure it out with respect to time. And again, iota here is the imaginary number, root over minus 1. And finally, the most important variable here is psi. This is a very unique kind of a symbol that you see, psi. It's pronounced as psi, p, s, i. This is known as the wave function. Now what is the wave function? We'll come to that maybe some later time. For the time being, it is some function that contains all the information about a particle's position, momentum, energy, everything that we can derive in the quantum mechanical theory. All of this combined gives us this particular equation. And this particular equation, as I just now mentioned, plays the same role in quantum mechanics that Newton's laws plays their role in classical mechanics. So for example, when we talk about Newton's second law, we know that the Newton's second law is what? F is equal to ma. Now what is f is equal to ma? f is a force, m is the mass, a is the acceleration. This is also nothing but a second-order differential equation. So if I know the force of the particle, if I know the mass of the particle and acceleration is nothing but d2r upon dt2, then this is just a second-order differential equation, which is a starting point for analyzing any kind of a classical mechanical system. So if I solve this differential equation properly, I end up getting what is known as RT. RT is basically the position vector, or it gives us an idea about the position. And once we know RT, we can figure out the particle's velocity. We can figure out the particle's trajectory. We can figure out the particle's kinetic energy, momentum, etc., etc. So in a way, if we can solve the Newton's second law in a classical system, we can figure out everything there is to know about that particle's evolution in time. It's trajectory, it's position, it's velocity, it's momentum, it's kinetic energy, etc., etc. Similarly, if there's a particle in the quantum mechanical theory and the particle is experiencing some kind of a potential v, what is v here? This is nothing but potential energy function. Now usually we deal with conservative systems. So in a way, if we are talking about a conservative system, f is equal to minus dv upon dx. So in a way, v gives us the information about what kind of forces the particle is experiencing. So if we can construct this particular equation and solve it, then that will give us a solution which is basically psi. And this psi is a function of x and time and this psi contains all the information about the quantum mechanical particle's position, velocity, momentum, energy, etc. Now although I have to give you a disclaimer, it's not as straightforward here as it is there. In the Newtonian perspective, in classical mechanics, the formulation of the theory is very simple and straightforward. You solve a differential equation, you get the exact physical quantities. Quantum mechanics are slightly weird and slightly different. We don't exactly get these quantities, but we get some idea about these physical quantities and what is the nature of those things, we will come to that maybe at a later point in time. For the time being, the Schrodinger's equation is all we have to get information about the particle's various physical quantities. You see quantum mechanical theory is supposed to be a more general theory. It is not just supposed to explain the microscopic particles and their behavior, but also everything else. And in a way, it is true. If we look at the Schrodinger's equation, then the Schrodinger's equation should give us the Newton's second law under the limit of macroscopic particles or macroscopic systems in the same way that relativity or special theory of relativity under the limit of v very, very less compared to the speed of light ends up giving us the same thing, the Newton's laws. So Newton's laws or classical laws are a special case of a more general quantum mechanical theory of which the Schrodinger's equation is a fundamental equation. Now I can say that, well, this is the starting point. Let's move on. But no, I just want to take some time to give you an idea how come this complicated equation even came in the first place. I want to give you a set of ideas that can be used to somewhat reach this particular conclusion. So in the second part of this video, I'm going to derive or provide you a set of arguments that can be used to come up with the Schrodinger's equation based on what we know from before, based on the De Broglie hypothesis, based on energy conservation principle, based on Planck-Einstein posture, et cetera, et cetera. We can come up with this particular equation. Let's see how. In November 1925, Schrodinger was giving a talk on the De Broglie hypothesis. Remember the De Broglie hypothesis? Which suggested that a moving particle has a wave-like character associated with it. And the hypothesis predicts the wavelength in relation to the momentum of the particle. So when Schrodinger was giving the talk, one of his colleagues asked him, if moving particles are indeed behaving like waves, then where is the wave equation? You see, if we have a wave associated with something, then there is a wave equation corresponding to it. So for moving particles, if we have a wave-like motion, then where is the wave equation? Then Schrodinger took this to heart, went into work, and just after a couple of months in January 1926, published a paper deriving the Schrodinger's equation that we just now saw in the previous few slides. So here we are going to kind of derive that Schrodinger's equation in a similar fashion, not exactly the same, but in a similar fashion, by taking into account the various arguments that points us in that particular direction. Now we have to keep in mind that because quantum mechanical theory is a more general theory and Newton's laws are a special case of a more general quantum mechanical theory, therefore we cannot actually rigorously derive the Schrodinger's equation from previous classical theories, which are essentially a special case of this more general quantum mechanical theory. Therefore, this is not going to be an exact derivation, but this is more a set of arguments pointing towards that direction. And at the end of the day, the success and the validity of the Schrodinger's equation is ultimately tested by repeated experimental evidences only and nothing else. So let's take a look at the ideas that can be used to come up with the equation more or less for some special situations. So first of all, the Schrodinger's equation is a wave equation. Just now I told you the story where he was trying to get a wave equation for the wave motion of a quantum particle, the Schrodinger's equation is a wave equation. Now we are familiar with wave equations. So let's suppose I am interested in studying waves which are moving in one D. Let's suppose along the x-axis, then the wave equation is essentially d2y upon dx2 is equal to 1 upon v square d2y upon dt2. Now this could be a wave equation for any wave, whether we are talking about the vibrations moving on a string or we are talking about sound waves or we are talking about light waves. All these waves has an equation of this particular form associated with it. So x represents that the motion is in the x-axis, t represents time, v is the speed of the wave and y represents that particular quantity that is waving. So for example, if we look at a guitar string or if we look at any string in which a particular motion of a wave is taking place, then y would represent the amplitude or the instantaneous displacement from the equilibrium from its original point. If we talk about sound waves, then y could represent the pressure differences that are traveling through the air. If we talk about let's suppose light waves which are essentially electromagnetic waves or oscillations, then y could represent the electric field or the magnetic field. So for example, for light waves, the wave equation is something that is very familiar to most physics students. So in terms of the electric field, it is d2e upon dx2 for a 1d wave along the x-axis, 1 upon c square, d2e upon dt2. And the solution of this wave equation is very simple. It is e is equal to e0 e to the power minus iota kx minus omega t or rather I should say plus iota here. Now this is basically how the electric field fluctuates as the wave is propagating, the light wave is propagating through space. Similarly, the most general solution for any kind of a wave equation essentially comes out to be y is equal to some constant a a to the power iota kx minus omega t. This is a very general solution for a wave. However, what we can do is because e to the power iota something can be decomposed into cosines and sines, we can decompose this particular function into let's suppose y is equal to a and then you have cos kx minus omega t plus iota and then you have sin kx minus omega t. Usually when we are looking at let's suppose ripples on the surface of a pond or vibrations on a string, we are not really interested in the imaginary component. We are only interested in the real part of this particular wave solution. So, majority of the time you must have seen this particular function used as a solution for a wave propagation. But mathematically speaking, this is a more general function which can be represented to be the solution of a wave equation which contains both the real part and the imaginary part combined together. Now because the Schrodinger's equation is essentially a wave equation, we want the solution of that equation to have somewhat of a similar look to it. But there are certain kinds of criteria. There are certain kinds of reasonable assumptions that this equation must satisfy if it is supposed to be the Schrodinger's equation. So, let us discuss all these various reasons. We have been talking meticulously about the origins of quantum mechanics. We have discussed the De Broglie hypothesis. We have discussed the Planck postulate. So, these are some things that we already know from experimental evidences arising in the first two decades of the 20th century. So, whatever Schrodinger's equation that we want to be the fundamental equation of quantum mechanics must be consistent with all of that. So, first of all, the equation that we are interested in, that we are trying to come up with, must be consistent with number one, the De Broglie hypothesis and second, the Planck Einstein postulate. What are these? Now we have been talking about these since many lectures. So, the De Broglie hypothesis simply states that if a particle is in motion, then there is a wavelength associated with the particle's motion. If the particle has a momentum p, then the wavelength is lambda, where lambda is related to momentum in this particular manner. That's the De Broglie hypothesis. And what is the Planck Einstein postulate? That is E is equal to h nu. Now, we can represent this slightly differently in terms of wave number and in terms of angular frequency is going to be helpful later on by supposing that if lambda is a wavelength, then there is something called the wave number k. What is k? k is basically 2 pi upon lambda. So, what is the wave number? The wave number is 2 pi upon lambda. So, this is equal to 2 pi lambda is equal to h upon p. So, what does this become? This simply becomes k is equal to h upon 2 pi is h cut. So, p upon h cut. So, on one hand, we have the De Broglie hypothesis that looks something like this. And we have basically here, E is equal to h upon 2 pi times 2 pi nu. So, this can be written as E is equal to h cut omega. So, this is the Planck Einstein postulate. So, whatever equation we are trying to come up with must be consistent with these particular relationships. So, if we try to come up with the Schrodinger's equation for, let's suppose, a free particle, a freely moving particle, and the freely moving particle has a wave associated with it, then I just now said that a wave equation has this kind of a general expression. The wave solution has this kind of a general expression. So, if we assume that for a quantum mechanical system also, whatever the wave is will have a similar kind of an expression, then we can say that the solution should represent something like this. So, for quantum mechanical systems, we have the wave function, which I am going to call as psi, which is a function of x and t. Don't yet know what that is. It's just a solution of the wave equation, which is the Schrodinger's equation that we are trying to find out. And we are assuming that it has a similar kind of a form, which is a solution of a wave equation. Let's say this is a is equal to e to the power iota kx minus omega t. Now, if I apply k is equal to p by h cut here, and omega is equal to e upon h cut here, this is simply going to look like psi x comma t is equal to a e to the power iota upon h cut px minus e t. So, this is the wave you can say for a quantum particle, which has energy e and momentum p. Now, what is the second condition that the Schrodinger's equation should satisfy? It should satisfy the energy conservation principle. That means the Schrodinger's equation is consistent with the conservation of energy. That means for any quantum mechanical system, the total amount of energy should be equal to the kinetic energy, half mv square plus the potential energy. The kinetic energy half mv square can be written as p square upon 2m. And the potential energy is simply v, let's suppose x comma t. So, here this is the total energy of the system. This is the kinetic energy of the system, which is by the way related to momentum, because the kinetic energy is supposed to be half mv square containing the velocity term that is related to momentum p square upon 2m term, where v is the potential energy expression. And for conservative systems, this is related to the force via f is equal to minus dv upon dx. So, the Schrodinger's equation should be consistent with this. Third, the Schrodinger's equation must be linear with respect to its solution psi, because if it is linear and we can apply the principle of superposition, then it automatically leads to situations where particles can experience interference. So, let me explain how, let me just rub this portion of the board first. The principle of linearity or superposition simply suggests that if one function, let's suppose psi 1 is a solution of a given Schrodinger's equation and another function psi 2 is also a solution of the same Schrodinger's equation, then a linear combination of these two wave functions must also be a solution of the original Schrodinger's equation. So, linear combination of them arbitrarily speaking, so psi is equal to let's suppose c1 psi 1 plus c2 psi 2, where c1 and c2 are arbitrary constants, must also be a solution of our Schrodinger's equation that we are trying to derive. Why is this necessary? This is necessary because this allows for the wave functions to interfere. You see, we have seen previously in the Davison-Germain experiment in the double slit interference experiment that electrons demonstrate interference. They demonstrate constructive interference or destructive interference. So, we must allow the theory to have that possibility for interference also. So, for example, in the double slit interference experiment, when electrons pass through one of the slits, we end up getting one particular distribution. When the electrons pass through another slit, we end up getting a different distribution. But when electrons pass through both the slits, we get interference. So, this possibility of interference arises mathematically from the principle of superposition when individual solutions arbitrarily combined together must also be a solution of the overall equation. This is a very important property because it allows for the wave-like behavior that we have seen for electrons where constructive and destructive interference is also possible in the distribution of electrons in some of these experiments. And lastly, the assumption that we are keeping in mind is that since we are dealing with Schrodinger's equation solution for a free particle, we must keep in mind that for a free particle, the force is essentially zero. So, we know that for conservative systems, the force can be derived from the potential in this particular manner if we are only interested in 1D. And if the potential is constant for a constant potential system, the force comes out to be zero. Now, in Newtonian mechanics, zero force simply means that a particle evolves with constant momentum and constant energy. So, since we are essentially trying to come up with the Schrodinger's equation for a free particle, for a very special situation, okay, for a free particle. So, even the Schrodinger's equation should give us a solution where the solution should convey the information that the momentum and the energy of that particular wave equation associated with the particle is a constant of time. So, because momentum and energy is a constant of time, therefore the wavelength and the angular frequency should also be a constant of time. That means for a free particle moving in constant potential, we should have basically some sort of a sinusoidal wave with constant wavelength and constant frequency. So, based on these assumptions, based on the assumption that the equation must be consistent with De Broglie hypothesis and Planck-Einstein relation, based on the assumption that it must be consistent with energy conservation principle, based on the assumption of linearity, based on this fourth point, we are now ready to finally derive for a very special situation of a free particle, the Schrodinger's equation. So, for that, we are going to start with the wave-mechanical solution. We just now said that Schrodinger's equation is a wave equation and wave equations have solutions that look like this. So, we are going to start with this. We are going to assume that whatever wave equation this Schrodinger's equation is, also has a solution that has some of this particular look to it. So, we are going to start with this particular thing and let's see what we can find out. So, we start with the wave solution for a moving particle having momentum p and energy e and let's see what we can do with it. Let's first differentiate it with respect to x. What do we get? If I differentiate it because psi, which is the wave function, is a function of x and time. So, it's a multivariate function. So, it's a partial derivative upon dx. This should give us something like a is a constant and then if I am differentiating it with respect to x, I should get another constant iota p upon h-cut and then I will have e to the power iota upon h-cut px minus e t. Let's differentiate it one more time. Differentiate again with respect to x. I should get something like d2 psi upon dx2 which is basically equal to a iota p upon h-cut square e to the power iota upon h-cut px minus e t. Essentially what does this become? A e to the power iota h-cut px minus e t is nothing but psi and then we have this constant here which gives us minus p-square upon h-cut square psi. So, now I can rearrange this and write down this expression. So, p-square psi is equal to minus h-cut square del 2 psi upon dx2. So, let's say this is point number one and keep it here. Now, let us differentiate psi again but this time with respect to time and see what we get. So, if I differentiate psi with respect to time, what do I get? Again, a comes out and because it's time I will have minus iota e upon h-cut and then e to the power iota upon h-cut px minus e t. So, I can combine a and the exponential term to get psi. So, this becomes minus iota e upon h-cut psi. So, finally, I basically have psi as a function of x and time. And this particular expression that e psi x comma t is essentially equal to if I take this to the left hand side, I should basically get minus h-cut upon iota which can be written as iota h-cut d psi upon dt. So, these are the two things I have. Now, let us apply point number one and point number two in the energy conservation principle. So, what does the energy conservation principle tell us? That the total amount of energy is equal to the kinetic energy which is p square upon 2m plus the potential energy. Let's multiply psi on both sides of the equation. So, I have e psi, I have p square psi and I have v psi. So, here p square psi can be substituted in this part, e psi can be substituted in this part to finally come up with this equation. So, e psi is essentially equal to iota h-cut del psi upon del t is equal to p square psi is essentially minus h-cut square upon 2m d2 psi upon dx2 plus potential energy which is a function of x and time and again psi which is a function of x and time. This is nothing but what is it? This is the Schrodinger's equation that I wrote down in the very first slide. So, this is the time dependent Schrodinger's equation. Now, keep in mind a few things that this is nothing but the energy conservation principle because what is this term? This term is essentially the total energy term. What is this term? This term is basically the kinetic energy term. What is this term? This term is nothing but the potential energy term. So, the Schrodinger's equation apart from being a wave equation is also essentially an energy equation and we can also show that it is consistent with the points that I mentioned before. It is consistent with the De Broglie hypothesis, the Planck-Einstein postulate, the energy conservation principle. It is also consistent with the principle of linearity because if I write the Schrodinger's equation for let's suppose psi 1. So, if I am interested in checking the principle of linearity, the Schrodinger's equation for psi 1 comes out to be let's suppose something like d2 psi 1 upon dx2 plus v psi 1 is equal to iota h cut d psi 1 upon dt. It means that psi 1 is a solution of the Schrodinger's equation and let's suppose that psi 2 is also a solution of this Schrodinger's equation. That means d2 psi 2 upon dx2 plus v psi 2 is equal to iota h cut del psi 2 upon del t. Then it is very easy to show from both these two equations that psi 1 plus psi 2 is also a solution for the Schrodinger's equation. For example, I can easily show mathematically that d2 upon dx2 of c1 psi 1 plus c2 psi 2 plus capital V c1 psi 1 plus c2 psi 2 is actually equal to iota h cut del upon del t c1 psi 1 plus c2 psi 2. What you do is multiply the first equation with c1, second equation is c2 and add them up you end up getting the Schrodinger's equation for c1 psi 1 plus c2 psi 2 which is a linear combination of psi 1 and psi 2. They are by proving that this equation is consistent with the principle of linearity. Now keep in mind that we have actually derived this equation for a very special scenario of a free particle experiencing constant potential or zero force. Now how can we say that this special equation for this special scenario is actually a general equation that can be applicable for all other scenarios? Because there are going to be scenarios in which the particle is going to experience a force. Particle is going to experience some potential. We are going to have cases like hydrogen atom, harmonic oscillator, etc. where the potential is not necessarily constant. Can we then say that what we have derived is actually going to be relevant there also? Well, that is where it becomes a postulate. You see what we have done here is not exactly a derivation. As I just mentioned at the very beginning, quantum mechanical theory is a more general, broader, fundamental theory and the classical laws that we knew before it are a very special subset. So we cannot use the classical laws to derive a more general theory. So this is where this equation becomes a postulate. We are postulating that this form of this equation is actually true for general situations and then we are going to check every time for different kinds of circumstances whether this yields a correct result and whatever result this equation yields is it experimentally verified. So in the end, the validity of this Schrodinger's equation rests on repeated experimentation over and over and over again. And in the last 100 years, it's 2023 right now. Schrodinger wrote his paper in 1926, almost 97 years back. The Schrodinger's equation has turned out to be remarkably accurate in terms of predicting how quantum mechanical systems behave. Now in the last part of this video, I want to resolve this equation into something simpler. You see this looks very complicated because this is at the end of the day a second order partial differential equation that contains both X and T as a variables, right? Unknowns. So I want to reduce it further into a time independent form which is going to be very helpful for us because in the later lectures when we talk about different potentials, there we don't have to deal with such a complicated equation we can deal with the time independent form of the Schrodinger's equation. So let us derive the time independent Schrodinger's equation from the time dependent form. All right, so we start with the time dependent Schrodinger's equation where V is the potential energy, right? Now the potential energy we have assumed is a function of X and quite possibly time but in most of the situations that we deal with in realistic scenarios the potential energy is a usually time independent. For example, if we look at a harmonic oscillator, if we look at a hydrogen atom these are all time independent potentials and majority of the situations that we are going to deal with in quantum mechanics we are going to deal with time independent potential. So let's make that assumption that for time independent potential let's suppose Vx, how would the Schrodinger's equation look like? So under this particular condition what we are going to see is that the Schrodinger's equation contains wave function psi which is also by the way a function of X and time. All right, but luckily we can write this as a product of the function of X and the function of time. So let's suppose I have psi which is a function of X and phi which is a function of time then we can write the wave function as a product of the function of X with the function of time and this is known as separation of variables method. Now is it valid or is it not? Well actually it turns out that this is a valid approach because if it was not valid we would see that it is not valid in our further calculation but it turns out that this is actually one of the ways in which we can represent the wave function. So this is known as separation of variables method. Here keep in mind that I am kind of using the same symbols here. You see this, what is this symbol? This symbol is like a curly psi and what is this symbol? This symbol is like a very straightforward psi. So this is basically the capital psi and this is the small or the lower case psi. Now majority of the books this is how they represent it. This will be some kind of a capital capitalized bold version of a psi and this is like a lower case not bold version of a psi and that is how we are going to distinguish it where this one will be the function of X and T both and this one will be the function of only X. So now let us plug that into our original equation, equation number one and see what we get. So this gives us minus h cut square upon 2m and here I am going to have d2 upon dx2 of psi X and phi T plus v which is just a function of X. Now psi X and phi T which is equal to iota h cut d upon del upon del T of psi X and phi T. Now let us see what we get from these two derivatives. So now if we just are interested in del 2 upon del X2 of psi X and phi T because X and T are independent variables and this is a function of only T and this is a function of only X and this is a derivative only with respect to X. So I can take the function which depends only on T out. So it will behave like a constant. So this is simply nothing but phi T del 2 psi X upon del X2 but now because this is just a function of X so now it is not a multivariate function anymore so I do not need to write partial derivatives. I can just write ordinary derivatives here. So I can simply write d2 psi upon dx2. Similarly for del upon del T of psi X and phi T because this is a function of time and this is a function of X the function of X will simply behave like a constant because with respect to the derivative with respect to time. So therefore I can take this out psi X and this becomes an ordinary derivative T phi upon dt. If I substitute all of this in equation number 1 or equation number this particular equation then what we get is quite simply something that looks like this minus h cut square upon 2 m phi T d2 psi upon dx2 plus VX psi X phi T is equal to iota h cut psi X d phi upon dt. Now let us divide the entire equation by psi X times phi T. What will we get? If we divide the entire equation by psi X times phi T what we end up getting is phi T phi T will get cancelled. We will have on this side 1 upon psi X and within brackets minus h cut square upon 2 m and d2 psi upon dx2 plus here psi X phi T will cancel but because I am keeping 1 by psi X common here I can write this as VX psi X alright on the left hand side and on the right hand side I am going to have something like 1 upon phi T iota h cut d phi upon dt. Now look at this what we have done. We have taken all the functions which are depending on X to the left hand side and all the functions which are depending on time to the right hand side. Now the left hand side is completely independent of time and the right hand side is completely independent of X but X and T are independent variables. So how is it possible that we have an equation where in the left hand side it is entirely dependent on X and the right hand side is entirely dependent on time. This is only possible if both these two equations are essentially equal to some kind of a constant. Alright so essentially what we have done is we have separated the time dependent partial second order differential equation into two ordinary differential equations which are equal to one another which are functions of X and T separately. So in a way the time dependent Schrodinger's equation is a coupling of this and this. So now let's see what we can further get from here. So let's suppose the constant I am going to call this constant as C. So if I call this constant as C what do we get? On one hand we have the left hand side. The left hand side simply gives us minus h-cut square upon 2m d2 psi upon dx2 plus vx psi x is equal to this constant C times psi x. Alright and on the right hand side what do we have? In the right hand side we have basically iota h-cut d phi upon dt is basically equal to the constant C times phi t. Alright let's say that this is equation number 2 and this is equation number 3. Let's first analyze equation number 3 that we have. So iota h-cut d phi upon dt is equal to C phi. I want to solve this and see what we get. So I can say d phi upon phi which is a function of t is equal to now I have C upon h-cut and then i goes to the right hand side I get minus iota here. If I integrate this with dt and I integrate this left hand side what do I get? This is nothing but ln phi is equal to minus iota C upon h-cut time t. So this simply becomes phi is equal to e to the power minus iota Ct upon h-cut. So finally we have a solution for what phi is. What is phi? Phi is a function of time and we have now obtained its actual form. So phi is equal to e to the power minus iota Ct upon h-cut. You see this? This is the time dependent solution of the Schrodinger's equation. Now if you look at this particular function what is this e to the power iota C being a constant t is time h-cut is a constant. Essentially this is an oscillatory solution. e to the power iota something it looks like basically the wave solution that we obtained earlier it is an oscillatory solution. Now when we talk about oscillatory solution we talked about the Planck-Einstein postulate initially. Initially we said that the solution is essentially some sort of a sinusoidal variation having a wavelength and an angular frequency. What was the angular frequency? We said that e is equal to h-cut omega where the angular frequency was essentially equal to omega is equal to e upon h-cut. But what is the angular frequency of this oscillatory solution? The angular frequency of this oscillatory solution is if you look at it carefully this should look like something like e to the power minus iota omega t. This is an oscillatory solution. So what is omega here? Omega is equal to c upon h-cut which is equal to e upon h-cut. So what is c? c is nothing but e. c is the energy. c is the constant energy of the particle. Are you getting what I am saying? Because this is an oscillatory solution with an omega is equal to c upon h-cut and we know from the Planck Einstein's postulate that omega is actually e upon h-cut. So by making a comparison we see that c is nothing but e which is the energy of the particle. So if a particle is in motion and it has an oscillatory wave mechanical solution associated with it then the particle's energy which is constant by the way, capital E for constant potential but here I am just assuming it is the energy e at the moment is actually this particular constant that we got when we used the separation of variables method. So finally we can write the oscillatory solution as well as the LHS equation in this particular manner. Let me rub the board first. So essentially the time dependent solution phi t takes the form of e to the power minus iota capital E t upon h-cut where capital E is the energy and then the LHS this equation simply becomes minus iota sorry minus h-cut square upon 2m d2 psi upon dx2 plus V psi is equal to E psi. This is the time independent Schrodinger's equation. So essentially we have reduced the time dependent Schrodinger's equation. So what was the time dependent Schrodinger's equation? It was minus h-cut square upon 2m del 2 psi upon dx2 where psi was a function of x and time plus V psi x and time is equal to iota h-cut del psi upon del t where psi was a function of x and time we have reduced this into number 1 this equation and number 2 basically this is the time dependent part. So phi t is equal to e to the power minus iota E t upon h-cut. Now what is the solution of this time independent Schrodinger's equation? The solution of this is simply psi x which we basically call sometimes as eigenfunction. We call this as eigenfunction. So whenever we have a problem whenever we are tackling a quantum mechanical system and we want to study it, we usually do not try to solve the time dependent Schrodinger's equation because it's a partial differential equation containing a wave function as a function of x and time. So it's kind of complicated. It's a little bit more laborious. So we do not start from there. What we do is for majority of the scenarios the potentials are time independent. So we start from the time independent Schrodinger's equation and when we obtain its solution then we say that the wave function which is psi a function of x and t is equal to this eigenfunction which is a function of x which is a solution of the time independent Schrodinger's equation times the power minus iota e t upon h curve. There you have it the sort of theoretical framework for the wave mechanical formulation of quantum mechanics which is essentially the Schrodinger's equation or rather the simplistic step by step I have seen. So whenever we tackle a quantum mechanical problem we do not start with the time dependent Schrodinger's equation. We start with the time independent Schrodinger's equation we get its solution then we multiply this with the time dependent solution to end up getting the actual wave function solution. So the actual wave function solution is a product of the space dependent function and the time dependent function. So you can see that this is actually a complex function because you have the iota here but what are the implications of this iota what is the implication of the fact that the wave function is a complex function what does it actually mean to have a complex function as a solution how is it related to the physical quantities of the particle like velocity, momentum, kinetic energy etc etc or trajectory and all these things. These things are a question and these things we will discuss in the coming videos. So I hope you have understood what we have tried to discuss here one of the most fundamental equations of non-relativistic quantum mechanics is the Schrodinger's equation. I have tried to derive it or rather just point towards it for certain very special scenarios by certain kinds of assumptions that we have seen previously in our previous lectures and then I have tried to simplify this equation into its time independent version and this here represents the starting point of our actual quantum mechanical theory. In our coming lectures we are going to look at what is the probabilistic interpretation of quantum mechanics we are going to look at what is the wave function we are going to look at expectation values we are going to actually look at various potentials and how we can solve the Schrodinger's equation for different kinds of potentials and we will see the interesting, the weird, the bizarre and the sort of surprising world that quantum mechanics reveals to us. So that is all for today I am Divya Jyothidas. Thank you very much. I will see you next time. Take care. Bye bye. It is a special day because we are finally discussing the Schrodinger Schrodinger Schrodinger Schrodinger equation.