 Now that we know that Bohr model is not going to work because it violates uncertainty principle the field is set for us to discuss Schrodinger equation and as you will see even Schrodinger equation has its beginning in classical mechanics. As we said Bohr model is too deterministic. So, now knowing that it is time to talk about a few more very path breaking a kind of discoveries that were made in that point of time. One thing was what got Einstein's Nobel Prize photoelectric effect. In physical optics in physics you have studied hygienist experiment for example which establishes the wave nature of light. Newton believed that light has corpuscular nature. In the explanation of photoelectric effect Einstein established that light has corpuscular nature particle nature. So, what does that mean that light can behave like wave it can behave like particle and whether you see the wave nature or particle nature is determined by what kind of an experiment you do. We will talk about diffraction later on in this course. If you do a diffraction kind of experiment then you get to see the wave nature of light. If you study photoelectric effect then you get to see the particle nature of light. It is all about what kind of experiment you perform. So, this was already known. So, where are we now? Bohr model the problem is it is too deterministic uncertainty principle is violated that is one side of the story. The other side of the story is that for light this wave particle duality had been established already. At this point of time in came de Broglie who was at that time a PhD student and I am sorry for the mistake in spelling here it is not meter wave it is matter wave. But well matter wave is the major mother of our current understanding of this quantum world. So, it is Freudian slip but perhaps it is not a bad idea to call it meter wave also. So, de Broglie's thesis is perhaps the shortest in the history of mankind. I encourage you to do a Google search and find out how many pages were there in that thesis. But that small little thesis was a completely disruptive phenomenon in our understanding of what everything is made up of and it involves very sophisticated mathematics. We will skip all that and we will just share with you the philosophy of de Broglie hypothesis. Well the correct pronunciation is apparently de Broglie but then well I am not good at pronouncing a European names I will just say de Broglie. So, de Broglie sort of said is that nature manifests itself in two forms energy and matter. Energy light has a dual nature sometimes you can see the wave nature sometimes you can see particle nature and then de Broglie made this philosophical statement that nature likes symmetry. So, particles matter should also have wave like nature. What does that mean absolutely mind boggling on the face of it it sounds very esoteric, philosophical and impossible to understand agree with that. But let us see what de Broglie has to say what he said was taking a lesson from light he could work out an expression for wavelength associated with this so-called matter waves and that wavelength turns out to be h by mv. And from here if you take this lambda equal to h by mv and plug it back you will see mv are equal to n h by turns out to be an essential condition the thing is this what is the circumference 2 pi r. So, in this 2 pi r you should have an integral multiple of half wavelengths n lambda by 2 should be there in 2 pi r and what is lambda h by mv substitute you are going to get this mv are equal to n h by 2 pi that kind of a relationship. And one can calculate the de Broglie wavelength so this is a calculation on electron moving at 10 to the power 6 meter per second you see lambda turns out to be 7 into 10 to the power minus 10 meter and this was experimentally verified as we are going to say very so. So, mathematics mathematically there is no problem you can have wave nature in this small little particles. If you calculate the de Broglie wavelength of a cricket ball moving at say what is the speed at which boomerables 140 kilometer per hour something. So, you can work out what that frequency what that wavelength is going to be and you can understand what it will be here in lambda what will happen is that v for electron is 10 to the power 6 meter per second fine in the in the place of 10 to the power 6 you are going to have maybe 10 to the power 2 4 orders of magnitude less. But what about mass the mass here is 10 to the power minus 31 kg. So, if you use a mass of 1 kg that is 10 to the power plus 31. So, the denominator will be 10 to the power 31 minus 6 10 to the power 31 minus 6 10 to the power 25 or so. So, that lambda is going to be very very small and as a preliminary discussion we can say that the lambda will be so small that you will not be able to see wave nature. Of course, this is the beginning of the discussion not the end try to think of yourself what will happen if that cricket ball is at rest or if it moves very very slowly should we see the wavelength should we see wave nature actually we will not see even then but think about it a little bit I leave it as an open question for you to ponder upon. But the thing is that it was established that this wave nature can actually be seen for atoms and electrons and so on. So, forth what you see here is real data for helium atoms scattering you take a stream of helium atoms pass them through a dispersing agent like a grating and you see this kind of a diffraction pattern. So, manifestation of wave nature of matter is very much there this is these are two other examples where this wave particle duality of electrons was manifested I leave it to you to find out what these two experiments are called and who these two scientists were. But suffice to say that wavelengths of electrons where found out experimentally to be of close values to those expected from de Broglie equation. So, matter waves what are matter waves I mean still you can do the math but it does not seem to make sense what are matter waves nobody can understand. So, in came Schrodinger and even Schrodinger did not really understand at the time what matter waves were. But what he had in his hand was that well particles and waves I mean particles can be waves and waves can be particles. And one thing that was understood at that time is that you need some kind of a new theory and this theory has to be probabilistic and non deterministic since you cannot really talk about a precise value of position and momentum you can only talk about things like average value most probable value. So, some kind of statistics would be required and it would be a deviation from Newtonian mechanics from Newtonian mechanics and since there is wave nature of matter what Schrodinger thought was well for any kind of wave wave that you see in the sea or wave that is produced on the surface of Tawla when we play it for all waves there is something called a classical wave equation that essentially describes what is going to be the displacement for main position amplitude as a function of space as well as time. So, Schrodinger thought sort of was that can we write classical wave equation for de Broglie wave. In many books you will see a so called derivation of Schrodinger equation please be advised that Schrodinger equation cannot be derived it can be arrived at it is a postulate. But even postulates have some basis the basis of Schrodinger equation is that it is a classical wave equation for de Broglie waves to start with and I will not even show you what the classical wave equation is if required you study it in physics course. But when he wrote it like this classical wave equation for de Broglie waves Schrodinger got an equation that looked like this psi here is aptitude maximum displacement from mean position and psi is dependent not only on spatial coordinates say x, y, z but also on time. So, Schrodinger equation turned out to be like this i h cross what is h cross h cross is equal to h by 2 pi h cross is fundamental quantity in quantum mechanics as you are going to see h cross is simply h divided by 2 pi it is a little shorthand notation to write h by 2 pi that is all this is what h cross is. So, i h cross del del t of psi is equal to minus h cross by 2 m del square plus v v is potential energy operating on psi del square is del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2 right now it looks very intimidating and well I do not know what you are talking about perhaps but we will see we will make sense of it. Something that we have done here is that we have written this operator in time and operator in spatial coordinates in different colors what is an operator an operator is something that operates on a mathematical function and transforms it in some way or the other we will have a lot to say about operators in the discussion to come. But now the first thing we should try to do is try to separate these variables try to simplify the situation because right now it is a mix up of x, y, z and t if we can separate them a little bit have smaller equations with fewer number of independent variables will be nice. So, to separate the variables we use something that is again well known by that time from mathematical treatment of this kind of equations we use a solution of this equation where the solution is a product of two parts a space dependent part psi n and a time dependent part phi psi n which is a function of x, y, z multiplied by phi which is a function of t plug it into the equation. Now what will happen see on the left hand side this del del t is the operator it is going to operate on phi of t fine but psi n is in terms of x, y and z. So, as far as del del t is concerned it is going to be a constant right. Suppose I give you x, y or maybe x square y plus 2 x, y square and ask you to differentiate with respect to x what will you get I have forgotten what I said just now. So, I will just write whatever comes to my mind y, x square plus say 3 x, y square how do you find the differentiation it is like this y will come out because it is not a function of x. So, del del x will operate on x square to give you 2 x plus again y square will come out as it is a constant then 3 differentiation of x with respect to x if you want. So, this is your answer. So, that is the same thing we are doing here nothing esoteric nothing very new. So, we are trying to separate the variables while doing that we use this wave function which is a product of a space dependent and time dependent part when we plug it back this is what we get psi n of x, y, z multiplied by i h cross del phi del phi t del t well here I might as well write d phi of t dt equal to phi of t multiplied by minus h cross by 2 m this whole thing but operating only on psi n. Now the way to proceed is to try to get everything in time on one side of the equation and everything in x, y, z to another side of equation if you can do that then things will be a little easier to handle how do I do it one easy way of doing it would be to divide the equation both sides by psi n yeah because as you know psi n n is equal to a product of a space dependent part and a time dependent part. So, what happens if I divide this whole thing by I will just write psi and phi okay psi phi on this right hand side what happens if I divide by psi and phi well phi and phi cancel on the left hand side psi and psi cancel. So, what do you have on the left hand side then i h cross d phi dt divided by phi. So, left hand side is only in terms of phi right hand side is only in terms of psi since it is looking ugly let me delete what I had written here in any case this is what I get left hand side is only in terms of time right hand side is only in terms of spatial coordinates. So, since one is in time was one in spatial coordinates both have to be equal to a constant otherwise it does not make sense how will you equate something that is in time and something that is in spatial coordinates. So, this constant is called a separation constant and you plug that in the first equation becomes very very simple right it becomes d phi t dt is equal to w multiplied by phi of t divided by i h cross or you can write d phi t divided by phi t is equal to i h cross w dt and then when you integrate this is what you get right this is the first equation the second equation when you integrate the left when you try to solve the left equation phi of t turns out to be e to the power minus i w t by h cross right it is not difficult to understand I hope. So, I know something already I know what the temporal part of the wave function is time part e to the power i w t divided by h cross what I do not know is what is w what is w the answer comes from an inspection of this operator in spatial coordinates this minus h cross square by 2 m del square plus v is known from classical mechanics to be the Hamiltonian operator and Hamiltonian operator in classical mechanics represents total energy. So, when the Hamiltonian operator operates on psi it is known that it gives us e multiplied by psi. So, w is actually either total energy of the system that is point number 1 point number 2 is that what you have got here is an eigen value equation what is an eigen value equation it is something like this you have an operator a hat an operator remember is something that something like d dt that operates on a function. So, I will write here a hat it operates on phi phi is a wave function to give me same wave function phi multiplied by a constant a this is called an eigen value equation in this equation the function wave function is called an eigen function please forgive my bad handwriting first of all it is bad secondly I am writing on this smooth screen which does not really help so I hope you are able to read what I am writing. So, when operator operates on this function if it is an eigen function then you get back the same thing multiplied by a constant which is called an eigen value. So, for energy of a quantum mechanical system what we get from Schrodinger's equation is that you can write an eigen value equation where the operator is Hamiltonian and the eigen value is energy extrapolating from here one can work out what is called the or one can formulate the postulates of quantum mechanics. But before that let us just realize something what it says is that each wave function until now we were saying wave function is just a displacement from mean position but now wave function gets a little more significance each wave function is associated with a particular value of energy En. So, it is called a particular energy eigen state and this eigen state is a stationary state stationary state is defined as a state in which a state whose energy does not change with time and even this term stationary state actually came from Bohr model remember Bohr had said stationary state so we have not really thrown out everything have we achieved quantization we have not all we have achieved is that we have learnt that you can write an eigen value equation for energy and we have learnt that wave function contains the information about the energy which is brought out by making the Hamiltonian the total energy operator operate on the wave function this is what we know we have not obtained quantization right how does quantization come we will learn that in module after next I think. But with this background we can now talk about postulates of quantum mechanics which are sort of the ground rules again cannot be derived that is the beginning it comes from well common sense and observations and intelligent assumptions the ground rules of quantum mechanics is something that we can start talking about once we know Schrodinger equation