 In this module, we will try to understand the wave nature of matter and it sounds quite maddening because it is very difficult to believe that matter can have wave nature. But then as we have seen that uncertainty principle sort of last nail in the coffin of board theory. So, such a deterministic approach of describing atomic structure is clearly not going to work and then in photoelectric effect the wave particle duality of light was already established. So, the question was does such a duality exist for matter as well. And first let us talk about the experiments that led to the realization that indeed matter can have wave like nature and those experiments are of electron diffraction. As we have said in the last module diffraction is a sure short sign of waves only when something has wave nature can it exhibit diffraction can it give you fringes. So, the experiment that was done well we are talking going to talk about two experiments the first well very similar experiments actually the first one Davidson-Germain experiment in this a beam of electrons from an electron gun and when I see electron gun one can think of a cathode ray tube with a hole at an end so that the cathode rays that are generated would go through the electrode and come out of the other end. So, the beam of electrons collimated beam of electrons was made incident on a nickel crystal and then the dispersed electrons were detected by a detector that could be rotated through an angle theta. And it was found that a diffraction pattern was obtained according to this n lambda or what is written here is m lambda equal to 2 d sin theta this is what tells us that even electrons can actually give diffraction and therefore have wave like nature. The other experiment quite similar was performed by G.P. Thomson. In G.P. Thomson experiment electrons from an electron source were accelerated towards a positive electrode in which a hole was drilled as I said a little while ago the resulting narrow beam of electron was directed through a thin film of nickel. So, the difference between this experiment and that the earlier experiment Davisen-Germain experiment is that in Davisen-Germain experiment it was done in a reflective mode. Here G.P. Thomson's experiment is performed in a transmission mode. Since it is done in a transmission mode one takes not a crystal of nickel but a thin film of nickel. So, what happened then is that for the electron beams that went through a screen was placed after the nickel film and a clear diffraction pattern was obtained. So, these two experiments indicate that matter manifested in electrons can have wave like nature. What does it mean? We do not know yet but then at that point of time De Broglie came in and what De Broglie did is that he said that well nature likes symmetry and nature manifests itself in two forms energy and matter. Since energy has dual nature there is no surprise that matter also has dual nature well that is only the philosophy of it rest of it is non-trivial mathematics and after doing that De Broglie arrived at this expression which one can arrive at very easily for photons for example where lambda is given by h by p where p is the momentum. So, for anything De Broglie proposed that wave nature is of course associated with a wavelength that wavelength of anything that anything can be electron cricket ball, Jupiter or dinosaur for anything this lambda is equal to h by p which is h by m v. In our tutorials we are actually going to work out the De Broglie wavelengths of things that are very light like electron and things that are seemingly like say a dust particle and maybe that is something that is a little heavier like a cricket ball and when we do that we will see for ourselves what happens to these De Broglie wavelengths and that will give us an answer to why is it that we do not see wave nature manifested in U, Me, Chair, Table all these things De Broglie wavelength is lambda equal to h by m v. So, for an electron which moves at 10 to the power 6 meters per second 10 to the power 6 meter per second is a very high speed but if we consider the speed of light there is 3 into 10 to the power 8 meters per second. So, the speed of this electron here is about 100 times almost 100 times less than the speed of light. So, it is very high but achievable. So, for an electron moving at 10 to the power 6 meters per second lambda one can calculate to be is 7 into 10 to the power minus 10 meter angstrom and that is why one can see this diffraction patterns using an atomic lattice because for diffraction to occur D separation between the planes of a lattice have to be comparable with the wavelength. If we take a lattice where the spacing is very small then we do not see diffraction. If we take a lattice where the separation is too much compared to the wavelength light just passes through. So, this value of lambda is very appropriate to be seen using an atomic lattice. So, now think of a human being weighing 50 kg moving at 10 meters per second one can work out what the wavelength will be and if you work out you will see that the wavelength is such that one cannot have any grating physically that can give you diffraction of that human being and show diffraction patterns. We will understand it better when we work out the assignment problems. But well remember what we said we had quoted Max Planck experimental results are the only truth without showing experimental results I cannot claim anything. So, here we show the experimental results of helium atom scattering and here the diffraction pattern is such for helium atoms moving at 2347 meters per second and impinging on a silicon nitride transmission grating which has 1000 lines per millimeter. So, all parameters are very well defined the calculated de Broglie wavelength is 42.5 into 10 to the power minus 12 meters. So, even for helium it is 42.5 into 10 to the power minus 12 meters rather small. So, for de Broglie wavelength for particles that are bigger you require something that is even smaller. However, for electrons in these experiments that we talked about a little while ago in both the experiments the wavelength that was calculated turned out to be in very very close agreement with that which we expect from de Broglie equation. And this is the vindication of de Broglie theory the experimentally observed result matches very nicely with what de Broglie theory predicts with all this background Schrodinger tried to develop a treatment for atomic structure. So, what did Schrodinger have in his hand? First of all particles can be waves waves can be particles and there is duality and when we say it like that nobody understands anything anyway. So, what he realized that you need a new theory if you want to explain the behavior of electrons atoms molecules and so on so forth. Also since we are handling things that are waves as well as particles deterministic theory is not going to work deterministic theory too much of determinism was the reason for downfall of Bohr theory remember. So, what we need really is a probabilistic theory and that is what is going to eventually lead to our understanding of what is the meaning of this wave nature of matter. This of course is expected to be non-Newtonian in nature but then what Schrodinger really wanted to do is he wanted to write a wave like equation for describing subatomic or atomic systems. Now the tools are all there because you know that total energy of a system like that would be a sum of kinetic energy and potential energy and then we know that E equal to h we know de Broglie relation. So, with this what Schrodinger did was the classical wave equation that existed at the time and is used even now for any kinds of waves sound waves for example any kind of wave when someone plays the tabula the wave that is generated on the diaphragm even for that wave one can use the classical wave equation. Del 2 psi del x2 is equal to 1 by c square del 2 psi del t2 well what are we what did we just say this is what we said this psi that we have written here something that keeps on coming back in quantum mechanics is the amplitude displacement from the mean position not the maximum amplitude amplitude. So, this displacement from mean position for any wave as we know from our knowledge of basic physics is a function of not only space but also time. So, what we are saying is that and this is something that is established in classical mechanics second derivative of this space and time dependent amplitude which is also called wave function with respect to space is equal to 1 by c square multiplied by second derivative partial derivative of course of the amplitude or wave function which is space and time dependent with respect to time. And in fact we are going to work this out in any case psi of x t is equal to c into e to the power i alpha where alpha equal to 2 pi multiplied by x by lambda minus nu t this is the phase this was known and de Broglie relationship was known to Schrodinger. So, what he did is he proposed an equation which is essentially a classical wave equation for de Broglie waves. Let us not forget that Schrodinger's equation is really a postulate it cannot be derived. Sometimes in some books derivation of Schrodinger equation is given that is not the right terminology you can arrive at Schrodinger equation cannot derive it in the same way that you cannot derive any of Newton's postulates their postulates they are the starting points and they hold because they are the universal truth. So, Schrodinger equation which is the classical wave equation for the de Broglie waves looks like this i h cross multiplied by del del t of space and time dependent wave function is equal to h hat we will come back to what h hat is h hat operating on psi which of course is a function of space and time what is h hat it is important to understand that h hat is the Hamiltonian operator and it is given by minus h cross square by 2 m del square by del square is del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2 plus v of x where v is the potential energy. Now here for the first time in this course we are introduced to the concept of operators. What is an operator? An operator is something that well it sounds a little silly if I put it that way but it still is the best way of putting it. An operator is something that operates on a function and transforms it in some way. So, the operators that we are going to encounter most of the time are del del x del 2 del x 2 so on and so forth. So and it is important to also understand that this concept of operators is not something that came with quantum mechanics. The concept of operators was already there in classical mechanics. In fact the Hamiltonian operator was there in classical mechanics and this is the strength of Schrodinger's treatment. The entire problem we have had so far is that it seems that this when you go down to atomic subatomic scales when you talk about quantum mechanics everything is very enigmatic and what we have in those length scales is nothing like what we see around us so there seems to be a disconnect. So, the strength of Schrodinger's approach is that Schrodinger took very well established classical mechanical tool and combined it with De Broglie's expression. So, the logic of classical mechanics actually is taken forward as far as possible. If you remember in Bohr theory the quantum numbers fell from the sky. They were invoked just like that whenever there was a requirement. In Schrodinger's equation there is no quantum number. Quantum numbers will arise later on and we will see why they come. But in Schrodinger equation per say there is none. It is just a classical wave equation written for matter waves that is all that it is. So, this here is your Schrodinger equation in 3 dimensions. Del square is Laplacian del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2. Now in the next step to simplify the situation a little bit we can write this space dependent and time dependent space as well as time dependent wave function psi which is a function of x, y, z special coordinates and t the time coordinate this we can write as a product of 2 functions small psi henceforth we will just call this psi small psi which is a function of special coordinates x, y and z and the other function is phi which is a function of t and the use of the small psi and small phi is quite popular. So, very often we do not even write these coordinates in brackets we just write capital psi equal to small psi multiplied by phi. The thing to remember here is that capital psi is dependent on special coordinates as well as time small psi depends only on special coordinates phi depends only on time. Why can we write like this because x, y, z, t x, y, z is special coordinate t is temporal coordinate and we can consider them separately space has got nothing to do with time if you work in the non-relativistic domain that is what we are doing now there is something called relativistic quantum mechanics but in this course we are not going to get into it here we work in the domain where time and space are completely separable. So, since they are completely separable we can write the wave functions as product of a space dependent part and a time dependent part. So, now we go back to the equation h operates on capital psi to give you ih cross multiplied by del del t of capital psi. So, we can write like this Hamiltonian operator H H hat operates on the product of small psi and phi to give you ih cross multiplied by del del t of the product of psi and phi. Now, the thing to understand is the Hamiltonian operator H what is it remember it is some constant multiplied by del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2 plus potential energy time independent potential energy. So, the entire operator has no term in time it is expressed completely in special coordinates. So, we can say safely that the Hamiltonian H operates only on psi and del del t operates only on phi. And this thing is going to hold for stationary states we come back to the stationary states defined by your by by Reynolds Bohr H operates only on psi and del del t operates only on phi. So, we can write like this we can take on the left hand side we can take phi out and write phi H psi on the right hand side we can take psi out and write psi multiplied by ih cross del phi del t and then we can divide both sides by the product psi phi this is what we get H psi divided by psi on the left hand side on the right hand side we have 1 by phi multiplied by ih cross multiplied by del phi del t. Now, see the left hand side is a function of coordinates of space sorry I have missed of space here L H is a is a function of spatial coordinates I should have said and RHS right hand side is a function of time. So, they can be dimensionally consistent only if they are equal to a constant we will write that constant as W. So, we write it as W and then we can get 2 equations one is this H psi equal to W psi the other is ih cross del phi del t is equal to W phi. So, what we have been able to do so far is that we have been able to break down the time and space and time dependent Schrodinger equation into 2 equations one of which is completely space dependent the other is completely time dependent and that is what makes our life a little simple not only because the math becomes simpler but also because this space independent sorry time independent space dependent Schrodinger equation is the one that we are only going to use for the rest of the course because we are all working with stationary states stationary states are such that psi psi star is independent of time such that E is independent of time. So, there is no need for us to worry about the time dependent part of the wave function anymore in this course what we should worry about is what is this W? Well we have obtained what is called separation of variables. Well just for the records if you just solve the time dependent part of the wave equation that is very simple and that turns out to be e to the power minus i Wt by H cross. And to answer what is W we take the lesson from classical mechanics as we said in classical mechanics Hamiltonian represents total energy it is the total energy operator. So, when Hamiltonian operates on the function and gives you W multiplied by psi from classical mechanics itself we know that this W has to be equal to E energy. So, H psi equal to E psi this is the form of Schrodinger equation that is most popular among chemists because this is what we are going to use for the rest of the time. So, we can expand it a little bit and write minus H cross square by 2m del to del x2 plus vx operating on psi x equal to E psi x. So, this is Schrodinger equation. And now we have reached an interesting juncture in our discussion because it turns out that Schrodinger equation is a special kind of equation it is called an eigenvalue equation. What is the meaning of eigenvalue equation eigen in German means one you can think unique. So, eigenvalue equation is one is such that an operator operates on a function to give us back the same function multiplied by a constant. In this situation if this is the situation then this function is called an eigenfunction of the operator in question and this multiplier that we get that is called the eigenvalue. So, what we see here is that the total energy operator Hamiltonian operates on the space dependent wave function psi of x to give us an eigenvalue equation H psi equal to E psi. So, first of all the space dependent time independent wave function is an eigenfunction of the Hamiltonian operator. Secondly, the energy of the system is obtained as an eigenvalue psi of x is the eigenfunction of the energy operator and E is the eigenvalue that you get in the eigenvalue equation that we get we obtain when the total energy operator Hamiltonian operates on psi of x. So, in Schrodinger from this understanding of Schrodinger equation we see that this eigenvalue equation and eigenvalues eigenfunctions they can have rather important implications in quantum mechanics. So, Schrodinger equation is an eigenvalue equation. From here quantum mechanics really opens up because what one can do knowing this is that one can now propose the laws of quantum mechanics or as they are commonly called postulates of quantum mechanics. So, let us not forget once again that postulates cannot be derived postulates are beginning points. How do we propose this postulates by looking around and seeing what how nature manifests itself. What we have obtained so far is Schrodinger equation which is an eigenvalue equation total energy operator operates on the wave function to give us an eigenvalue equation in which energy turns out to be an eigenvalue. From this equation it is extended further and this postulates are proposed the first one is that for every classical variable say position momentum kinetic energy potential energy. There is a corresponding operator one can use in quantum mechanics. For example, for position the quantum mechanics quantum mechanical operator will be just x that is multiplied. For momentum it turns out to be minus i h cross d dx kinetic energy if you look at the Hamiltonian operator we know already what kinetic energy operator is it is minus h cross square by 2m d to dx 2 in one dimension. In 3 dimensions it is minus h cross square by 2m del 2 del x 2 plus del 2 del y 2 plus del 2 del z 2 potential energy operator actually is the potential energy depending on what kind of system it is the expression is going to change. So, first law of quantum mechanics first postulate of quantum mechanics is that there is always an operator for every classical variable every classical property of the system that you might want to know how do we know. We know the value by making the corresponding operator of that physical property operate on the wave function we will get an eigenvalue equation like we did for Schrodinger equation and the eigenvalue is going to be the value of that property. So, operators and eigenvalue equations have a very central role to play in quantum mechanics. So, if you look at this expression we can think like this that the wave function contains all the information of the system if you ask it is going to give you the information. How do you ask a question to a wave function by making the appropriate operator operate on the wave function and how does it give you the answer in the form of eigenvalue that we get. So, we can only know the value we can only have a specific value of the property if one can write an eigenvalue equation with the corresponding operator for the wave function that is characteristic of the system. So, in the next module we will start from here we will have a further discussion of what kind of operators what are the properties of operators that are acceptable in quantum mechanics and what are the other postulates of quantum mechanics and after that we will try to make sense of this entire wave business of matter because until now we have been we have an equation that describes the wave. We have sort of obtained postulates that can work in quantum mechanics. What we do not have so far is what is the meaning what is the real meaning of this wave what is the real meaning of this wave function that is what we are going to get at in the next maybe couple of modules.