 We have discussed the free particle, we have discussed what happens when we put this free particle into confined space in one dimension that is a particle in a box problem. Now let us see what happens if we increase dimensionality of the problem what happens if we put this particle in a 2D box what happens if we put the particle in a 3D box and when we do that we will see that to depict the wave functions we will need to draw a very good looking pictures like the one that we see here. Also we are going to learn a very very important technique which we have already used but this is something that we are going to elaborate a little further upon in this discussion. So, when we say we put particle in a 2D box what we mean is that it is free to move in say on the surface of a table that kind of a situation or you can think of we have talked about this conjugated poly in earlier and we said that very approximately you can treat the and a particular electron in those conjugated molecules using a particle in a 1D box model. Now think of something like graphene graphene is a flat two-dimensional molecule think of a p-electron in graphene a conduction electron in graphene it is free to move over the entire surface that would be the chemical application of particle in a 2D box model. To start with let us talk about the square box which means Lx is equal to Ly. So, from 0 to Lx that is the range in x and from 0 to Ly that is the range in y. This is the space in which the electron can move freely outside it is the wave function is 0 it cannot go there. All right what will the wave function be and what will the equation be? We use here the technique of separation of variables that we had also used when we talked about separating the spatial part of Schrodinger equation from the time dependent part. So, we write the Hamiltonian as Hx plus Hy here we can do this because x and y are independent of each other and we are talking about kinetic energy only. If potential energy was involved then we would not have been able to separate it so easily we would not have been able to write H equal to Hx plus Hy we can we get away doing it because the only kind of energy that a particle has here is kinetic energy. Otherwise it would have been impossible we are going to encounter situations later on where you cannot write things like this. Wave function is a product of psi of x and psi of y this is not difficult to understand. First of all H contains the Hamiltonian contains second order derivative that kind of an operation. So, if you take a product the derivative is a sum we want the energy to be a sum that is why it makes perfect sense. Also something in x and something in y they are independent you cannot add them. You can only multiply them because we are talking about displacements along x and displacement along y. There is no way you can just add them multiplication is fine. So, when we do that this is the kind of wave function that we get in fact we have been we have written something like this root over 2 by L multiplied by sin nx n pi x by L multiplied by root over 2 by L sin n pi y by L 2 particular in a box wave functions here 1 in y x and 1 in y then the constant turns out to be 2 by L sin n pi x by L sin n pi y by L please check whether this is a normalized wave function or not. But remember when you try to do that this is going to be a double integral when you integrate this it will be an integral with respect to x integral with respect to y also. So, it will turn out to be a normalized wave function all right. What about energy kinetic energy as we said it arrives p square by 2 m and p is essentially a vector sum of p x and p y. So, naturally energy is also going to be sum of e x and e y we have written e nx and e ny. So, this is what it is going to be nx square h square by 8 ml square plus ny square h square by 8 ml square since the length is the same along x and y direction equal to L we can take it out right. So, we get h square by 8 ml square multiplied by nx square plus ny square where nx and ny are 1 2 3 4 so on and so forth independent of each other. So, we have learned separation of variables and we have learned how to tackle the problem when the dimensionality increases a little bit. Now, let us learn how to draw the wave function how do we draw the wave function well 2 depictions are shown here. So, along x side let us say I am talking about 0 1 1 wave function. So, what is it it is a sine wave along x I will draw here and it is a sine wave along y sorry for the distortion I meant this to be a square box for whatever reason it has become distorted. Now, what happens when I multiply this by that do not I get a hill that looks like this a very symmetric looking hill yeah 0 at the boundaries maximum at the center of the box and the hill rises like this product of 2 sine functions. So, this diagram I think we understand nicely what is the meaning of color we will come to that, but what is the meaning of these lines. So, here what I have done is these lines are these lines generally these lines are basically joining all points that have the same value of psi. So, these are called contour lines contours join all points here with the same value of psi. Those who know how to read survey maps would be familiar with contours in context of those maps. There it is those contour lines add join not all points with the same value of psi, but all points with the same height from mean c level you might have seen maps like this in atlases and geography books. So, these are the contours what will the contours look like you join everything it is going it is a box right. So, it is going to look like squares that are rounded at the ends and this point at the middle that is the maximum point. So, this is your nx equal to 1 ny equal to 1 this is nx equal to 2 ny equal to 2 if nx not ny equal to 2 I am sorry that was just inertia nx equal to 2 ny equal to 1. So, nx equal to 2 means your wave function will be something like this in x direction wave function will remain what it was in y direction. Now, when you multiply what will happen you consider this line here at x equal to L by 2 you are multiplying whatever it is there in the y part of the wave function y dependent part of the wave function by a 0 this point is called in node a node is where the wave function goes through 0 and changes sign sign of wave function is minus here and plus here. So, well I have drawn it the other way because I have written plus here and minus here but anyway does not matter it changes sign. So, what will you get from between 0 and L by 2 along x direction you will get one hill and the hill now will not be completely symmetric on both sides it will longer on the y side and it will be half as long on the x side because it is going up to L by 2 what happens beyond x by 2 the sign of your x dependent part of the wave function has become negative. So, you are going to get a trough and that is the meaning of the color here red denotes plus sign blue denotes minus sign does it mean the wave function is actually red and blue no these are pseudo colors these are colors that we add to help us understand what the shape is once again if you go back and look up an atlas or a map you will see colors are used in addition to contour lines to denote height or depth of terrain here height and depth is in terms of wave function what about this wave function this is n equal to 2 y equal to 2. So, now I will draw it in the right manner change in sign and here also we have a change in sign this is that my diagram is not good. So, we have 4 quadrants a peak a trough a peak another trough and this is how you draw the contours remember in contours in order to show what is the sign you either use different colors or use explicitly you write the signs or better still you do both. So, this is how you depict three dimensional surfaces. So, here y has it become three dimension because the Lx because x and y these are the two special dimensions third dimension is wave function again this will become very very important when we talk about wave functions of hydrogen atom later on. One more thing we are this is wave function everywhere I am very sorry about that now what happens if we have a rectangular box instead of a square box everything else is same the only difference is now Lx is not equal to Ly. So, energy expression is h square by 8m multiplied by nx square by Lx square plus ny square by Lx where nx and ny again are 1 2 3 4 so on and so forth. So, what is the difference let us say I try to draw the energy levels of a square box and a rectangular box we will draw both for a square box what is the lowest energy level 1 1 what is the next energy level 1 2 but what is the energy of 2 1 well 1 2 means nx equal to 2 Ly ny equal to sorry nx equal to 1 ny equal to 2 what happens if nx is equal to 2 and ny equal to 1 the energy is the same because remember what the expression is h square by 8 ml square multiplied by nx square plus ny square how does it matter whether I write nx first or ny first it is the same. So, here for a square box we have degeneracy degeneracy means we have energy levels whose energies are the same what is the next level both can be 2 what is the level after that well it can be 2 3 or it can be 3 2 this is 2 2 2 3 and 3 2. So, what we see is that we have alternate levels that are doubly degenerate and single what about a rectangular box well 1 1 is the lowest energy level fine the next level is let us say 1 2 what is the level after that is 1 2 and 2 1 are they the have the same energy do they have the same energy no 1 2 and 2 1 will have different energies and whether 1 2 will be lower or 2 1 will be lower will depend on what is the relationship between Lx and Ly is Lx greater than Ly or is Lx less than Ly that is what will determine which one is lower which one is higher. So, degeneracy is not there I think I have something written here so I am just going to erase all this. So, what we learn from here is that symmetry and degeneracy go hand in hand you have more degeneracy when you have more symmetry if you break symmetry then degeneracy is lost and this is something that we encounter in say a metal ion complexes think of a free metal ion is a perfectly symmetric object it is a sphere. So, all the d orbitals have the same energy put it in an octahedral field what happens energy symmetry is a little less and now you have d orbital splitting E g and T 2 g perform yarn teller distortion then what happens energy decreases even more and so the degeneracy even within this T 2 g and E g sets is lifted to certain extent. So, the symmetry and degeneracy going hand in hand is a very important concept in chemistry and we encountered it for the first time in our discussion from this particle in a 2D box model. Now, let us talk about 3D box everything else is same wave function is a product of psi of x and psi of y and psi of z energy is a sum of energy along x along y along z fine. So, these are the expressions problem is how do I draw this because now x y z 3 dimensions are there and psi is a 4th dimension I do not know how to draw a 4 dimensional graph the way you draw it is that you assign color to the 4th dimension the way these are generated is that I have decided that I want particular value of psi actually mod psi let us say I want value of psi to be something like 0.2 units. So, wherever value of psi is 0.2 unit I put a dot and then all these dots generate a shape that is my depiction of wave functions. So, for psi 111 I will get a sphere not very difficult to understand what about psi 211. Now I am going to get a I am going to get a node and that node is going to show up like this I will join all the points that have value of say 0.2 and I will join all the points that have value of minus 0.2. So, you get something that reminds you of p orbital does not it similarly for 2 to 2 and so on and so forth. So, this is how we can draw a 4 dimensional picture in 3 dimensional space by using color or by using point for a particular value of the 4 dimension which is psi here. So, in our discussion of particle in a box what have we learned first of all we have encountered a situation where Schrodinger equation is exactly solvable in free particle as well as particle in a box free particle is less satisfying because you get a wave function that is not all that great in particle in a box you get a wave function that is better. What is more important for the first time in our discussion we arrive at quantization that arises automatically from application from imposition of boundary conditions. So, we learned that boundary conditions lead to quantization. Remember Schrodinger equation is a perfectly classical equation for de Broglie waves that is where it started from and then we forgot about de Broglie waves later on that it is a different issue. But there was no quantum number there. Quantum numbers arise when we use bond interpretation and apply boundary conditions. One thing that we have seen is that if you have more nodes in wave function higher is the associated energy remember particle in a box wave functions n equal to 1 no node n equal to 2 1 node n equal to 3 more nodes and so on and so forth. So, this is an important thumb rule that we use often in chemistry that a wave function that has more nodes is associated with higher energy. We learned about eigen function of linear momentum operators is this wave function of particle in a box an eigen function of linear momentum operator no it is not. But we showed that we can express it at sums of two eigen functions of linear momentum operator. So, when we make a measurement we will see either plus h cross k or minus h cross k. Average value of linear momentum is going to be 0 because the overall wave function is not an eigen function it is a linear function linear sum of two eigen functions. The simple model as we have seen yet it does find application as a starting point for many interesting chemical systems. We have started working on separation of variables which is going to be very important tool in our subsequent discussion and finally we have learned that symmetry and degeneracy go hand in hand I absolutely love this symmetry and degeneracy go hand in hand. So, we get an hint that symmetry is going to be an important parameter in chemistry. We are not going to dwell too much upon this in this course but just be advised that there are entire subject this is an entire subject in itself symmetry in chemistry. And also we learned how to plot functions that are more than three dimensions. This has proved to be a testing ground for most sophisticated treatment which we are not going to get into here. With all this background we are going to talk about hydrogen atom. How do you perform a quantum mechanical treatment for hydrogen atom? And there one important question that we are going to pose and hopefully learn the right answer to is what is an orbital? So, that would be the theme of the next maybe three or four lectures.