 Welcome back to the lectures on quantum mechanics and quantum chemistry. In the past few lectures we have been concerned with the basic mathematics of quantum chemistry or quantum mechanics. In the last couple of lectures we have been looking at the differential equation method for solving some of the model problems and one of the most important problems the as a model in quantum mechanics is the model of harmonic oscillator and we were looking at the Hermite polynomials and the solution of the Hermite equation which leads to Hermite polynomials and so on. And by the end of the lecture I made some statements regarding the special nature of these polynomial solutions to the harmonic oscillator. In the first part of today's lecture of this lecture we shall see a little more on the mathematics and also the physical consequences, but harmonic oscillator problem will also be done in this course by using another method known as the operator method. And therefore without spending too much time on the physical consequences we will wait until that part is done when the entire harmonic oscillator model will be dealt with in detail. But in today's lecture we shall look at again the functions that we wrote down and the energy level expressions that we had for the harmonic oscillator. The first thing is if you recall h psi is equal to E psi for harmonic oscillator becomes minus h bar square by 2 m d square psi by dx square plus half k x square psi is equal to E psi psi is a function of x is a function of x ok. And we solved this by writing psi of x as by an integer n psi n of x and we wrote down the solution as a normalization constant which is specific to the n and exponential minus alpha x square by 2 and a Hermite polynomial of alpha x ok. x is the position coordinate or is it the amplitude of the oscillator about the equilibrium value x naught which we did not include we assume that to be 0. And therefore the dimension of x is the length and root alpha has a dimension 1 over length and the energy level expression for this was also written in the last lecture as h nu into n plus a half where nu was written as 1 by 2 pi square root of k by m ok. You recall that we had polynomials which had odd or even character depending on the value of n n being an integer odd n meant that the polynomials were odd functions even n meant that the polynomials were even functions ok. But they were polynomials with a leading term of x raise to n for a given n. Therefore the functions increase as x increases what works against to them is this exponential to ensure that the wave function actually goes to 0 at x is equal to minus infinity to plus infinity and plus infinity this is our boundary condition or the domain of x ok. And we had psi of x goes to 0 as x goes to plus x r minus psi of x also goes to 0 at some finite points also at a number of points in between or number of values of x those values are called nodes of the function. They are called nodes of the function they do not really mean anything except that the probability of finding the system very near the node is very small because the probability depends on psi squared and the probability is never given at a particular point for a continuous system it is in a small domain and therefore around the nodes in a small interval the probability is very low. Otherwise the nodes have no other role as far as this particular course is concerned ok. But what is interesting is that the classical model of the harmonic oscillator if you remember and if you write the coordinate the position coordinate or the amplitude as plus and minus infinity the potential energy half k x square if we write v of x ok. The potential energy is a nice parabola well not really this side is not that even let us see if I can do it somewhat ok. This width k is this width is a function of the k so if k is very large this is a narrower parabola if k is very small this is a shallower broader parabola which is an indication of how much the potential energy is binding the pair of atoms in the case of a harmonic oscillator model for a diatomic molecule how closely or how rigid the atoms are as a spring as a molecular spring. So, a tighter or a very high value for the bond energy means a large value for k and therefore this potential energy is for a k much larger than the k that I have drawn this potential energy will look something like this for a k which is very loose for a non rigid molecule where the vibrational amplitude can be very large and the bond energy can be very significantly small the k will be different. So, the parabolic nature of the potential energy is also indicative of the molecular system that we are looking at whether the molecule as a the bond bond is a very strong bond or is it a floppy or a weak bond and so on. So, that is the association of the harmonic model to the bond of a diatomic molecule during the molecular vibrations. Now, what this means that is the energy E n H nu is equal to n plus or half means is that even when n is 0 even when the quantum number n please remember this n came from the fact that some lambda by alpha minus 1 was put equal to 2 n and this lambda and all forward constants associated with the mass and the force constant ok. So, the physical parameter has led to the quantum condition in order for the wave function to be meaningful and when n is 0 you still have an energy E 0 which is half H nu. If we mark this on a scale here this as 0 we have half H nu scale of energy total energy and n equal to 1 n equal to 0 the quantum number n equal to 0 for the quantum number n equal to 1, 2, 3, 4 etcetera you can see successively that it is 3 halves, 5 halves, 7 halves, 9 halves and so on. Therefore, the harmonic oscillator model gives rise to energies which are equidisposed ok, roughly ok, 3 by 2 H nu, 5 by 2 H nu, 7 by 2 H nu, 9 by 2 H nu and so on n equal to 1, n equal to 2, n equal to 3, 4 and so on ok. The eigenvalues remember H psi is equal to E psi is an eigenvalue equation and for this we have obtained the solution H psi n of x is equal to E n psi n of x where psi n and E n were obtained already. The eigenvalues and eigenfunction the eigenvalues are discrete the eigenfunctions correspond to the eigenvalues they are non degenerate there is one eigenfunction for one eigenvalue this is a one dimensional model ok no degeneracy whatsoever. Therefore, the harmonic oscillator energy levels are non degenerate at the wave function associated with this is psi 1 of x this is psi 0 of x psi 2 of x psi 3 of x and so on ok. If you look at the function carefully and if you also look at the classical expression for the potential energy you see that the potential energy is binding half k x square. Now at any extreme points if we take this x max for this energy this v and at that v the energy this energy if we take it as a total energy it is easy to see that the potential energy here as marked by this value half k x square x max x max is the potential energy is equal to the total energy and the oscillator has 0 kinetic energy that is what is true at the extreme ends of the oscillator the oscillator goes to 0 kinetic energy then the potential energy is the driving force for it to come back and at this point where the amplitude is 0 the oscillator has 0 potential energy, but has maximum kinetic energy and therefore at this point the total energy is all kinetic energy. So, you can consider in a classical system the energy to be any value continuous either this or this or this ok. The potential energy which is the maximum at this point is the total energy and all energies in between between these two extremes that is the here all these energies are now sums of the potential and kinetic energy such that the total energy is this or if you take this length of the total energy is e and so on any arbitrary value is possible here no the harmonic oscillator energy is given by a specific quantity n plus a half times h e ok. So, that is discretization or that is a quantization. The second part is that with the wave functions let us draw it here itself. Let us take the first wave function the 0 quantum number wave function corresponding to the n value n equal to 0 psi 0 of x is alpha by pi not square root sorry alpha by pi to the 1 by 4 e to the minus alpha x square by 2 and e 0 is half region ok. Now, the plot of that function psi of x we do that as a function of x positive and negative. We let us mark the potential energy half k x square which we can always write as a classical form for any value of k of course, k fixes this nu therefore, k and nu are related to each other. So, for that particular energy let us write the half k x square let us draw the half k x square as well that is ok that is all right. Now, let us mark the energy half h nu as this therefore, the energy level is that it is p is equal to h nu ok. If we plot the function e to the minus alpha x square by 2 this is x is equal to 0 the function is maximum given by root of alpha by pi to the 1 by 4. So, if you mark that since the Hermite polynomial associated with psi 0 is a constant and there are no contributions the function is a simple gaussian and it goes on like that ok. I have drawn all these things using the mathematical formula program drawing programs such as Maple or Mathematica in the lecture notes you will find all these things drawn to scale ok. This is psi of x if you plot psi squared of x in the same graph psi squared phi naught is alpha by pi to 1 by 2 e to the minus alpha x square. So, it is a slightly narrower gaussian because the coefficient on the exponent is large, but it is alpha by pi to the one half. So, something like this you can if you do that the graph looks more like that important. The square of the function outside of the classical potential barrier maximum value of the potential where for this value this potential energy corresponds to the total energy and therefore, in a classical model any potential energy any value of x larger x corresponds to the corresponding potential energy of half k x square here and that means the kinetic energy for the classical system is negative which is not permitted ok. Quantum mechanics tells you that it is possible for us to find the harmonic oscillator outside of the classically forbidden potential barrier because of the finiteness of the absolute square of the wave functions outside the potential limits maximum limits. This phenomenon leads to what is known as tunneling quantum mechanical tunneling. We will not do more than that many it is necessary we will read we will worry about this. Calculating tunneling probabilities etcetera is usually done in a standard course on quantum physics, but here when we do with a finite one dimensional potentials we will be able to do that. Tunneling is a basic new concept. Even though the quantum system is given a specific energy of half h nu the quantum system is likely to be found in areas where as a classical system it is forbidden that is with amplitudes larger than what a classical system would permit that is what is called tunneling ok. And what happens as we check this for not just half h nu, but 3 halves 5 halves 7 halves and so on ok. Thus the tunneling persists for if the oscillator has higher and higher energies. The answer is yes because for larger values I mean of the quantum number you have the Hermite polynomial which modulates this exponential and therefore, you have more oscillatory features because of the power series and so on. It is oscillatory because this term keeps driving everything down for large values of x. Therefore, even though the power series may tend to increase the exponential will tend to cut it down to size to be precise and to bring it to 0 for a larger and larger values of x. If you put higher order Hermite polynomial the function may go to 0 for very large values of x, but not for small values of x. But at the end of the day the exponential at the end of all of this calculation the exponential is the master the Gaussian is the master kills everybody if you are sufficiently far away from the x. But the point is it is never 0 exactly 0 except at a few points where the polynomials are 0 and also for all points outside of that potential barrier if there is still a finite probability. The only nice thing is that as the n increases to larger and larger and larger values the contribution of this probability to the total probability which is the area under this graph the square graph the contribution of this segment these two segments symmetrical is smaller relatively and it goes to 0 as n goes to very large values meaning tunneling becomes less and less important as the energy of the oscillator increases or the oscillator is tending to behave more and more like a classical harmonic oscillator which is a more correspondence principle that for very large values of energy that the system the quantum mechanical system will have all the features of the corresponding classical system and it reaches the behavior of the classical system asymptotically for large values of energy right that is still true here. But most important at low energies at low temperatures the quantum behavior is very striking and therefore the harmonic oscillator is fundamentally important in being a model different from the particle in a box model where now you understand that putting an infinite potential barrier ensured that there was no leaking of the probabilities outside of the barrier. This is like a leak in the probability of the system being conserved within this finite region from being conserved within this finite region and that becomes smaller and smaller as you go higher and higher ok. We will just see two forms of the harmonic oscillator wave functions before we stop this part of it and move on ok. Let us look at the next one we will have the potential energy still like this the next we will take a look at 3 half h nu which is the psi 1 e 1 is equal to this energy is 3 by 2 h nu and psi 1 or n equal to 1 as the functional form psi 1 of x is 1 by 2 2 alpha by pi to the 1 by 4 e to the minus alpha x square by 2 and h 1 of root alpha x which of course is 2 root alpha x 2 root alpha x or lumping the constants together root 2 alpha to the 3 by 4 pi to the 1 by 4 all of that e to the minus alpha x square by 2 times ok. This is an odd function it is psi 1 n is 1 the harmonic the Hermite polynomial is an odd polynomial ok. Since it is an odd function and since it is multiplied by x if you try to plot psi of x starts with 0 because of x is equal to 0 the function is 0 and as x increases the function increases ok, but the exponential minus alpha x square by 2 will try to bring it down. So, it reaches a maximum and then it goes down to 0 very quickly. Being an odd function of course it has exactly the opposite form the negative form here. So, let me draw this carefully is like that and if we do that it sort of goes to 0 ok. So, this is the function what is the square of the function this is psi 1 of x. The node here is at x is equal to 0 because that is the point where the function is 0 and only other place where the function is 0 is at x is equal to infinity or minus infinity ok. So, again here you have the same thing that the function is present outside of the classical potential barrier corresponding to this x max this x max because this is the value of x max is where the total energy is equal to the potential energy ok that is classical, but in quantum mechanics you see that even though the energy is this the wave function is still outside of that it is non-zero. What about the square of this? Obviously the negative side becomes positive and the maximum or the same place ok, but it goes to 0 much faster. Still this part is non-zero as we bring the function here this part is still non-zero so there is an energy there is a probability component outside of the potential barrier even for this energy of the oscillator which is non-zero meaning that the oscillator can if we try to find out its wave its position the possibility that the oscillator is outside of the potentially forbidden range classically forbidden range it can be found that probabilities finite. So, this is the node for the second wave function. Let me plot only one more and then I can illustrate the general trend psi 2 of x. We look at psi 2 of x is 1 by 2 root 2 alpha 2 by phi 2 1 by 4 e to the minus alpha 2 by 2. And h 2 of root alpha x is 4 alpha x square minus 2. If we plot that technically I should plot it some energy here 5 half h nu 5 half h nu e to if you do that this function is 4 alpha x square minus 2 goes to 0 at x is equal to 1 by what 1 by root 2 alpha plus or minus yeah. So, there are 2 points the alpha is of course the physical parameter associated with the model and therefore, the location of this points will depend on the oscillator itself what is its mass and what is its depending on its mass and depending on its constants therefore, the alpha is not a fixed point it is the oscillator dependent point. And also at x is equal to 0 the answer is minus 2 times whatever that constant that you have here alpha x square term is 0. So, the oscillator is something like this and it increases it reaches let me draw the function like this and then it increases to a maximum because the x square term will continue to increase will be this whole term will be positive and increasing as x is greater than plus 1 by root 2 alpha or x is less than minus 1 by root 2 alpha this term is positive, but it keeps increasing, but this guy ensures that the whole thing comes back to 0 and the corresponding it is an even function therefore, it looks exactly the same one. Now, you can see that f of x is equal to f of minus x the way I have drawn it appears that if I take the square of the function psi squared then the negative part becomes positive the psi squared becomes like that and the probability increases probability density increases to a larger value and then it comes back to 0 increases to a larger value and it comes back to 0. They are symmetric the point is this is a smaller segment than the sum of these two meaning that for larger energy the oscillator in a classical sense the oscillator spends more time towards the edges far away distances because you see the potential energy is also very large for larger energies the x max is large therefore, you can see that the oscillator spends more of its time farther away from the center and quantum mechanically also you get the same picture that the the probability of finding the oscillator farther away from the center is more, but there is always this problem that quantum mechanics tells you that the probability of finding the oscillator near the middle or right where the classically we would say that it spends the least amount of time therefore, it is least likely to be spotted the quantum mechanics tells you that it is not 0 it is reasonable there are things that you have to be worried and it is also dependent on whether it is half h nu the first wave function or second wave function or the third wave function and so on. So, all these quantum wiggles or the quantum properties are so important at low energies, but not very high energies because what happens is if I may draw the picture for a very large n and if I plot let us assume that this energy scale is some 100 times or 1000 times that scale and we have a quantum number n equal to say 100 or 200 or whatever if we do that and we put in the potential energy barrier something like this. What you will see is that for very large n when the energy is given by this this is this is e and also v of x then what you have is you will see depending on odd or even you will see a smaller wiggle slightly a larger wiggle and even larger and even larger and finally this whole thing will come down to 0 how many I have 1, 2, 3, 4, 5 1, 2, 3, 4, 5 this segment this segment that is outside this region becomes smaller and smaller compared to the probabilities that you compute in this segment and if you see this oscillations carefully as you increase the n value to larger and larger as the oscillator has more and more energies what you will see here is a nice amplitude which looks like this as the probability which means the probability of locating the oscillator and the time relative time that the oscillator spends in the intermediate region in a classical sense if you compare these two you will see that they both approach one another as the energy of the oscillator increases to larger and larger values and finally you would see when n is so large that you would see this middle oscillation is almost like a non-existent small therefore this graph goes through a minimum which is exactly what you will get in a classical sense that the time spent by the oscillator at the middle the relative time is the least therefore the in the quantum mechanical sense the oscillator is least likely to be located probable to be found in that region for very large of energy this is the Bohr's correspondence principle that quantum systems and classical systems approach each other or the quantum system behaves more like a classical system as the quantum number increases or as the energy of the quantum system increases relative to the rest of the energies. So it has all these very nice features and of course that is far more to it than our other operator methods and so on but as far as the chemistry is concerned we will stop at this point with respect to understanding the harmonic oscillator or what are called the basic characteristics and I shall write you the expression that you must also probably verify the orthogonality into the normalization expression I know I have written that earlier but it is worth remembering that when you write the orthogonality property psi n of x psi m of x the star is not written here because it is a real function dx which is given by minus infinity to plus infinity one sorry we have written the normalization constant n normalization constant m e to the minus alpha x square h n of root alpha x h m of root alpha x dx and that is equal to delta m m chronicle delta that is when n and m are the same the oscillator wave function is normalized when n and m are different the oscillator is the wave functions of the oscillator of the orthogonal to each other ok. So, these are the most important things as far as the harmonic oscillator is concerned and instead of starting a new lecture let me stop at this point of this discourse on the basic mathematical characteristics of the harmonic oscillator. In the next lecture what we would do is to skip this differential equation a little bit and look at coordinate transformations from Cartesian to spherical polar coordinate systems and then take up the differential equation after we have done a little bit about the hydrogen atom and the transformation of the hydrogen atom Schrodinger equation to spherical polar coordinates after that we will take up the differential equation again and then look at the solutions for the angular part of the hydrogen atom and this we will follow with the radial parts of the hydrogen atom. Now, what I would do in the next two or three lectures on the hydrogen atom is slightly more mathematical than what I have earlier done on the hydrogen atom in an engineering chemistry course whose videos have been alreadythere for quite some time I shall attach those videos to the present set of videos in my lecture notes for those of you who want to have some continuity. These two lectures will appear to have quite a lot of common material, but the algebra is taken a little more in this set of lectures than the introductory lecture that I gave on the engineering chemistry. Therefore, it is important for you toconnect both of these lectures and put the mathematical details at the appropriate points if you want to. If you do not need the algebraic details you just want to know the characteristics of the hydrogen atom solution then those engineering chemistry lectures with all the animations have them already and I would suggest that you go through the four lectures on the hydrogen atom without much of the mathematics. What I would do in the remaining the next three four lectures on mathematics is to take us through the algebraic details and also the transformation details and finally the solution in the form of spherical harmonics for the angular part and the log air polynomials and the exponential for the radial part rigorously we will derive it rigorously to the extent that the rigoris needed for the physicists and chemists particularly chemists we will not worry too much about the convergence and other properties there are of course layers at which you have to stop. So, let me stop at this point and we will meet again to study the coordinate transformations from Cartesian to spherical polars in the next lecture. Until then thank you very much.