 Hello, this is lecture 3 in this segment of the course on nano electronic device fabrication and characterization. And to recall this part of the course deals with nano materials, nano systems and tries to understand how to deal with them. Now in the last session we were introduced to the basic concepts of quantum mechanics the Schrodinger equation and the concept of a potential barrier. And we began to work with tunneling concepts and we will continue with that and then we go and go on to the so called potential well and quantization of energies that will be the agenda for today. Now to come back to the basic precept of quantum mechanics namely the time independent Schrodinger equation. As we said yesterday it is a second order differential equation where psi represents the probability amplitude or the so called wave function that describes the motion of a particle in a quantum mechanical system. The Schrodinger equation can be written as an operator equation as we said yesterday where h is the Hamiltonian operator which is given by minus h square by 2 m delta square by delta x square plus v x where v is the potential that is a time independent potential. And the solution of the Schrodinger equation yields the so called Eigen functions which are solutions to the equation. And the corresponding values of energy which are Eigen values of energy these are the possible probability amplitudes and the corresponding energies. And therefore, the probability of finding a particle at a position x then would be given by the probability integral namely psi x psi star x dx integrated over space that should give you unity and the probability itself is given by the product psi x square modulus. Now to repeat the Schrodinger equation is a second order differential equation. Therefore, the solution for psi requires two constants of integration and these are provided by requiring that both psi and the first derivative of psi are continuous across any potential barrier through which the particle might be moving. Now again to repeat what I have said yesterday psi and d psi by dx that is the first derivative if they are not continuous if they change abruptly at a boundary then the implication is that effectively infinite sources of energy might be required and this is of course physically unrealistic. Therefore, the requirement for solution uses the boundary conditions that psi is continuous across a boundary as well as the first derivative derivative of psi is continuous that gives us two constants of integration which allows us to solve for psi exactly. So, this is the basic principle of solving the Schrodinger equation in the time independent case. Now let us look at it further the Schrodinger equation is a linear differential equation that is the Hamiltonian operator which we have just shown here in one dimension is a linear operator the implication is that suppose you have solutions psi 1 psi 2 psi 3 which satisfy this second order differential equation. Then a linear combination of psi 1 psi 2 psi 3 such as a 1 psi 1 plus a 2 psi 2 plus a 3 psi 3 would also be a solution of the second order differential equation where these constants a 1 a 2 a 3 etcetera can actually be even complex numbers because what is really physically meaningful is only the square of the probability amplitude and therefore, you can have the quantities like a 1 to be complex. Now, this is called the principle of superposition and that is applicable to all kinds of wave motion and so on. So, essentially it is the characteristic of a second order differential equation that is a linear differential equation. So, the principle of superposition allows us to construct various possible solutions out of the linear combination of the solutions that we might find to the Schrodinger equation. Now, having dealt yesterday with some of the basics of the quantum mechanical principles. Now, we come to another very important quantum mechanical principle that you would have heard about namely the Pauli exclusion principle. Yesterday we dealt with the Heisenberg uncertainty principle and the Pauli exclusion principle is another very important part of quantum mechanics. Now, consider two identical particles let us say two electrons particle 1 is in state psi 1 that is this is a particular jargon of quantum mechanics psi 1 for example, depends on the position of the particle in one dimension. So, in a particular for a particular value of the position then you would call the particle as being in a state psi 1 where x for example, has a particular value. So, we call particle to 1 to be in the state psi 1 and particle 2 to be in state psi 2. Now, as these two particles are identical and they are indistinguishable it might as well be that particle 1 is in state state psi 2 and particle 2 is in state psi 1 you cannot tell between those two particular possibilities. Thus the two particle wave function if you had one particle the wave function is psi 1 if you have two particles then you have the product of those two functions psi 1 and psi 2. So, the two particle wave function in this case can be written either as psi 1 psi 2 plus that is psi 1 with particle 1 in that state psi 2 with particle 2 in that state and psi 1 with particle 2 in that state and psi 2 with particle 1 in that state as shown here. Now, this is a linear combination as we have just said linear combinations are valid wave functions, but you can either have a plus sign for the linear combination here or a minus sign for the linear combination. It turns out that whether there is a plus sign or a minus sign in such a case where we are considering identical particles describing the states of an identical particle or set of identical particles. It makes a big difference whether there is a sum of this and or a difference of these two these two terms. The plus sign corresponds to so called bosons and the negative sign corresponds to so called fermions. Bosons have integral spin the spin quantum number is a very special quantum mechanical property of particles. So, if the spin is integral then it is a boson if the spin is half integral then it is a fermion you would have learned about it in your earlier classes. So, you have both bosons and fermions particle particles therefore, with odd spins where the quantum of spin is equal to h bar by 2. So, if the if there is an odd multiple of h bar by 2 for a particle that is if the spin angular momentum of a particle is an odd multiple of h bar by 2 then it is a fermion. An example is electron of course, a famous example whose spin is equal to h bar by 2 electrons or fermions. Helium 3 nuclei are also fermions the spin there is 3 h bar by 2 whereas, helium 4 helium 3 is an isotope of helium the normal isotope is h bar I mean helium 4. So, this nucleus has a spin of 4 h bar by 2 and therefore, it is a boson. Now, as I said a moment ago the spin or the spin angular momentum is a quantum mechanical phenomenon properties quantum mechanical property with no equivalent in classical mechanics. Now, the poly exclusion principle states that two identical fermions cannot be found in the same state that is the broad general statement of the poly exclusion principle. If we consider two such fermions with the probability amplitude then the probability amplitude must change sign when the particles are exchanged that is we have written here the wave function for this two particle system as we have said in the previous slide. So, if you have this negative sign between these two terms these two product terms of wave functions then you have fermions as we have said. Now, you can see by the form of this particular wave function that if you exchange these two that is if psi 1 r 1 is the same thing as psi 2 r 1 then the product becomes 0 rather the factor becomes 0 the two factors in the sum they cancel each other therefore, the wave function becomes identically 0. So, if there is an exchange of particles between psi between the state psi 1 and psi 2 then you have a 0 wave function and therefore, this is not possible and that means that the poly exclusion principle statement that two identical particles two identical fermions cannot be found in the same state reduces to mathematically this statement that the wave function should be of this kind of this particular form. And if you exchange those particles then you would if they if they are in the same state then the wave function psi r 1 r 2 is 0 and that is not possible. So, that is the mathematical statement of the Pauli exclusion principle bosons on the other hand those that have integral half spins are not similarly restricted that is an arbitrary number of bosons can occupy the same quantum state for two bosons the wave function is the sum of these two terms as we have discussed earlier. And of course, there is no way this is going to be 0 under these conditions therefore, wave functions of all combinations of particles in different states are added which means that there is an increasing probability of two or more particles occupying the same state. And this is the phenomenon of Bohr's condensation which is observed at very low temperatures among other conditions. Now, it is important to remember that apart from helium for which is an example of a boson as we just mentioned bosons the most important class of particles for which are bosons are photons. We do not think of photons as having spin, but this spin for photons is actually plus or minus h bar sorry there is a slight error here is a plus or minus h bar corresponding to the left and right circularly polarized light we know of polarized light, but that is the equivalent of a spin for the photon. Now, one aspect that I mentioned in the verifax lecture is the statistics of nano systems. Now, when you are dealing with quantum particles then one has to formulate quantum statistics to deal with such particles or ensemble such of such particles. From the statement of the Pauli exclusion principle that we just went through it is clear that the Pauli exclusion principle affects profoundly the statistics of an ensemble of fermions. Because after all no two fermions can occupy the same state whereas such a restriction or a constraint does not apply to bosons. Therefore, one can see from here although elaboration is necessary and we will try to attempt that later on the statistics for ensembles of bosons and fermions have to be different because of this fundamental difference between the bosons and fermions namely the fermions have to obey the Pauli exclusion principle. As a result quantum statistics is divided into two parts namely Fermi direct statistics which describes fermions, Bose Einstein statistics that describes bosons. We will return to that in a later part of this segment. Let us come back to what we are dealing with yesterday in the previous class that is namely quantum mechanical tunneling where we have said that we have particle with an energy e which is less than a potential barrier v that it encounters in its motion. So, on the left of this diagram for x less than 0 we have a particle moving with energy e and to the right is a potential barrier of height v which is greater than e. As we said yesterday under these conditions the particle of the electron let us say in this case with energy e less than v would simply bounce off the barrier it will come back that is is going forward this way and then it will just bounce back and go back along the negative x direction because it cannot surmount the barrier. Now what we did yesterday was to consider this as a quantum mechanical problem and in the solution to the Schrodinger equation we found that the solutions are of the form of e to the i k x, but in this case k which is defined as square root of 2 m into e minus v over h bar square this becomes a complex number imaginary number. Therefore, the wave function e to the i k x really becomes now a real function because k itself is complex and therefore, what we have is a e to the minus i k x i kappa x where kappa is a real number. Therefore, what you have is an exponential decay of the function on the other side of the barrier that is you have here a sinusoidal function as the wave function over here e to the minus i k x whereas on this side what you have is an exponential decay, but still a real solution to the Schrodinger wave equation. So, we can depict the solution to the problem of a potential barrier in the following manner namely on the left side the probability of finding the particle namely the psi square over here is just e to the i k x minus e to the into e to the minus i k x which really is a constant. Therefore, the probability of finding the particle to the left of x equal to 0 is constant as shown here whereas on the other side psi is now an exponentially decaying function. Therefore, in quantum mechanics unlike in classical mechanics this particle has a finite probability of being found on the other side of the barrier, but the probability of finding it goes down exponentially as a function of distance on the other side of the barrier. This phenomenon whereby a particle with energy less than a potential barrier that it encounters goes to the other side with a finite probability is called tunneling. In this case if you are dealing with electrons you have electron tunneling and many phenomena in quantum mechanics with practical applications depend on such tunneling phenomena. Can we have an estimate of the distance over which tunneling takes place? Now, as we have said to the right side of the barrier the probability is the square of the wave function namely a square e to the minus 2 kappa k x 2 kappa x for x equal to 0 then therefore, this probability is a squared. On the other side let us take a value of x equal to 1 over 2 kappa. So, we are just going to equate x equal to 1 over 2 kappa when we do that the probability for that distance is equal to a square divided by e that is at x equal to 0 the probability is a square and at x equal to 1 over 2 kappa then the probability is a square divided by e. So, the distance over which the probability falls to 1 over e of its value at the boundary is 1 over 2 kappa. Now, actually the problem of the potential barrier is a very important problem for example, a very common problem too you have electrons in metal so called free electrons. The electrons are bound to the metal and they cannot come out because there is something known as something you would have learned about as the work function you have to provide so much energy for the electron to come out of the metal. But even if you do not provide that kind of energy there is a finite probability for the electron to come out of the metal and this estimate over here. Suppose you take a metal of work function 5 electron volts then using these numbers here using this equation where kappa is given by this equation and substituting the value for the mass of the electron and so forth and v minus e equal to 5 electron volts then you can find that 1 over 2 kappa is 0.25 4 5 angstroms that gives a feeling for the order of magnitude of the distance over which electron tunneling would take place in such a common case of free electrons in a metal. So, it is generally negligible although as we will show later this tunneling of electrons out of a metal is actually the basis of the development of the scanning tunneling microscope we will return to return to that later. Let us now come to a problem a very well known basic problem called the particle in a box problem in quantum mechanics. It is a colloquial way of representing a problem whereby you have a particle that is boxed what is meant by that it is there is no potential within the box, but the box has two walls of course and those walls are infinitely high. So, you can think of it as an infinitely high barrier or an infinitely deep potential well in which the particle is situated. So, this diagram represents the it is a schematic diagram of where you have a particle in a box the potential is 0 inside the box. Now, the Schrodinger equation then is become simple because v is equal to 0 from x equal to 0 to x equal to l and v is infinite outside these boundaries. So, we have to solve the Schrodinger equation to find the solutions for the problem namely what are the energy levels that the electron would assume if it is an electron that is confined to this potential well. Now, the barrier is infinitely high and therefore, the electron cannot go outside the box. So, the particle is confined how does it move within the box what are the energy levels of the particle within the box to do this of course to know this the Schrodinger equation needs to be solved by applying the appropriate boundary conditions. The Schrodinger equation for the particle in the box is simplified because v is equal to 0 and that equation is shown here. This is a simple equation for which we have already obtained the solution because this is the so called free particle. In the previous session we showed that the solutions are of this form psi x is equal to A e to the i k x which is a plane wave and k is given by this equation is related to the mass and the Planck's constant. This is really just a restatement of e is equal to h bar square k square over 2 m. So, that is the equation from which this one comes. Now, let us recall what we said a while ago namely that the general solution is a linear combination of waves travelling in the positive and negative x directions e to the i k x is a solution e to the minus i k x is also a solution. Therefore, the general solution is a linear combination of e to the i k x and e to the minus i k x and there are two constants A and B. Now, remember this is a second order differential equation. Therefore, there are two constants and as we have said earlier you can solve for these things by using the boundary conditions of where psi is continuous and d psi by d x is also continuous at the boundaries. So, this is the sort of a textbook problem for solving the Schrodinger equation. Now, one can go through the algebra of determining the constants and so on and as I said this is a textbook problem. What the solution comes out to be is that there is quantization that deterministic solution. There is a quantization condition that is the solutions are restricted to values of k such that k is equal to an integral multiple of pi divided by L or k is equal to n pi divided by L where n is equal to 1, 2, 3 etcetera integral numbers positive integral numbers. Now, let us note that k is inverse length over here 1 over length and therefore, it is so called reciprocal length or reciprocal wave vector. Now, from the equation e is equal to h bar square k square by 2 m we get therefore, that the energy is that the particle can have in this box is equal to e n is equal to n square pi square h bar square over 2 m L square. So, as I said this is the standard textbook treatment of the particle in the box where one solves Schrodinger equation and systematically obtains this solution. There is actually a simpler so called inspection method for knowing the solution let us say this is instructive. So, one can deduce the solution by inspection knowing that the free particle in the box has wave functions of the form psi is equal to e to the i k x as we already said. Now, e to the i k x has both sine and cosine terms, but remember that psi must vanish at x equal to 0 and x equal to L which are the boundaries of the box. Now, the cosine function is not appropriate here because the cosine function does not vanish at x equal to 0. Therefore, we can limit our consideration to the sine function. So, we can write psi x is equal to a into sine k x where a actually is determined by the so called normalization condition that is you integrate psi x multiplied by psi star x the complex conjugate over all of its domain from minus from 0 to L x equal to 0 to L and that gives you the condition that determines the value of a which is the normalization factor. Now, coming back as you said only sinusoidal functions are valid solutions for this and you see that k L for example, this has to vanish at x equal to L. Therefore, k L must be equal to n pi because it is sinusoidal function. Therefore, the quantization condition becomes k L is equal to n pi where n is equal to 1 2 3 etcetera. So, we get the same answer k is equal to n pi divided by L that can be obtained by a more elaborate treatment and of course, the Eigen values of energy are once again n square h square pi square over 2 m L square. So, these are the Eigen values of energy and we see that these are discrete or quantized. The normalized wave functions as I said can be obtained by going through this integral where you integrate between 0 and L which is the domain of this particle and the quantization this normalization condition yields the constant a to b square root of 2 divided by L. Therefore, the wave functions or the Eigen functions of the particle in a box particle in an infinitely deep well are given by psi n x equal to square root of 2 by L sin n pi x by L where n is integral. Now, suppose you consider the difference in energy for such a particle in a box between 2 consecutive levels as we have said here the energy levels are given by h bar square h bar square pi square into n square over 2 m L square. So, the value of e increases quadratically as the quantum number L as it is called this quantum number is gradually increased from 1 to 2 to 3 etcetera. So, if you take the difference in energy between 2 consecutive levels namely n and n plus 1 then that is given by 2 n plus 1 into h bar square pi square over 2 m L square. Now, notice that L is in the denominator therefore, the smaller the value of L that is the smaller the box the narrower the box then this you get a larger value for delta e that is the difference between 2 consecutive levels is greater when the box is narrower. This is a sort of a qualitative I would not say explanation although I have put it down that way it is a qualitative indication for the finding that the energy band gap in semiconductors which you have all heard about of course, you have learnt in the earlier part of the course. This course the energy band gap of semiconductors increases when the size of the crystal is reduced to a few nanometers. So, if you take L to be a few nanometers then you can see that the difference in the energy levels of consecutive states is larger when L is smaller. So, this is the so called blue shift in the band gap of semiconductors which are reduced in dimensions to nanometer levels we will come back to this in a more systematic fashion at a later time in this segment. Now, let us consider some examples an electron in different sized boxes that is to illustrate the point I have just made. Suppose L the size of the box is 1 nanometer then you can do the arithmetic to find that E n is given by 0.05 into n square in electron volt units. If n is equal to 5 the energy Eigen energy is 1.25 E v approximately if n is equal to 4 it is 0.8 E v. Now, one point I want to once again note for you here which I forgot to mention is that the difference between successive energy levels of the particle in a box increases as n increases that is as the quantum number increases successive energy levels are separated by larger and larger differences of energy. So, coming back to this particle in a box of 1 nanometer electron in a box of 1 nanometer the transition from the level E 5 level 5 to level 4 produces radiation in the infrared because the difference in wavelength is about 0.65 E v which is in the infrared. But suppose you know about a macroscopic box 1 centimeter sized box which is really our physical experience. We are familiar with these sizes then the same formula gives you that E n is given by 10 to the power of minus 15 times n square in electron volt units. That is at macroscopic sizes of confinement so called confinement the energy levels though they are quantized are extremely low and very closely spaced unless the quantum numbers are very large that is suppose n is a small number 1 or 2 or 3 or 10 or something like that a small number then you can see that E n is really very small in magnitude almost immeasurably small. But if the quantum number is large let us say of the order of 10 to the power of 7 then you begin to approach electron volt kind of differences in the energy. Now coming back to such a case where the energy levels are spaced very closely but are still quantized as in the case of a macroscopic object. What you have here is quantization but really extremely small differences between successive energy levels. This is the so called quasi continuous state of quasi continuous distribution of energies where even though there is quantization it is essentially impossible experimentally to learn that and that is really how Newtonian mechanics comes into play for large sized object or macroscopic objects. We say Newtonian mechanics that energy levels are continuous but really what it is is that these energy levels are quasi continuous they are so close to one another for macroscopic objects that they are in practice they are continuous. Now so far we have dealt with the infinite potential well so the barrier is very large or infinite the particle simply cannot escape but that is really a theoretical construct in practice and in practical situations and problems the potential well to which a particle is confined is of a finite height that is the potential well or the potential barrier has a height of v naught which is not infinite it is finite f u volts for example as in the case of the work function of a metal. Now what we learned earlier if you go back to the treatment of the potential barrier we saw that in such a case you can have tunneling that spreads the wave function outside the barrier even though the barrier height is less than the energy of the particle. So you can see that immediately when you have a finite potential well you can possibly have the wave function spread outside the barrier outside the well and therefore a finite potential well would be different from an infinite potential well because of the tunneling that makes the wave function outside the box finite. So if actually one goes through such a finite barrier of height v naught then the energy level diagram for such a case is altered slightly. What is shown here is the dotted lines show the energy levels of the infinite potential well e naught the ground so called ground state 2 e 4 e naught because that is the next excited state because remember the energies are proportional to n square. So if n is equal to 2 the energy is 4 times as much as that of the ground state. So these dotted lines represent the energy levels of the infinite potential well when the potential is finite what I want you to note is that the resulting energy levels which one can compute through simulations is lower than the energy for the infinite well every energy level is diminished with respect to the infinite potential well. The physical reason for that is that the wave function spreads outside the barrier on both sides therefore some of the energy is dissipated outside and therefore the energy of the particle inside the box is smaller than the energy would be for an infinite well. So each energy level is lowered with respect to the infinite well. Now there is a very nice book called a picture book of quantum mechanics in which problems of the potential well and so forth are simulated to illustrate the basic and important aspects of quantum mechanics. What is shown here is the simulation of a potential well finite potential well sorry this one is the simulation of an infinite potential well where the wave functions for different values of the quantum number n equal to 1, n equal to 2, n equal to 3, 4 and 5 are all shown as you can see these are all sinusoidal functions. The first wave function is just a half sine wave, the second one is a full sine wave and then 3 halves 2 and 2 and half and so on. So all of these are sinusoidal functions, these are the wave functions of the particle in an infinite box infinitely deep potential well. The bottom part of the simulation shows the variation in the probability of finding the particle at different positions within the box. Remember that the probability is a square of the wave function. So this is the probability amplitude, the wave function is the probability amplitude and this is the probability itself the modulus of the square of the wave function. So what I want to see here is that the probability for n equal to 1, n equal to varies considerably over the spread from x equal to 0 to x equal to l. As you increase, as you increase the value of n as you go to higher and higher values of n what I want to do note is that the variation in the probability of finding the particle within the box from x equal to 0 to x equal to l that variation becomes smaller and smaller in amplitude. So this as I said is the case for n equal to 5. You can imagine that as you go to higher and higher values of n much larger values of n let us say 100 or 1000 or something this variation is smeared out essentially to become effectively constant across the potential barrier within the potential barrier. So as the wave as the quantum number increases as the energy level of the particle increases because the quantum number automatically means higher energy level as the energy of the particle increases within the box then the probability of finding the particle within the box is essentially constant throughout the box that is what the simulations show. Now let us go to the case of the finite potential well of the same width but different depths. So what are shown here are four cases of a potential well each one deeper than the next one. So here you have a very small shallow potential well deeper deeper and deeper but all of them have the same width l is the same. So what this is showing is that if the potential well is finite also very shallow then it can only accommodate one energy level as shown here only one energy level is possible within this shallow potential well. But what is really important to note here is that this wave function phi x psi x over here the wave function over here because the potential barrier is very short so to speak means that the particle is not really very much confined the wave function spreads outside the box considerably it has a significant amplitude outside the box and also a significant extent outside the box. When any equal to 2 you can see that two levels rather when the potential well is less shallow when it is deeper than the first case then two levels can be accommodated and the wave functions still spread outside but that spread is now that amplitude of the waves outside the box is now reduced compared to the previous case and so on. If we come to a case where the depth is greater now four levels are accommodated in this one and you can see that the spilling over of the wave functions outside the potential box potential well is now reduced. What I want to also point out is that for the ground state for example the wave function is more confined to the potential well than the wave functions of the higher energy levels that is they tunnel out more outside the box than the ones that correspond to the ground state or a low quantum number. What I want to point out is that such wells of finite depth are really practically very important. Let us consider a well where there are two energy levels that are given over here just two one can imagine a transition from a particle or an electron in particular at the upper level to the lower level to the ground state. If the value of the L that is the extent of the potential well and the depth of the well if they are all appropriate then this transition between the upper level and the lower level in this two level box would then become the basis of a laser for example. So, wide emitting devices can be therefore fashioned out of potential wells of this sort which can actually be realized in material structures of a special kind and if time permits we will illustrate some examples of such potential wells and their applications. Now let us consider a different case where you have potential wells of the same depth, but different widths. In the previous case they are they all had the same width, but different depths. So, this alternative case is where they have different widths, but the same height or the same depth. You can see that when the well is narrow then there are only in this particular case there are only two possible energy levels within this narrow well. As the width of the well is increased what I want you to note is that the ground state energy is steadily reduced. The ground state energy is high over here compared to the bottom of the well. It is now lower over here even lower even lower and so on. That is the minimum energy of the particle is greater when the confinement is greater. Confinement greater meaning the well is narrower. This is a direct result of the uncertainty principle. When the confinement of the that is when delta x is small delta p is large as we saw yesterday and therefore delta p corresponds to the momentum of the particle and delta p large means the energy is greater. That is this is a direct illustration of the Heisenberg uncertainty principle where when you squeeze the box so to speak when you confine the particle to a greater degree then you raise the energy levels of the particle. One can also deal with remember these are simple functions where v is equal to 0 within the box and v is a constant value outside the box. But it is possible to simulate cases where you have a linear variation of the potential across a length x. So, you can see that you know this is sort of a saw tooth potential where the potential is increasing along x within the box. So, it is not constant within the box. So, in such cases one can obtain simulated solutions and you know these have a functional form that is different slightly different from the simple case of where the potential is constant within the box. But all these are possible through numerical solutions of the Schrodinger equation applying the right boundary conditions in a case like this. We have here an asymmetric linear potential on this this case and a symmetric linear potential in this case over here. So, here it is you can see that there is a difference in the nature of the solutions. Function forms are slightly different, but they are largely sinusoidal. So, and then once again there are energy levels that are different and what I want you to note is in this case the separation between energy levels successive energy levels decreases with the quantum number. Remember in the case of the infinite potential well the energy difference increases as n square here it is diminishing with n. So, the case physical case is quite different. This is the linear harmonic oscillator the v is equal to k x square function and the solutions are as you probably know or e n is equal to n plus half n plus half h nu where n is a an integer positive integer and therefore, the energy levels are equidistant from one another that is e n plus 1 minus e n is equal to h nu. So, if you compare this case what you can see is that this is coming close to the case of the linear harmonic oscillator where this is actually constant in this case it is not constant, but it is becoming so. So, what you can imagine is that one can think of this linear harmonic potential k x square potential as a kind of a sum as a kind of a sum of these triangular potentials and then you would actually see that the harmonic potential can be approximated through linear piecewise potentials. So, what you have illustrated is that one can simulate different potentials and learn about the differences between how the particle in a box in different shaped boxes that is you can have a simple box of the infinite height you can have a finite height potential where you have really square walls vertical walls. So, to speak on the other hand you can have a v shaped sort to shape to potential and such potentials can be used to simulate an actual quadratic potential of the simple harmonic oscillator where the solutions are well known the energy levels are quantized and equally spaced with respect to one another. So, what we have shown today is that the Schrodinger equation can be solved for simple cases exactly analytically, but in the case a more complicated case cases such as this sort of potential and piecewise linear potential and so on the solutions sometimes have to be approximate and they can be simulated, but the solution for the linear harmonic oscillator I must point out is actually exact. So, these are exact solutions involving so called Hermite polynomial as the wave functions for the simple harmonic oscillator. So, what we have done today is recapitulate the work that was discussed in the previous session then when we have gone on to discuss the Pauli exclusion principle and how that has bearing on the statistics that are applicable to fermions and bosons different kinds of statistics of these ensembles of bosons and fermions. Then we have returned to the case of the potential barrier and illustrated the case of tunneling across a barrier and how tunneling across a barrier becomes practical and practically important when you have a particle in a finite potential well as opposed to an infinite potential well. So, what we will come back to next time is how such simulations can be used to build periodic potentials that is a particle moving in a periodic potential. How does it behave when it is subjected to periodic potential? We will come to that in the next session. Thank you.