In this video, we will discuss the tight binding method. So let us consider two identical potential wells with a finite barrier between them, so finite in both height and width. So if the two wells are completely separated from each other by an infinite barrier, infinite in either height or the width, then the two wells are completely separated and they wouldn't interact at all, and we can simply solve each of the potential wells separately and find a solution. And because the potential wells are identical, they should have identical solutions. With a finite barrier however, these wave functions will stretch out into the adjacent well and there will be some interactions. And therefore both the wave functions and the eigen values of their energy will change. Now in this case it is difficult to use the degenerate part of Asian theory because it's difficult to set up the problem appropriate for the perturbation theory that is, it is difficult to set up the Hamiltonian with known solution, unperturbed Hamiltonian with known solution plus probation term. So instead what we do here is to formulate a solution constructed by or made of the eigenstates of the uncoupled potential well, individual potential well. So this method is called the tight binding method. So here is the problem, so these are the two finite potential well separated by a finite barrier here. Now, for the left well only, if we consider the left well only then it's a single finite potential well problem which we know how to solve, we have to solve numerically but we know how to solve it. And you will show us the energy level here, this great energy level and the wave function for that. We're just showing the lowest energy state there and for the right potential well, the solution will be identical except that the wave function is shifted so that it is centered on the right well, the energy level will be identical to each other. Now, these are our basis functions with which we will construct the final wave function. And the final wave function, we already know what it will look like because this potential profile is symmetric about this central point. Because of this symmetry, we know that the eigenstates of this problem should have a definite parody, the wave function should be either even or odd. So, if we construct the total wave function using these two left well wave function and the ground state wave function for the left well, and the ground state wave function for the right well, using these two. If we construct the total wave function, then we can imagine that we could construct a symmetric combination to make an even function using these two way functions down here, showing in the red curve. Or we could construct an odd function anti symmetric function as shown here in the blue curve. So that's what we generally expect what the final solution would look like. So let us proceed to actually construct a wave function and calculate the energy level of these coupled well system. So let us denote the potential for the left potential well as V left of Z and the V left of Z looks like this. We choose the potential, outside the potential well as zero and inside the potential well the potential will have some negative value then. So that profile is this V's of left of Z, and the solution for this is shown here in the green curve that we denote as size of left of Z, and the associate energy level is E1. Now we can do the same thing for the right potential well and the v right of Z is zero outside the potential well and inside the potential well has some negative value same as this left potential well of course. And the solution once again is psi of right, shown here as a green curve and the energy Level is a one should be equal to, this is a one for the left potential well because they are identical. So the solution, once again, we know how to solve it, we need to solve it numerically, but we know how to do it. So the total potential of the coupled potential well is given by the simple summation of this v left and v right, and this is because we have chosen v equal zero outside the potential well. So that allows us to simply write the total potential is the sum of the left and right potential wells. The Hamiltonian, then can be written like this, and we solve the problem by using the left potential well, ground state wave function and the right potential well ground wave function as the basis set. So the eigen of the total couple of potential well is expressed as a linear combination of these two wave functions psi left and psi right. So these two functions, as we have seen in the previous slide are primarily confined in their respective well, psi's of left is primarily confined in the left well, psi of right primarily confined in the right well. And hence the term tight binding the finite potential barrier in between allows the wave function to stretch through the barrier and reach the other side. This leakage is the origin of the coupling between those two potential wells. So now we can write down the Schrodinger equation in a matrix form using side, left hand side, right as the basis function a and b are the coefficients for psi left and psi right, respectively. And the Hamiltonian matrix is constructed using these two once again as the basis sets. So index 1 denotes the left potential well and index 2 denotes the right potential well. So H sub 11 is the matrix element for H using the left wave function or the expectation value of H with the left wave function as shown here. A key approximation in this is that we assume that the amplitude of the left wave function is negligibly small in the region of the right well, right potential well, and similarly, the amplitude of the right wave function, psi of right is negligibly small in the region of the left potential well. So if we do that, make that approximation then in the region of the right potential well where the V sub right of Z is non zero, left wave function is 0, amplitude is 0. So in this integration, this part the contribution by the right potential well is 0 and similarly for other combinations. So we can simplify these matrix element evaluation significantly. And this is a good approximation when the barrier is relatively thick or tall so that the coupling is relatively weak. So writing down the 11 element of the Hamiltonian Matrix H sub 11, we ignore the last term V sub right, and we're only left with this, and this simply, this part is the Hamiltonian for the left potential well. And therefore this integration simply gives you the energy eigen value for the left potential well is a one. Similarly, H22 gives you the energy eigen value for the right potential well which is again is a one, identical. Now we're left to evaluate the off diagonal elements H sub 12 and H sub 21, H sub 12 can be exclusively written like this. So once again the Index 1 represents the left wave function and index 2, right wave function, and here is the Hamiltonian. And because the right wave function has negligible amplitude in the region where we left is non zero, that part makes a negligible contribution to the integral. When the v right is non zero, your left wave function has negligible amplitudes. So this part also, this term also makes negligible contribution and you are left with simply this, integrating this kinetic energy term only sandwiched between the left wave function, complex congregate and right wave function and this integral is now over the barrier region only. This is only the region where neither of these wave functions have non negligible amplitude. Similarly H 21 the other off diagonal element have a very similar form. The only difference is that these wave functions are now switched so there right wave function is complex conjugated and multiply from the left and left wave function is multiplied from the right. So we only need to basically evaluate these two integral and from the symmetry, we already know that H12 is equal to the complex conjugate of the other off diagonal element H 21, and let's call that delta E. Then the Schrodinger's equation, the matrix equation then becomes very simple, two by two matrix eigen value equation. The diagonal elements are both E sub 1, the off diagonal elements are delta E and delta E complex conjugate. So to find the eigen value, you set up the secular equation and solve for the eigen value, energy eigen value. The unperturbed energy level, energy level of the isolated separated well now is split into two. E1 plus delta E1 minus delta E, corresponding eigen functions by calculating the cushions A and B. We find that the Eigen function corresponding to the negative sign here is the symmetric combination of the two functions and the Eigen function corresponding to the plus sign here is the anti symmetric combination here and that's what's depicted here on the right, bottom right. So the symmetric combination here shown as the red curve has the lower energy, E sub 1 minus delta E, and the anti symmetric combination shown as the blue curve here has a higher energy E sub 1 plus delta E. So this bullet just repeat what I just said, lower energy state is a symmetric combination and the higher energy state has the anti symmetric combination of the wave function. The wave functions are no longer confined in one or the other of potential well, but they are shared by the two wells. So this properly represents the coupled potential well system. And the bringing two identical systems, in this case, potential well, finite potential well close together results in an interaction between the two and consequently splitting the originally degenerate high energy eigen value and into a one higher and one lower energy state. This is a very commonly observed phenomenon in quantum mechanics. So you can use this approach to chemical bonding, and in that case the finite potential well that we described here will be replaced by atoms. And you will be making linear combination of atomic wave functions to produce these symmetric and anti symmetric combinations, and those are the molecular wave functions, or sometimes called the molecular orbital. So by bringing two identical atoms together, for example, two hydrogen atoms close together, the hydrogen atom state will couple together and will perform a one lower energy state, which we call the bonding state, and one higher energy state, which is called the anti bonding state. You can imagine that we can extend this process to more than two atoms, three, four atoms, in fact, we can extend this to many, many atoms forming a solid, crystalline solid. And the tight binding approach is one of the commonly used techniques in calculating the energy band structures in solid.