In this video, we will discuss the variation method. Now the perturbation theory that we have discussed before is one of the most popular approximation methods, very widely used. Obviously a very powerful technique, however, it is only useful when we know the solutions of the unperturbed system. If we don't know the solutions of the unfiltered system, then we cannot use the perturbation approaches. The variation method that we will discuss here we will introduce in this video allows us to estimate the ground state energy for a system. We don't know the exact solutions for and specifically what we do is to attempt to estimate the energy expectation value. Expectation value of your hamiltonian operator for a certain trial wave function feet. Since the hamiltonian is presumably a permission operator, the elgen functions of the permission operator forms a complete set and if for each I can function size of end, we have the associated value S. And it should be true even though we may not know exactly what they are. So for simplicity, we assume that all the energy and values are not degenerate here. Now we express an arbitrary function here fee as a linear combination of the organ functions of the hamiltonian. So we can do that because once again this idea functions form a complete basis set. And we further require the state here fee is normalized, which simply gives us this equation here the absolute value square of all these corruptions and summed over all. Again, state should be equal to to what this is the normalization condition. Now we can write down the energy expectation value for the state fee here and so we send which H hamiltonian operator with fee. And by substituting this into these fee, we obtain, okay. This simply is equal to the a seven squared absolutely square the corruption. Absolutely square times the energy elgen values summed over all elgen states. Now let us arrange the terms in the summation in such a way that the lowest energy again valley smallest energy again, value comes first and then in the increasing order of the energy and values. So the question here then we ask is what is the smallest possible expectation value of energy that we can have for any arbitrary state or trial way function feet. Well, without doing any calculation, we know that the smallest possible energy expectation that you must be the smallest I can value, smallest energy value is the one. So this is the case when the fee itself is actually the elgen state for this lowest energy Elgen value. Or in the series expansion in the linear combination expression ace of one is equal to one and all other coefficients are zero. So this sounds like a trivial case though. Now let's make one other term non zero. Let's pick just any arbitrary against J and let's make that caution ace of J zero and all other cautions are still zero. So the only non zero coefficients are ace of one for the lowest energy against state and ace of J some other energy against state. So the energy expectations value is then given by this simple two sum of two terms. And the normalization condition requires that the sum of absolute valley square of this caution A seven is some 21. What that means in this case is that ace of one square plus, ace of J squared is equal to one and therefore ace of one absolute bell square here can be written as 1 -1 a sub J, absolute fairly square. And we arranging this, we can show that, okay. This energy expectation value is east of one, the lowest energy I can value plus the ace of J coefficients squared. And the difference between the two energy and this term is positive because each of one is the lowest energy elgen value is of J is always greater than is of one. so this is positive, this is non zero, therefore this quantity is always great and is a one. This is once again a tribunal result because the lowest possible energy of the expectation value for a system which has the ease of one as the lowest energy, lowest energy. Again value that expectation value must always be greater than is the one lowest energy again value. However, noticing this allows us to construct an approximate solution for the ground state and its energy, this is the heart of the variation of method. Specifically, we choose a trial way function and minimizes energy expectation value. Variation of method does not tell us how to choose the trial way function or what function we should choose as a trial way function. So we're free to choose whatever we want. And this requires some physical intuition because when choosing trial way function, we should try to capture the key features of the ground state. For example, higher energy state tends to develop nodes in their way function, lowest energy state tends to not have any nodes. So try to capture some key features of the ground state in your trial wave function. It should also be mathematically convenient for the minimization process. If you construct the function that is very difficult to handle, then the minimization process will become very difficult. And for energy minimization process, energy expectation value minimization process. The trial wave function usually contains a parameter to be varied. And we find the parameter value that minimizes the energy expectation value and choose that value to obtain the energy expect lowest energy ideal value and the and the approximate wave function for the lowest energy state. And therefore because of this typical introduction of the variation parameter, we call this the variation all method. Now, unlike the probation method, which allows us to track the errors through the other parameter in the variation method, we do not know how accurate our result is. We simply know the lower the energy the better. So we keep trying to lower the energy expectation value in some way. And this is because we just don't know the exact solution of the unfiltered camel Tonia. So in some sense, we are taking a guest and we are increasing the possibility that our trial wave function, our guests is close to the real solution. So, if you chose your trial wave function cleverly then the results will be sufficiently accurate with a relatively simple algebra. So that's the goal of the variation of method. You can use variation of method to calculate the excited state energies. By minimizing the trial wave function while keeping the trial wave function or orthogonal to the ground state wave function. You can imagine that that's quite difficult. At least a lot more difficult than minimizing for the ground state. So usually the variation of method is used to find the ground state energy. So here we give an example of the electron in an infinitely deep potential, infinite potential. Well, we know the solutions already. So the ground state wave function is here and the ground state energy is shown here. The we use this coastline function here, meaning that here we have chosen x equals zero to be the center of the potential. Well, Now, let us for a moment pretend that we don't know the exact solution and we try to find a trial wave function to calculate the lowest possible energy of this system. To do that, we should consider some key aspects or key features that we expect for the ground state wave function. First, it should vanish the wave function should vanish at the edges or end of the infinite potential that's the boundary condition. So at x equals plus and minus L over to the wave function should be zero. And it shouldn't have any note in the middle because again, developing a note is a characteristic of an excited state. And also it should be symmetric about the center of the well because your potential is symmetric about the center of the potential well. So with those considerations, we can choose a simple trial wave function, as shown here, this is just the upside down proble. And by choosing this constant here to be well over two squared, we satisfy the boundary condition. That require the wave function to be zero at the ends of the edges of the potential well, then we can calculate the energy expectation value. The hamiltonian here is simply the kinetic energy part inside the potential well. So integration is across the width of the potential well, and the hamiltonian is just the second derivative multiply by this constant. We have this term in the denominator for normalization. And if you do the algebra then you get this quantity here, compare that with the actual ground state energy of the infinite potential well, piece of one. This value is roughly 1% of the actual value, which is pretty good, very simple trial function that we tried heels. The very accurate results with an error of the order of 1%, which means that we made a very good guess. We chose a very good trial way function that resembles the actual ground state wave function. But we haven't really tried any variation. So true to the spirit of the variation of method, we introduce a parameter here that we can vary. So this here is a simple variation of the trial function that we used instead of choosing squared. We choose these exponents to be the variable primary lambda. And this, you can check that this wave function still satisfy the boundary condition and goes to zero at the edges of the potential well. Now we can plug it this into the expectation value equation to calculate the energy expectation value as a function of lambda this time and we obtain this equation. Now we take the derivative of this expression with respect to lambda to find the value of lambda that minimizes this quantity. And if you do that, you find that this expectation value reaches minimum when λ is equal to roughly 1.72. So you can see that our choice of exponents of two was already pretty close to the the value that leads to the real minimum. The corresponding minimum value of the energy expectation value is then then given by this and that leads to 1.298 times the actual ground state energy. So this time, by introducing a variable parameter and minimizes the wave function we have reduced the error even further, and now the error is only about 0.3%.
