In this video, we will develop the time-dependent perturbation theory. Let us recall the time evolution equation in the interaction picture. Here, the coefficient c_n, which we use to describe the quantum state using as the basis of the eigenstates of the unperturbed Hamiltonian. These are c_ns or coefficients in that series. We can find the rate of change of those coefficients if we know this matrix. This matrix is the constructed matrix of the perturbation Hamiltonian V constructed by using the unperturbed eigenstate as the basis there once again. Now we have a general equation, but usually we can't really solve this equation exactly. The notable example is the two-level system for which we we can obtain exact solution, but most other cases, we cannot solve them exactly and that means we have to resort to an approximation approach. We can try a perturbative approach where these coefficients c_ns are expressed as a perturbation expansion. Here is a zeroth order, first-order or second-order, and so on and so forth. Now, let us define a time evolution operator in the interaction picture. This U_I is the time evolution operator in the interaction picture. If you operate this operator onto a quantum state ket Alpha in the interaction picture at time t_0, it evolves the time into t, which is larger than t_0. Now, let's use the Schrodinger-like equation in the interaction picture, which governs the time evolution of the state kets in the interaction picture. So the Schrodinger-like equation in the interaction picture was here. Instead of H that we normally use in the Schrodinger picture, in time-dependent Schrodinger equation. In the interaction picture, we use V_I here. If you plug in this definition of the time evolution operator here for Alpha of t, you can obtain the time evolution equation governing the time evolution operator itself in the interaction picture. You can solve this equation with an initial condition given here. This simply says that the t equals t_0 is set as our reference time point so that at that point your time evolution operator is one and your time evolution operator will describe the behavior of your state ket or operator as time moves away from this reference point t_0. That's simply is a initial condition. The general solution can be found first by converting the differential equation in the previous slide into an integral equation as shown here. You can convince yourself that by comparing this integral equation and the differential equation in the previous slide, they are equivalent. The solution to this differential integral equations can be found in the form of an infinite series, which is often called the Dyson series. The Dyson series is obtained by successively substitute this integral equation for the time evolution operator included inside the integral. If you substitute this whole equation for this U_I inside the integral, you obtain this very first line here, and if you keep on doing it then you get infinite series. You can of course truncate this infinite series at some point to obtain an approximate solution. Then it becomes a matter of just performing these multiple integrals of the perturbation potential term in the interaction picture. Once we found the time evolution operator, then we can consider this time evolution of an eigenstate of the unperturbed Hamiltonian H_0. Let's say that this i here is one of the eigenstate of the H_0 and it had time has evolved. It's at a certain time and we're expressing that state in the interaction picture with a subscript I shown here. From the definition of the time evolution operator, so this time evolved state at t can be expressed as the initial state operated on by this time evolution operator and the initial state I just added a subscript S here because at t equals 0, the state ket in the interaction picture is the same as the state ket in the Schrodinger picture. They are identical. We're going to just drop this subscript here. We assume that all these kets represents the state at t equals 0. This term here can be further modified by multiplying this operator here, the outer product of the eigenstate of the unperturbed Hamiltonian summed over all eigenstate. This, if you recall, is just an identity operator. We can multiply this operator to anything without really changing anything. This equation hold, but if you write the time evolved eigenstate in the interaction picture in this form, you can immediately see all we have done here is to express this state as a linear combination or superposition of the unperturbed eigenstate, and the coefficients in the superposition is given by this matrix element of the time evolution operator. To show it explicitly, we can express any eigenstate as a linear combination of the eigenstate of the unperturbed Hamiltonian, and the probability is determined by these coefficients in the expansion in the linear combination. Comparing this and this, we can see that C_n is simply equal to this matrix element of the time evolution operator. The matrix element gives you the transition probability amplitude between two eigenstates or if you take the absolute value square of that quantity, we call transition probability amplitude, you actually get the transition probability. This represents the probability of finding the system in state n when the system was initially in state i after a certain time t has elapsed. What is the probability of finding the system in another eigenstate of H naught n. That probability is given by this quantity here. From the Dyson series, we can find the probability or the coefficients immediately. The zero-order is just a delta function which simply restates the initial condition; the system was initially in state i. The first-order coefficient is given by this integral here and this quantity here, V_ I is the perturbation Hamiltonian interaction picture, and if you spell it out using the definition of the operators in the interaction picture you can write it out like this. V_ni is the matrix element of the time-dependent perturbation V using the eigenstates of H naught as the basis set. If you construct that matrix element, that's this V_ n and i, and there is this exponential factor containing Omega_ni and this Omega_ni, of course, is the energy difference between level n and level i divided by h bar. Now, for the first order, you have only one integral. For the second order, according to the Dyson series, you have double integral summed over all eigenstates of H naught. In the end, you truncate the summation, the Dyson series at some point, find the coefficient C_n. Just add up, however, many terms that you want in the Dyson series, truncate at some point and calculate the absolute value squared that gives you the probability of finding the system in state n at time t, of course. Now, we have developed a perturbation theory using the interaction picture and the solution was found by using the Dyson series. You can follow this exact same approach that we have used for time-independent perturbation theory. You can set up a perturbation series solution into the Schrodinger equation. Equating the same orders, you can obtain the first order equation, second order equation, as we have done before with a time-independent perturbation theory. In this case, we are strictly staying within the Schrodinger picture. The results, of course, are identical. In this case, you will obtain the arbitrary state cap at time t in Schrodinger picture is given by this equation here. There is the time-dependent coefficients and the phase factor associated with the eigenstates is added explicitly here. If you use this, plug it into the Schrodinger's equation using the perturbation series as shown here, then you will find that these coefficients in the perturbation series is related to each other by this. Specifically, the time derivative of the k plus 1 order of the coefficient is related to the kth order coefficients through this equation. If you look at this carefully and compare with the integral equation, the Dyson series solution that we have looked at before, you will be able to see that the solution of this is given by the same integral equations that we have found using the interaction picture. Those two approaches are identical. They give you the same equations.
