In this video, we will introduce the interaction picture. The question that we want to answer is, how to deal with time-dependent potential. Let's consider a Hamiltonian that can be split into two parts, H naught that is time-independent, and V of t which is time-dependent, and we assume that we know the solutions for these time-independent part H naught. Write down the Schrodinger equation. Here, we know the energy eigenvalues of n and the corresponding energy eigenkets. In general, we know we can describe the time evolution of quantum state using the time evolution operator constructed from its Hamiltonian. However, when the Hamiltonian itself is time-dependent, we can write down the time evolution operator like this. In general, the time-dependent potential induces transition from one state to another state. The answer that we're trying to find here is what is the probability of that transition induced by the time-dependent potential. To be specific, let's express the initial state at t equals 0, our Alpha here, as a linear combination of the unperturbed Hamiltonian eigenstates, and here, so because the eigenket n should form complete orthonormal basis set, we can do this and these coefficients can be determined by the standard algebraic techniques. Now, we attempt to find the state at a later time t greater than 0. We can express that state, Alpha of t as this. This coefficient here contains the time evolution information. It changes over time. We inserted this exponential factor containing the energy eigenvalues of the unperturbed state. We could, of course, include these e^i, E_n t over h bar factor into these time-dependent coefficients C. C depends on time anyway. This is another part of time dependence, we can lump it into it, but we will see later that separating them out leads to a simpler equations that we have to handle. Now, let's consider a physical system in a state Alpha at t equals 0, initially. At a later time, the state ket is in this state. In this notation, we have Alpha of t, but we added a subscript S, indicating that we are using Schrodinger picture. If you recall, in the Schrodinger picture, time evolution is entirely described by the state ket, whereas the operators are time-independent. Now, we introduce another state ket with a subscript I indicating this is the state ket in the interaction picture. This state ket in the interaction picture is given by the state ket in the Schrodinger picture multiplied by this exponential factor containing the unperturbed Hamiltonian. At t equals 0, the two states are the same because the exponential factor simply is one. For operators representing physical observables, we define the operator in the interaction picture as this. This here, A_S is the same operator in the Schrodinger picture. Once again, in the Schrodinger picture operator has no time dependence. This is time-independent quantity. The operator in the interaction picture has time dependence, given by this exponential factor multiply to A_S from left and from right. Therefore in the interaction picture, the time-dependent potential V that we have introduced at the beginning can be written like this. Now, we compare this with the Heisenberg picture. In the Heisenberg picture, the state cat has no time dependence, and only the observables, operators are time-dependent. The time evolution of a quantum system in the Heisenberg picture is entirely described by the time evolution of the operator and no time dependence is associated with the state cat. The relationship between Heisenberg picture and the Schrodinger picture, the state cats in the Heisenberg and Schrodinger picture, and the operators in the Heisenberg and Schrodinger picture are given here. This is something that we discussed earlier, and now we write down the time dependence in the interaction picture, and to see the time-dependence of interaction picture, we try to calculate the time derivative of the state cat in the interaction picture. We just introduced the definition of this state cat in the interaction picture, it's given by this, the state cat in the Schrodinger picture multiplied by this exponential factor containing unperturbed Hamiltonian h-naught. We then use the chain rule to calculate the time derivative and we get these two terms. The first term just remains here. For the second term, we have a time derivative of the state cat in the Schrodinger picture and this time derivative of state cat in the Schrodinger picture should obey, satisfy the time-dependent Schrodinger equation. Substitute this time derivative with this term here according to the time-dependent Schrodinger equation and we can see that this term and this term cancels out and we're left only with this term here. This exponential factor multiplied to V multiplied to the state cat in the Schrodinger picture, that's the result and here we insert it at these two terms which should cancel out. Multiplying these two should not impact the results, and we do that so that we can obtain an equation of operators perturbation term and the state cat, both in interaction picture. Noting that this here is the V, perturbation term in the interaction picture, and this here is the state cat in the interaction picture, we can write it down compactly like this. This here is an equation governing the time evolution of state cat in the interaction picture. Sometimes it's called the Schrodinger-like equation because this looks very much like the time-dependent Schrodinger equation with the substitution of V_ I in place of the Hamiltonian H. Now, we can derive a similar equation, time evolution equation for the operators in the interaction picture, and we can obtain this equation here, which looks very much like the Heisenberg equation in describing the time evolution of operators in the Heisenberg picture. Once again, here, we replace the Hamiltonian H in the original Heisenberg equation with H-naught, the time-independent part. In the interaction picture, the time evolution of the state cat is determined by the time-dependent perturbation term, whereas the time evolution of the operators is determined by the time-independent part, H-naught, of the Hamiltonian. Here is the quick summary of these three different pictures. In the Heisenberg picture, we already know that state cat is time-independent. Time dependency is entirely with the operator's observables and the time evolution there is determined by the Hamiltonian H. In contrast, in the Schrodinger picture, the operators are time independent. Time evolution of the quantum system is described entirely by the time evolution of the state ket and the time evolution of the state kets, once again determined by the Hamiltonian H. In the interaction picture, both state kets and the operators evolve over time have time dependence. The time evolution of the state kets determined by the V_I, the time dependent part of the Hamiltonian in the interaction picture, whereas the time evolution of the operators are governed by the time independent part of the total Hamiltonian H naught. We could continue by constructing these state ket in the interaction picture by expressing it in terms of the linear combination of the eigenkets of H naught as a basis set and these eigenkets of H naught which satisfy this equation, still forms the complete orthonormal basis set so we can express any ket as a linear combination of this time dependence is included in the coefficients here. Now, we multiply the bra n to the Schrodinger-like equation that we obtained before. The left-hand side simply is the time derivative of this inner product, which should give you this quotient here. The right-hand side is this. Now we insert this term here, the outer product of these eigenkets summed over all possible eigenkets, and this, if you recall the outer product of these eigenkets, some deliverable kets in interfaces set is an identity operator. This is simply equal to this. By doing this, now, what we can do is we can write down the matrix element here for V_I, spell out the definition of V_I here. V of t is the time-dependent part of the Hamiltonian given to us initially from the problem and the time evolution part, the phase factor or the exponential factors including H naught is multiplied here according to the definition of V_I. Then we can, because n and m are the eigenkets of H naught here, these exponential operators are simply converted into these exponential factor, including the energy eigenvalue E_n and E_m as shown here. What we are left with is this matrix element of V of t, the time dependent part of the Hamiltonian in the matrix element of that constructed by these energy eigenkets of H naught. We call that V_mn and to explicitly express the time dependence, we include these of t here and then the phase factor, including the corresponding energy eigenvalues of E_n and E_m. That's the right-hand side here and of course, this inner product here and here is simply the expansion coefficients in this. We can write down that equation compactly like this. The left-hand side is the time derivative of the expansion coefficient C_m. The right-hand side here has the matrix element of V, time dependent perturbation and these exponential factor containing the energy difference between the energy eigenvalue n and energy eigenvalue m, Omega here is defined as this. This here is the coefficient itself. We can write down this in a matrix form as shown here. If we know all of these matrix elements, and we can calculate the matrix element V_mn given V of t, and given all the eigenstates of the unperturbed Hamiltonian H naught. We can do that and of course, the exponential factor simply can be calculated from the energy eigenvalues of the H naught. We can do all that and if we construct this matrix, then we can calculate the time evolutions of the coefficient C_n's using this by solving this matrix equation and therefore determine the time evolution of the quantum system.
