Lecture Notes on Time Dependent Perturbation Theory | PHYS 6210, Study notes of Quantum Mechanics

Download Lecture Notes on Time Dependent Perturbation Theory | PHYS 6210 and more Study notes Quantum Mechanics in PDF only on Docsity! Physics 6210/Spring 2007/Lecture 34 Lecture 34 Relevant sections in text: §5.6 Time-dependent perturbation theory (cont.) We are constructing an approximation scheme for solving ih̄ d dt cn(t) = ∑ m e i h̄ (En−Em)tVnm(t)cm(t), Vnm = 〈n|V (t)|m〉. For simplicity we shall suppose (as is often the case) that the initial state is an eigen- state of H0. Setting |ψ(0)〉 = |i〉, i.e., taking the initial state to be one of the unperturbed energy eigenvectors, we get as our zeroth-order approximation: cn(t) ≈ c (0) n = δni. We get our “first-order approximation” c(1)n (t) by improving our approximation of the right-hand side of the differential equations to be accurate to first order. This we do by using the zeroth order approximation of the solution, c(0)n in the right hand side of the equation. So, we now approximate the differential equation as ih̄ d dt cn(t) ≈ e i h̄ (En−Ei)tVni(t), with solution cn(t) ≈ c (0) n + c (1) n (t) = δni + 1 ih̄ ∫ t 0 dt′e i h̄ (En−Ei)t ′ Vni(t ′). Successive approximations are obtained by iterating this procedure. We shall only deal with the first non-trivial approximation, which defines first-order time-dependent perturbation theory. We assumed that the system started off in the (formerly stationary) state |i〉 defined by H0. Of course, generally the perturbation will be such that |i〉 is not a stationary state for H, so that at times t > 0 the state vector will change. We can still ask what is the probability for finding the system in an eigenstate |n〉 of H0. Assuming that n 6= i this is P (i→ n, i 6= n) = |1 h̄ ∫ t 0 dt′e i h̄ (En−Ei)t ′ Vni(t ′)|2. This is the transition probability to first-order in TDPT. The probability for no transition is, in this approximation, P (i→ i) = 1− ∑ n 6=i P (i→ n, i 6= n). 1 Physics 6210/Spring 2007/Lecture 34 TDPT is valid so long as the transition probabilities are all much less than unity. Other- wise, our approximation begins to fail. The transition probability for i → n is only non-zero if there is a non-zero matrix element Vni “connecting” the initial and final states. Otherwise, we say that the transition is “forbidden”. Of course, it is only forbidden in the leading order approximation. If a particular transition i → n is forbidden in the sense just described, then physically this means that the transition may occur with very small probability compared to other, non- forbidden transitions. To calculate this very small probability one would have to go to higher orders in TDPT. Example: time-independent perturbation As our first application of TDPT we consider what happens when the perturbation V does not in fact depend upon time. This means, of course, that the true HamiltonianH0+V is time independent, too (since we assume H0 is time independent). If we can solve the energy eigenvalue problem for H0 +V we can immediately write down the exact solution to the Schrödinger equation and we don’t need to bother with approximation methods. But we are assuming, of course, that the problem is not exactly soluble. Now, in the current example of a time-independent perturbation, we could use the approximate eigenvalues and eigenvectors from time independent perturbation theory to get approximate solutions to the Schrödinger equation. This turns out to be equivalent to the results we will obtain below using time dependent perturbation theory. It’s actually quite a nice exercise to prove this, but we won’t bother. Assuming V is time independent, the integral appearing in the transition probability can be easily computed and we get (exercise) P (i→ n, i 6= n) = 4|Vni| 2 (En − Ei)2 sin2 { (En − Ei)t 2h̄ } . Assuming this matrix element Vni does not vanish, we see that the transition probability oscillates in time. The amplitude of this oscillation is small provided the magnitude of the transition matrix element of the perturbation is small compared to the unperturbed energy difference between the initial and final states. This number had better be small or the perturbative approximation is not valid. Let us consider the energy dependence of the transition probability. The amplitude of the temporally oscillating transition probability decreases quickly as the energy of the final state differs from the energy of the initial state. The dominant transition probabilities occur when En = Ei, i 6= n. For this to happen, of course, the states |n〉 and |i〉 must correspond to a degenerate or nearly degenerate eigenvalue. When En ≈ Ei we have P (i→ n, i 6= n,En = Ei) = 1 h̄2 |Vni|2t2. 2