Time-Dependent Perturbation Theory: Understanding Rates of Transitions, Exams of Quantum Mechanics

Download Time-Dependent Perturbation Theory: Understanding Rates of Transitions and more Exams Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004, Mar. 16, 2004 16th March 2004 1 Miscellaneous 1. Continue Shankar Ch. 18 for time-dependent perturbation theory. 2. I will be 5 minutes late to class on Thursday. I am out of town next week (exam on Tuesday, we will schedule a makeup class). Interest in an exam review session on Sunday afternoon? 3. Today: (a) Review sheet for mid-term #2. Exam is on TUESDAY, MARCH 23 (1 week from today). (b) Hand out key for HWK #7, problem #3 and review HWK #8 while men- tioning degenerate perturbation theory. (c) Start on time-dependent perturbation theory. Last time, we used the sum ∑∞ k=1 1 (2k)2−1 . I think someone pointed out in class that this sum collapses. Here is how: ∞∑ k=1 1 (2k)2 − 1 = 1 2 ∞∑ k=1 [ 1 2k − 1 − 1 2k + 1 ] = 1 2 ( 1− 1 3 + 1 3 − 1 5 + . . . ) = 1 2 . 2 EXAM #2 NOTES Again, review the handouts, your own notes, and understand how to solve each home- work problem. You can also practice with the problems in Shankar. The format of the exam will be quite similar to the first exam. Again, a table of formulas will be provided. 2.1 Reminder of topics The topics for this exam are the variational method we studied, the WKB approxima- tion, and time-independent perturbation theory. Time-dependent perturbation theory will not be on the exam. Note that we relied on Griffiths for the WKB approximation, especially. So be prepared to answer questions about 1 • Variational method for computing approximate wavefunctions - the principles and applications. • WKB method: how to derive and the conditions for its application. • Use of the Airy function in deriving connection formulas. • Quantization conditions derived from WKB. • Applications of the WKB method, including computing bound state energies and tunneling rates. • Time-independent perturbation theory: how to arrange the expansion. • Perturbation theory in general: uses and dangers. • First-order nondegenerate perturbation theory: when can it be used and how to use it. Computing corrections to the energy and to the wavefunctions. • Second-order nondegenerate perturbation theory: conditions for use, sign of cor- rection for the ground state, and uses. • Degenerate perturbation theory: why is it that perturbation theory must be carried out more carefully if there are degeneracies? How to choose states for computing first-order corrections to degenerate states. • Extra stuff that came from solving homework problems: Airy function, wave equation in presence of magnetic fields, transmission coefficients. 3 Degenerate perturbation theory The main problem when there is degeneracy in the unperturbed problem is that the perturbed states are not necessarily continously connected to the unperturbed states, if the unperturbed states are poorly chosen. The perturbation selects a preferred basis among the degenerate states of the original Hamiltonian. This is called “breaking the degeneracy”. Perturbation theory relies on the states and energies changing smoothly with varying small parameter. To get this to happen, the trick is to choose combinations of unper- turbed degenerate energy eigenstates that are eigenstates of the perturbation. This can be done by diagonalizing the perturbation over the degenerate states or by finding an operator that commutes with the perturbation and choosing degenerate states that are eigenstates of this other operator. In any case, choosing the proper basis gives states that change continuously as the perturbation is turned on. Last time we looked at an example of this with a 2 × 2 Hamiltonian. When the per- turbation had only off-diagonal terms, the naive application of first-order perturbation theory gave no change to the energy eigenvalues. By rotating the basis, however, the perturbation became diagonal and the first-order corrections were easily read. You should review and understand this example and of course HWK #8. 2 which has solutions df = δfi − i h̄ ∫ t 0 dt′ 〈f |H ′|i〉eiωfit . This is the basic first order result that will get us going on several topics. 4.4 Types of perturbations 4.4.1 Direct computation The first-order formula for df can be directly applied, as Shankar does in Eqns. (18.2.10) through (18.2.13). 4.4.2 Sudden Perturbations Shankar has a section on sudden perturbations that actually has several types of changes in the Hamiltonian: • A brief, finite H ′: in the limit that the duration of the perturbation is short, there will be no change in ψ(t). This results from the fact that for finite total H , the change in ψ over an interval of duration  is zero as → 0. • A brief “infinite”H ′. Consider a δ-function over time change inH , withH ′(t) = δ(t− t0)∆H . Then consider the interval [t− , t+ ]: ψ(t0 + ) = ψ(t0 − ) + ∫ t+ t− dt′ ∂ψ(t′) ∂t′ = ψ(t0 − ) + 1 ih̄ ∫ t+ t− dt′H(t′)ψ(t′) = ψ(t0 − ) + 1 ih̄ (∆H)ψ(t0) . This is not clearly solvable in general, as ψ(t0) is not well defined (ψ(t) is chang- ing very rapidly). Taking the perturbation to be small however, allows us to assume to first order that ψ(t) is not changed by much. The amplitude to be in a state |f〉 at t > t0 is then df = 1 ih̄ 〈f |∆H|i〉 . • A sudden “permanent” change in H at a time t0. Again, over a short time inter- val, ψ(t) will not change for finite H , so ψ(t0 + ) = ψ(t0 − ). For example, if a system is in its ground state for t < t0 and the Hamiltonian is rapidly changed (much faster than the frequencies in the original H), the probability that the sys- tem will be in the ground state of the new Hamiltoinan is simply |〈ψ+0 |ψ − 0 〉|2 , where ψ−0 is the ground state wave function for t < t0 and ψ + 0 is the ground state for the Hamiltonian when t > t0. 5 4.4.3 Adiabatic perturbations This is a very important case that applies when the changes in H(t) are very very slow. Ifψn(t) are the eigenstates forH(t), then, when the conditions of the adiabatic theorem apply, dn(t) is constant: if the system is in an eigenstate at a given time, it remains in the continuously connected eigenstate at later times! For example, suppose a particle is in a box and is initially in the ground state. If the walls (boundary conditions) are moved very slowly compared to the characteristic velocity of the particle, then the particle will remain in the ground state for all times. We will go over the adiabatic theorem in more detail, soon. 4.4.4 Corrections to adiabatic perturbations See Shankar (Eqns. 18.2.28 through 18.2.31) for how first-order time-dependent pertur- bation theory can be used to rederive first-order time-independent perturbation theory. [Use H(t) = H0 + et/τH ′ for large τ and first-order time-dependent theory to find dm(0).] 4.4.5 Periodic perturbations An extremely important case, which leads to Fermi’s golden rule. This is especially important in understanding the interactions between matter and radiation. This will be covered in more detail after adiabatic perturbations are covered. I list it here for completeness. 6