Suppose you don’t assume H'_aa = H'_bb = 0

(a) Find C_a(t) and C_b(t) in first-order perturbation theory, for the case C_a(0) = 1 C_b(0) = 0. show that |C_a⁽¹⁾(t)|² + |C_b⁽¹⁾(t)|² = 1, to first order in Ĥ'.

(b) There is a nicer way to handle this problem. Let

Show that

where

So the equations for and are identical in structure to Equation 11.17 (with an extra factor e^i∅ tacked onto Ĥ').
(c) Use the method in part (b) to obtain C_a(t) and C_b(t) in first-order perturbation theory, and compare your answer to (a). Comment on any discrepancies.

Transcribed Image Text:

da =et ₁₁a (1) dt ca. dh = et So H₁b (1) dt' Ch. Hea (11.26)