This problem is added by team 8;
Source: Introduction to Quantum Mechanics,D. Griffiths,Problem 4-34.
Problem: An electron is at rest in an oscillating magnetic field
where
and
are constants.
(a) Construct the Hamiltonian matrix for this system.
(b) The electron starts out (at t = 0) in the spin-up state with respect to the x-axis
[that is,
]. Determine
at any subsequent time. Beware.' This
is a time-dependent Hamiltonian, so you cannot get
in the usual way from
stationary states. Fortunately, in this case you can solve the time-dependent
Schr/Sdinger equation directly.
(c) Find the probability of getting
if you measure
(d) What is the minimum field
required to force a complete flip in
?
Solution:
(a)
(b)
with
, so
(c)
(d)
The argument of
must reach
(so P=1)
, or