- #1
Haorong Wu
- 413
- 89
Homework Statement
An electron is at rest in an oscillating magnetic field
$$ \mathbf B = B_0 cos\left( ωt \right) \hat k $$
where ##B_0## and ##ω## are constants.
What is the minimum field (##B_0##) required to force a complete flip in ##S_x##?
Homework Equations
$$H=- γ \mathbf B \cdot \mathbf S $$
$$ c^{(x)}_{-} = χ^{(x)†}_{-} χ$$
The Attempt at a Solution
I have solved that
$$ χ(t) = \begin{pmatrix} \frac 1 {\sqrt 2} exp \left( -i \frac {γ B_0} {2ω} sin ( ωt ) \right) \\ \frac 1 {\sqrt 2} exp \left( i \frac {γ B_0} {2ω} sin ( ωt ) \right) \end{pmatrix} $$
Also, the probability of getting ##- \frac \hbar 2## by measuring ##S_x## is
$$ P(- \frac \hbar 2 ) = {| c^{(x)}_{-} |}^2 = sin^2 ( \frac {γ B_0 sin(ωt)} {2ω}) $$
The statement of "to force a complete flip in ##S_x##" troubles me. Since English is not my first language, I failed to find out the meaning of the statement at google. From the solution, it seems that the probability ## P(- \frac \hbar 2 ) =1 ## relates to something that forces a complete flip.
So, what is a complete flip?