Spins 1/2 and Time-Dependant Perturbation Theory

[Erythro73]

Homework Statement

We consider two spins 1/2, $\vec{S_{1}}$ and $\vec{S_{2}}$, coupled by an interaction of the form $H=\alpha(t)\vec{S_{1}}*\vec{S_{2}}$. $\alpha(t)$ is a function of time who approches 0 for |t|-->infinity and takes appreciable values only in the interval of $[-\tau,\tau]$ near 0.

a) À t=-infinity, the system is in the state |+->. What is the probability of finding the particle in the state |-+> for t=infinity with the first order perturbation theory.

b) Prove that we can solve exactly this problem with the Schrodinger equation. Suggestion : you can start with the exact equation
$\displaystylei\hbar \frac{db_{n}}{dt}=\sum_{k} exp(iw_{nk}t)W_{nk}b_{k}(t)$

for the b(t) coefficients. Comparing this result with part a), what conditions is required so that perturbation theory is correct?

Homework Equations

$\displaystyle \frac{d^{2}b}{dt^{2}}= \frac{db_{n}}{dt}(\frac{d\alpha}{dt}\frac{1}{\alpha}+C_{1}\alpha)+C_{2}\alpha^2 b$

have the solution

$b(t)=Aexp(C_{+}\int_{-inf}^{inf}\alpha(t')dt')+Bexp(C_{-}\int_{-infinity}^{t}\alpha(t')dt'$

The Attempt at a Solution

a)
Ok. So, the first thing I did is write $\vec{S_{1}}*\vec{S_{2}}=\frac{1}{2}(S_{1+}S_{2-}+S{1-}S{2+}+2S{1z}S{2z})$.
I calculated that
$\frac{1}{2}<-+|S_{1+}S_{2-}|+->=0$
$\frac{1}{2}<-+|S_{1-}S_{2+}|+->=\frac{\hbar^2}{4}$
$\frac{1}{2}<-+|Sz1Sz2|+->=0$

So, I used
$P=\frac{1}{2}\left|\int_{-inf}^{inf}dt'exp(iwt')\alpha(t)<-+|\vec{S_{1}}*\vec{S_{2}}|+->|\right|^2$

which finally gave
$P=\frac{\hbar^2}{4}\left| \int_{-inf}^{inf} dt' exp(iwt') \alpha(t') \right| ^2$

That's as far as I can go as I don't know $\alpha(t)$.


b) Well... there's the problem. I start with the suggestion,
$\displaystylei\hbar \frac{db_{n}}{dt}=\sum_{k} exp(iw_{nk}t)W_{nk}b_{k}(t)$

I have no clue of where to start from that. Basically, I have to jump from the last equation I wrote to the equations written in the relevant equations. So, I'll try to make what I think, but this could be very wrong.

$\displaystylei\hbar \frac{db_{0}}{dt}=exp(iw_{01}t)W_{10}b_{1}(t)$
Let's say that, for the sake of me not writing more in this already long post, that $W(t)=\alpha(t)*K$

We have $\displaystylei\hbar \frac{db_{0}}{dt}=exp(iw_{01}t)K\alpha(t)b_{1}(t)$

Now, I'm stuck. I can't take the derivative whit respect to time at each side, because I don't know the b's.

Have a clue? :S


Thank you for reading me!
Erythro73

 

[Erythro73]

If I don't have any answer yet... Is it because my thread is badly done (have I forgot to put something?) or because nobody has a clue? :S

I'm still searching on my part.