Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

I want to understand where the matrix:

$$ \left|\psi(t)\right> = \binom{a(t)}{b(t)} = \begin{bmatrix} cos(\Omega t/2)&-ie^{i\phi_L t}sin(\Omega t/2) \\ -ie^{-i\phi_L t}sin(\Omega t/2) & cos(\Omega t/2) \end{bmatrix}\binom{a(0)}{b(0)}$$

comes from.

I've already derived $$\dot{a} = \frac{\Omega}{2}e^{i\phi_L t} b$$ and its counterpart $$\dot{b} = \frac{\Omega}{2}e^{-i\phi_L t} a$$

where $\Omega = \frac{1}{\hbar} \left<1\right|-d\cdot\varepsilon\left|0\right>$ and, I think, $\phi_L$ is the frequency of the incident radiation. But how can I get from the above two relations to the matrix?

share|improve this question

1 Answer 1

This is a coupled linear differential equation. Just write it in a matrix form as $\dot{X} = M X$, where $X$ is a vector formed by $a$ and $b$, and $M$ is a matrix. The solution is similar to the one dimensional case, but you will get the exponential of the matrix $M$ in the solution. You can get the components of $\exp (M)$ by using the Taylor expansion of the exponential.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.