Created
July 29, 2015 23:09
-
-
Save emptymalei/36c5ba5cb8938c218c6e to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
\begin{align*} | |
\ket{\Psi_{m}(x)}= &\cos\theta_m \left( a_L \exp(-i \int_0^x \omega_m(x)/2 dx ) \ket{\nu_L(x)} + a_H \exp(i\int_0^x \omega_m(x)/2 dx) \ket{\nu_H(x)} \right) \\ | |
&+ \sin\theta_m \left( -a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx ) \ket{\nu_L(x)} + a_L^* \exp(i\int_0^x \omega_m(x)/2 dx) \ket{\nu_H(x)} \right) \\ | |
=& \left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x)/2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx ) \right)\ket{\nu_L(x)} \\ | |
& +\left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m a_L^* \exp(i\int_0^x \omega_m(x)/2 dx) \right)\ket{\nu_H(x)} \\ | |
=& \left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x) /2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx ) \right)( \cos\theta_m \ket{\nu_e} - \sin \theta_m \ket{\nu_x} ) \\ | |
& +\left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m a_L^* \exp(i\int_0^x \omega_m(x)/2 dx) \right) ( \sin\theta_m \ket{\nu_e} + \cos\theta_m \ket{\nu_x} ) \\ | |
=& \left[\left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x)/2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx ) \right) \cos\theta_m \right.\\ | |
&\left. + \left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m a_L^* \exp(i\int_0^x \omega_m(x)/2 dx) \right) \sin\theta_m \right]\ket{\nu_e} \\ | |
& + \left[ -\left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x)/2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx ) \right) \sin \theta_m \right.\\ | |
& \left. + \left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m a_L^* \exp(i\int_0^x \omega_m(x)/2 dx) \right) \cos\theta_m \right] \ket{\nu_x} | |
\end{align*} |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment