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Nt1330 Unit 1 Pdf

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Let us suppose that Hamiltonian of the system has a form $\hat{H}\big(p,q,\lambda \big)$. Here $p,q$ are canonical coordinates and $\lambda$ is the parameter. For the solution of Schrodinger equation $\imath \frac{\partial \Psi}{\partial t}=\hat{H}\Psi$ we implement following ansatz: $\Psi =\sum_{n}a_{n}\big(t\big)\varphi_{n}\big(p,q,\lambda\big)\exp \big\{-\imath \int_{-\infty}^{t}E_{n}\big(\lambda\big)dt\big\}$, where $E_{n}\big(\lambda\big)$ are the instantaneous quasi-energies that adiabatically depend on the parameter $\lambda$. After standard derivations for time dependent coefficients $a_{n}\big(t\big)$ we obtain iterative solution $a_{n}^{(1)}\big(t\big)=-\int_{-\infty}^{t}d\tau\sum_{m\neq n}\frac{\big\langle\varphi_{n}\big|\frac{\partial H}{\partial \lambda}\big|\varphi_{m}\big\rangle\dot{\lambda}}{E_{m}-E_{n}}\times a_{m}\big(-\infty\big)\exp \big\{-\imath \int_{-\infty}^{\tau}\big(E_{m}-E_{n}\big)d\acute{\tau}\big\}$. Adiabatic approximation is valid when the following criteria holds $\frac{a_{n}^{(2)}}{a_{n}^{(1)}}\sim \frac{\partial H}{\partial t}\frac{1}{\big(E_{m}-E_{n}\big)^{2}}$. Here $a_{n}^{(2)}$ is the second order correction to $a_{n}\big(t\big)$.

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