# 6.1: Time Dependence

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

The exact expression for the time dependence of a system with $$N$$ states required a set of $$N$$ simultaneous differential equations. One case where we can solve this problem exactly is when we have a small number of states. Consider a system which requires only two basis states. Say we prepare it in initial state $$|1 \rangle$$ and we want to know how long it will take to go to the other state $$|2 \rangle$$. From section 5, we have two coupled equations in the time dependent $$c_1$$ and $$c_2$$:

$i\hbar \dot{c}_1 = V_{11}c_1 + V_{12}c_2e^{i\omega_{12}t} \\ i\hbar \dot{c}_2 = V_{22}c_2 + V_{21}c_1e^{i\omega_{21}t} \nonumber$

where $$c_1(0) = 1$$ and $$c_2(0) = 0$$.

If the change is slow, we can use first order time-dependent perturbation theory. We thus replace the $$c_n(t)$$ by $$c_n(0)$$, and integrate whence:

$c_1 \approx \text{ exp}(iV_{11}t/\hbar ) \\ |c_1|^2 \approx 1 \\ c_2 \approx \frac{−i}{\hbar} \int^t_0 V_{21}e^{i\omega_{21}t} dt \nonumber$

Including the constant of integration for $$c_1(0) = 1$$.

This page titled 6.1: Time Dependence is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.