Let $\varphi$ be a normalized, well-behaved function that satisfies the boundary conditions of the problem (called test function). The variational theorem says that: \begin{equation}\label{ecu1} \int\varphi^{\ast}\hat{H}\varphi dq\geq E_0 \end{equation} Where $E_0$ is the energy of the state fundamental.


The variational integral given by the equation (\ref{ecu1}) can be written in the following form: \begin{equation}\label{ecu2} \int\varphi^{\ast}\left(\hat{H}- E_0 \right)\varphi dq\geq 0 \end{equation} Let $\varphi=\sum_{i} c_i \psi_i$, where $\psi_i$ is the eigenfunctions of the Hamiltonian operator $\hat{H}\psi_i=E_i \psi_i$ \begin{equation} \int\sum_{i}c_{i}^\ast \psi_{i}^\ast\left(\hat{H}-E_{0}\right)\sum_{i }c_{j}\psi_j dq=\sum_{i}c_{i}^{\ast}\sum_{j}c_j\int\psi_{i}^{\ast}\left(\hat{H}- E_0\right)\psi_j dq= \nonumber \end{equation} \begin{equation} \sum_{i}c_{i}^{\ast}\sum_{j}c_j\left[\int\psi_{i} ^{\ast}\hat{H}\psi_j dq-\int\psi_{i}^{\ast}E_0 \psi_j dq\right]=\sum\left|c_i\right|^2\left(E_i- E_0\right)\geq0 \end{equation}

We use cookies

We use cookies on our website. Some of them are essential for the operation of the site, while others help us to improve this site and the user experience (tracking cookies). You can decide for yourself whether you want to allow cookies or not. Please note that if you reject them, you may not be able to use all the functionalities of the site.