Generated by the Physics Derivation Graph. Eq. \ref{eq:2934848} is an initial equation. \begin{equation} k = \frac{2 \pi}{\lambda} \label{eq:2934848} \end{equation} Eq. \ref{eq:9214650} is an initial equation. \begin{equation} \omega = 2 \pi f \label{eq:9214650} \end{equation} Eq. \ref{eq:2949002} is an initial equation. \begin{equation} \hbar = h/(2 \pi) \label{eq:2949002} \end{equation} Eq. \ref{eq:1203491} is an initial equation. \begin{equation} p = h/\lambda \label{eq:1203491} \end{equation} Eq. \ref{eq:3499522} is an initial equation. \begin{equation} E = h f \label{eq:3499522} \end{equation} Divide both sides of Eq. \ref{eq:9214650} by \(2 \pi\); yields Eq. \ref{eq:2939402}. \begin{equation} \frac{\omega}{2 \pi} = f \label{eq:2939402} \end{equation} Substitute RHS of Eq. \ref{eq:2939402} into Eq. \ref{eq:3499522}; yields Eq. \ref{eq:2949821}. \begin{equation} E = \frac{h \omega}{2 \pi} \label{eq:2949821} \end{equation} Substitute RHS of Eq. \ref{eq:2949002} into Eq. \ref{eq:2949821}; yields Eq. \ref{eq:3741728}. \begin{equation} E = \omega \hbar \label{eq:3741728} \end{equation} Divide both sides of Eq. \ref{eq:3741728} by \(\hbar\); yields Eq. \ref{eq:4499582}. \begin{equation} \frac{E}{\hbar} = \omega \label{eq:4499582} \end{equation} Divide both sides of Eq. \ref{eq:2934848} by \(2 \pi\); yields Eq. \ref{eq:1039485}. \begin{equation} \frac{k}{2\pi} = \lambda \label{eq:1039485} \end{equation} Substitute RHS of Eq. \ref{eq:1203491} into Eq. \ref{eq:1039485}; yields Eq. \ref{eq:2901049}. \begin{equation} p = \frac{h k}{2\pi} \label{eq:2901049} \end{equation} Substitute RHS of Eq. \ref{eq:2901049} into Eq. \ref{eq:2949002}; yields Eq. \ref{eq:1039013}. \begin{equation} p = \hbar k \label{eq:1039013} \end{equation} Divide both sides of Eq. \ref{eq:1039013} by \(\hbar\); yields Eq. \ref{eq:4948325}. \begin{equation} \frac{p}{\hbar} = k \label{eq:4948325} \end{equation} Replace scalar variables in Eq. \ref{eq:4948325} with equivalent vector variables; yields Eq. \ref{eq:2948487}. \begin{equation} \frac{ \vec{p}}{\hbar} = \vec{k} \label{eq:2948487} \end{equation} Eq. \ref{eq:3940505} is an initial equation. \begin{equation} \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right) \label{eq:3940505} \end{equation} Substitute RHS of Eq. \ref{eq:3940505} into Eq. \ref{eq:2948487}; yields Eq. \ref{eq:2100421}. \begin{equation} \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right) \label{eq:2100421} \end{equation} Substitute RHS of Eq. \ref{eq:2100421} into Eq. \ref{eq:4499582}; yields Eq. \ref{eq:1291313}. \begin{equation} \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) \label{eq:1291313} \end{equation} Simplify Eq. \ref{eq:1291313}; yields Eq. \ref{eq:1305534}. \begin{equation} \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) \label{eq:1305534} \end{equation} Eq. \ref{eq:1039948} is an initial equation. \begin{equation} p = m v \label{eq:1039948} \end{equation} Eq. \ref{eq:1353583} is an initial equation. \begin{equation} E = \frac{1}{2}m v^2 \label{eq:1353583} \end{equation} Raise both sides of Eq. \ref{eq:1039948} to \(2\); yields Eq. \ref{eq:1432042}. \begin{equation} p^2 = m^2 v^2 \label{eq:1432042} \end{equation} Multiply RHS of Eq. \ref{eq:1353583} by 1, which in this case is \(m/m\); yields Eq. \ref{eq:2326309} \begin{equation} E = \frac{1}{2m}m^2 v^2 \label{eq:2326309} \end{equation} Substitute RHS of Eq. \ref{eq:1432042} into Eq. \ref{eq:2326309}; yields Eq. \ref{eq:3576787}. \begin{equation} E = \frac{p^2}{2m} \label{eq:3576787} \end{equation} Partially differentiate Eq. \ref{eq:1305534} with respect to \(t\); yields Eq. \ref{eq:2364546}. \begin{equation} \frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) \label{eq:2364546} \end{equation} Substitute RHS of Eq. \ref{eq:2364546} into Eq. \ref{eq:1305534}; yields Eq. \ref{eq:5345567}. \begin{equation} \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t) \label{eq:5345567} \end{equation} Substitute RHS of Eq. \ref{eq:3576787} into Eq. \ref{eq:5345567}; yields Eq. \ref{eq:2495835}. \begin{equation} \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t) \label{eq:2495835} \end{equation} Multiply both sides of Eq. \ref{eq:2495835} by \(i \hbar\); yields Eq. \ref{eq:3429538}. \begin{equation} i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t) \label{eq:3429538} \end{equation} Apply gradient to both sides of Eq. \ref{eq:1305534}; yields Eq. \ref{eq:5577584}. \begin{equation} \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) \label{eq:5577584} \end{equation} Simplify Eq. \ref{eq:5577584}; yields Eq. \ref{eq:3454565}. \begin{equation} \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) \label{eq:3454565} \end{equation} Substitute RHS of Eq. \ref{eq:3454565} into Eq. \ref{eq:1305534}; yields Eq. \ref{eq:5535257}. \begin{equation} \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \label{eq:5535257} \end{equation} Apply divergence to both sides of Eq. \ref{eq:5535257}; yields Eq. \ref{eq:4938589}. \begin{equation} \vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right) \label{eq:4938589} \end{equation} Simplify Eq. \ref{eq:4938589}; yields Eq. \ref{eq:1495034}. \begin{equation} \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) \label{eq:1495034} \end{equation} Substitute RHS of Eq. \ref{eq:5535257} into Eq. \ref{eq:1495034}; yields Eq. \ref{eq:1049553}. \begin{equation} \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right) \label{eq:1049553} \end{equation} Simplify Eq. \ref{eq:1049553}; yields Eq. \ref{eq:4959593}. \begin{equation} \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t) \label{eq:4959593} \end{equation} Multiply both sides of Eq. \ref{eq:4959593} by \(\frac{-\hbar^2}{2m}\); yields Eq. \ref{eq:4349493}. \begin{equation} \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t) \label{eq:4349493} \end{equation} LHS of Eq. \ref{eq:3429538} is equal to LHS of Eq. \ref{eq:4349493}; yields Eq. \ref{eq:1304924}. \begin{equation} \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) \label{eq:1304924} \end{equation} Eq. \ref{eq:2344324} is an initial equation. \begin{equation} \frac{-\hbar^2}{2m} \nabla^2 = {\cal H} \label{eq:2344324} \end{equation} Substitute LHS of Eq. \ref{eq:2344324} into Eq. \ref{eq:1304924}; yields Eq. \ref{eq:2456546}. \begin{equation} {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) \label{eq:2456546} \end{equation} Eq. \ref{eq:2456546} is one of the final equations.