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