% this tex file was generated by the Physics Derivation Graph 
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage[dvipdfmx,colorlinks=true,pdfkeywords={physics derivation graph}]{hyperref}
\usepackage{graphicx} % for including PNG files
% inference rules as newcommand for use in the body
\newcommand\declareinitialexpression[1]{Eq.~\ref{eq:#1} is an initial equation.}
\newcommand\dividebothsidesby[3]{Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.}
\newcommand\substituteRHSofexprintoexpr[3]{Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\replacescalarwithvector[2]{Replace scalar variables in Eq.~\ref{eq:#1} with equivalent vector variables; yields Eq.~\ref{eq:#2}.}
\newcommand\simplify[2]{Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\raisebothsidestopower[3]{Raise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.}
\newcommand\multiplyRHSbyunity[3]{Multiply RHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}}
\newcommand\partiallydifferentiatewithrespectto[3]{Partially differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.}
\newcommand\multiplybothsidesby[3]{Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.}
\newcommand\applygradienttoscalarfunction[2]{Apply gradient to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\applydivergence[2]{Apply divergence to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\LHSofexprequalsLHSofexpr[3]{LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\substituteLHSofexprintoexpr[3]{Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\declarefinalexpression[1]{Eq.~\ref{eq:#1} is one of the final equations.}
\title{derivation of Schrodinger Equation}
\date{\today}
%\author{a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}
\begin{document}
\maketitle
\begin{abstract}
Generated by the \href{http://allofphysics.com/}{Physics Derivation Graph}.
\end{abstract}

% step ID = 0001204921
\declareinitialexpression{24829834f445a7fd622eca653dfa73217a73048dc7912677751f088eb588cf6e}
\begin{equation}
k = \frac{2 \pi}{\lambda}
\label{eq:24829834f445a7fd622eca653dfa73217a73048dc7912677751f088eb588cf6e}
\end{equation}

% step ID = 0002919311
\declareinitialexpression{77477f431ef869637ada4946cd87eb04fc76d50bf544bb6ffc265fd3b06c565d}
\begin{equation}
\omega = 2 \pi f
\label{eq:77477f431ef869637ada4946cd87eb04fc76d50bf544bb6ffc265fd3b06c565d}
\end{equation}

% step ID = 0001294844
\declareinitialexpression{6e8e878f581a90ebbc0362b7ac7e4d7842d3439cc8a10d0fa536ddb01b0237d5}
\begin{equation}
\hbar = h/(2 \pi)
\label{eq:6e8e878f581a90ebbc0362b7ac7e4d7842d3439cc8a10d0fa536ddb01b0237d5}
\end{equation}

% step ID = 0009394842
\declareinitialexpression{7da046bbc410bb9c5af96410ccb2e21c475c74cb67c3bfbd858f824a178382c6}
\begin{equation}
p = h/\lambda
\label{eq:7da046bbc410bb9c5af96410ccb2e21c475c74cb67c3bfbd858f824a178382c6}
\end{equation}

% step ID = 0003934948
\declareinitialexpression{61bb0c7333431d34acf215337292b4ab805deb55ddd8ef6af57ba760b1ed1370}
\begin{equation}
E = h f
\label{eq:61bb0c7333431d34acf215337292b4ab805deb55ddd8ef6af57ba760b1ed1370}
\end{equation}

% step ID = 0003949482
\dividebothsidesby{2 \pi}{77477f431ef869637ada4946cd87eb04fc76d50bf544bb6ffc265fd3b06c565d}{ef60ed16bdc7f46063508c8f1e930614809309f3b8d11f082820a790ad63a5e8}
\begin{equation}
\frac{\omega}{2 \pi} = f
\label{eq:ef60ed16bdc7f46063508c8f1e930614809309f3b8d11f082820a790ad63a5e8}
\end{equation}

% step ID = 0001294945
\substituteRHSofexprintoexpr{61bb0c7333431d34acf215337292b4ab805deb55ddd8ef6af57ba760b1ed1370}{ef60ed16bdc7f46063508c8f1e930614809309f3b8d11f082820a790ad63a5e8}{f94e32dd28b5350b45fa8472b3643624ed7a2211fdbb9a4f1534f4fdbafd46a6}
\begin{equation}
E = \frac{h \omega}{2 \pi}
\label{eq:f94e32dd28b5350b45fa8472b3643624ed7a2211fdbb9a4f1534f4fdbafd46a6}
\end{equation}

% step ID = 0002930492
\substituteRHSofexprintoexpr{f94e32dd28b5350b45fa8472b3643624ed7a2211fdbb9a4f1534f4fdbafd46a6}{6e8e878f581a90ebbc0362b7ac7e4d7842d3439cc8a10d0fa536ddb01b0237d5}{22b7bcd1313e30b2ea209301c35211b3e40f2cae27ae2e3cadd8acf5fc070f4f}
\begin{equation}
E = \omega \hbar
\label{eq:22b7bcd1313e30b2ea209301c35211b3e40f2cae27ae2e3cadd8acf5fc070f4f}
\end{equation}

% step ID = 0003919384
\dividebothsidesby{\hbar}{22b7bcd1313e30b2ea209301c35211b3e40f2cae27ae2e3cadd8acf5fc070f4f}{5f28be326021d5606263f076692927f70fbf5024c8e0a4b850d1a85fb8b2ffc4}
\begin{equation}
\frac{E}{\hbar} = \omega
\label{eq:5f28be326021d5606263f076692927f70fbf5024c8e0a4b850d1a85fb8b2ffc4}
\end{equation}

% step ID = 0002900428
\dividebothsidesby{2 \pi}{24829834f445a7fd622eca653dfa73217a73048dc7912677751f088eb588cf6e}{349e9647c9eaab705a09d2d213fe47b67842a5629fe51e76e9d73eb5de2cdda3}
\begin{equation}
\frac{k}{2\pi} = \lambda
\label{eq:349e9647c9eaab705a09d2d213fe47b67842a5629fe51e76e9d73eb5de2cdda3}
\end{equation}

% step ID = 0001204945
\substituteRHSofexprintoexpr{349e9647c9eaab705a09d2d213fe47b67842a5629fe51e76e9d73eb5de2cdda3}{7da046bbc410bb9c5af96410ccb2e21c475c74cb67c3bfbd858f824a178382c6}{d8a8e2066b21dcf0f6754936299abef85aaadcc611bce207455f047ee2b197b9}
\begin{equation}
p = \frac{h k}{2\pi}
\label{eq:d8a8e2066b21dcf0f6754936299abef85aaadcc611bce207455f047ee2b197b9}
\end{equation}

% step ID = 0002939400
\substituteRHSofexprintoexpr{6e8e878f581a90ebbc0362b7ac7e4d7842d3439cc8a10d0fa536ddb01b0237d5}{d8a8e2066b21dcf0f6754936299abef85aaadcc611bce207455f047ee2b197b9}{df38feaa59cfd356b73bdbd3e495a9ff4bc1955c7baf1fe11beef28c6aa22542}
\begin{equation}
p = \hbar k
\label{eq:df38feaa59cfd356b73bdbd3e495a9ff4bc1955c7baf1fe11beef28c6aa22542}
\end{equation}

% step ID = 0002030624
\dividebothsidesby{\hbar}{df38feaa59cfd356b73bdbd3e495a9ff4bc1955c7baf1fe11beef28c6aa22542}{d5535648d302fd17f05884d064e7222ea64e3b8a1c3f0a83a451e7d8bff97381}
\begin{equation}
\frac{p}{\hbar} = k
\label{eq:d5535648d302fd17f05884d064e7222ea64e3b8a1c3f0a83a451e7d8bff97381}
\end{equation}

% step ID = 0001039774
\replacescalarwithvector{d5535648d302fd17f05884d064e7222ea64e3b8a1c3f0a83a451e7d8bff97381}{3773e0ca6a20d1927510533fa37908d7731b779fe1b168073c6afcf3d5b8eac1}
\begin{equation}
\frac{ \vec{p}}{\hbar} = \vec{k}
\label{eq:3773e0ca6a20d1927510533fa37908d7731b779fe1b168073c6afcf3d5b8eac1}
\end{equation}

% step ID = 0001923945
\declareinitialexpression{97a9edab1643aeb420eb6bbd0818dad12843ab0a4d924c8bee106beeb49c8919}
\begin{equation}
\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)
\label{eq:97a9edab1643aeb420eb6bbd0818dad12843ab0a4d924c8bee106beeb49c8919}
\end{equation}

% step ID = 0002938341
\substituteRHSofexprintoexpr{3773e0ca6a20d1927510533fa37908d7731b779fe1b168073c6afcf3d5b8eac1}{97a9edab1643aeb420eb6bbd0818dad12843ab0a4d924c8bee106beeb49c8919}{f9fb7b888af3c6d021b8e82263027cfcbaaefe15f35b00fd70a265a5161fdfa4}
\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:f9fb7b888af3c6d021b8e82263027cfcbaaefe15f35b00fd70a265a5161fdfa4}
\end{equation}

% step ID = 0001203100
\substituteRHSofexprintoexpr{5f28be326021d5606263f076692927f70fbf5024c8e0a4b850d1a85fb8b2ffc4}{f9fb7b888af3c6d021b8e82263027cfcbaaefe15f35b00fd70a265a5161fdfa4}{2767bff29fadc30a21c151d2c6b25d43667c8c7f2deab6d76ec16e197b13169a}
\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:2767bff29fadc30a21c151d2c6b25d43667c8c7f2deab6d76ec16e197b13169a}
\end{equation}

% step ID = 0001341141
\simplify{2767bff29fadc30a21c151d2c6b25d43667c8c7f2deab6d76ec16e197b13169a}{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}
\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:515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}
\end{equation}

% step ID = 0001204929
\declareinitialexpression{63b2f08f100632cbcce2438da3880de79a658c2ef19eb2b8a448fc7999384076}
\begin{equation}
p = m v
\label{eq:63b2f08f100632cbcce2438da3880de79a658c2ef19eb2b8a448fc7999384076}
\end{equation}

% step ID = 0001395335
\declareinitialexpression{635eae16f863aff79a60f21d038f248726ba79be0264633dd5509db173d009a4}
\begin{equation}
E = \frac{1}{2}m v^2
\label{eq:635eae16f863aff79a60f21d038f248726ba79be0264633dd5509db173d009a4}
\end{equation}

% step ID = 0002422434
\raisebothsidestopower{2}{63b2f08f100632cbcce2438da3880de79a658c2ef19eb2b8a448fc7999384076}{90343a4baa51c00bf0d2aee7113acc18d6b84058b5a911d2dc26f2d44dd76865}
\begin{equation}
p^2 = m^2 v^2
\label{eq:90343a4baa51c00bf0d2aee7113acc18d6b84058b5a911d2dc26f2d44dd76865}
\end{equation}

% step ID = 0002322434
\multiplyRHSbyunity{m/m}{635eae16f863aff79a60f21d038f248726ba79be0264633dd5509db173d009a4}{74c32a27f1c3234bb0a498ed69f0c83c64413b65cc65e65b98fe6611b1e7b6e6}
\begin{equation}
E = \frac{1}{2m}m^2 v^2
\label{eq:74c32a27f1c3234bb0a498ed69f0c83c64413b65cc65e65b98fe6611b1e7b6e6}
\end{equation}

% step ID = 0002449858
\substituteRHSofexprintoexpr{74c32a27f1c3234bb0a498ed69f0c83c64413b65cc65e65b98fe6611b1e7b6e6}{90343a4baa51c00bf0d2aee7113acc18d6b84058b5a911d2dc26f2d44dd76865}{b06d57d8dea7b73cb6dd4838e8369d8d01e486c99e4bfbf6ee570b6d7af2a5fe}
\begin{equation}
E = \frac{p^2}{2m}
\label{eq:b06d57d8dea7b73cb6dd4838e8369d8d01e486c99e4bfbf6ee570b6d7af2a5fe}
\end{equation}

% step ID = 0001347587
\partiallydifferentiatewithrespectto{t}{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}{53397b950405b42d76642300bdc1e5deca82a5671b7d993e6870115bdb01f05a}
\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:53397b950405b42d76642300bdc1e5deca82a5671b7d993e6870115bdb01f05a}
\end{equation}

% step ID = 0002464445
\substituteRHSofexprintoexpr{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}{53397b950405b42d76642300bdc1e5deca82a5671b7d993e6870115bdb01f05a}{4523c683a4d983195f22dba04627c073376a96ce8cbe08fd59390602758bf4c8}
\begin{equation}
\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)
\label{eq:4523c683a4d983195f22dba04627c073376a96ce8cbe08fd59390602758bf4c8}
\end{equation}

% step ID = 0003454353
\substituteRHSofexprintoexpr{4523c683a4d983195f22dba04627c073376a96ce8cbe08fd59390602758bf4c8}{b06d57d8dea7b73cb6dd4838e8369d8d01e486c99e4bfbf6ee570b6d7af2a5fe}{19d21f173508f836f6ba4c05b6150d0a8e0d915bc9ef035680eb986ec804eb9c}
\begin{equation}
\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)
\label{eq:19d21f173508f836f6ba4c05b6150d0a8e0d915bc9ef035680eb986ec804eb9c}
\end{equation}

% step ID = 0004358635
\multiplybothsidesby{i \hbar}{19d21f173508f836f6ba4c05b6150d0a8e0d915bc9ef035680eb986ec804eb9c}{67dc31740772af75abbee14e0431bcd127e608f0de63bc1f3b51a468cb7ed9bb}
\begin{equation}
i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)
\label{eq:67dc31740772af75abbee14e0431bcd127e608f0de63bc1f3b51a468cb7ed9bb}
\end{equation}

% step ID = 0002454535
\applygradienttoscalarfunction{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}
\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:515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}
\end{equation}

% step ID = 0005858694
\simplify{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}{84d9e9c70480967e07a7c9fe00f20029f87f348224b681488010f2c9c71704b7}
\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:84d9e9c70480967e07a7c9fe00f20029f87f348224b681488010f2c9c71704b7}
\end{equation}

% step ID = 0005354635
\substituteRHSofexprintoexpr{515af4a728a748e4120ba91430411733051a1b84a040f56d2618e4e80fce11d8}{84d9e9c70480967e07a7c9fe00f20029f87f348224b681488010f2c9c71704b7}{0e3be3e8448134af86b6f18f51f9cd034fdcf37b32d6dced781b71be074ed096}
\begin{equation}
\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)
\label{eq:0e3be3e8448134af86b6f18f51f9cd034fdcf37b32d6dced781b71be074ed096}
\end{equation}

% step ID = 0003294932
\applydivergence{0e3be3e8448134af86b6f18f51f9cd034fdcf37b32d6dced781b71be074ed096}{8b5859e448e2a9329f716e9bb00d8ef6cbdceb8f57f0e663de2a311477e7cd70}
\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:8b5859e448e2a9329f716e9bb00d8ef6cbdceb8f57f0e663de2a311477e7cd70}
\end{equation}

% step ID = 0002394495
\simplify{8b5859e448e2a9329f716e9bb00d8ef6cbdceb8f57f0e663de2a311477e7cd70}{1efbdb71ef1f6cf5781511bf7f50343fe9c60d7e2c4e18c7ba6d0114637a1ee0}
\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:1efbdb71ef1f6cf5781511bf7f50343fe9c60d7e2c4e18c7ba6d0114637a1ee0}
\end{equation}

% step ID = 0004059592
\substituteRHSofexprintoexpr{1efbdb71ef1f6cf5781511bf7f50343fe9c60d7e2c4e18c7ba6d0114637a1ee0}{0e3be3e8448134af86b6f18f51f9cd034fdcf37b32d6dced781b71be074ed096}{ec24a292b06993ed9f67c083eca3b954b5dac60b1d7ef7ac86ebc956c895cb43}
\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:ec24a292b06993ed9f67c083eca3b954b5dac60b1d7ef7ac86ebc956c895cb43}
\end{equation}

% step ID = 0004305953
\simplify{ec24a292b06993ed9f67c083eca3b954b5dac60b1d7ef7ac86ebc956c895cb43}{7a1e284aa30551298d3492e8afab7d8eeaca7d01ece3037724b8ff3987c480ab}
\begin{equation}
\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)
\label{eq:7a1e284aa30551298d3492e8afab7d8eeaca7d01ece3037724b8ff3987c480ab}
\end{equation}

% step ID = 0004939459
\multiplybothsidesby{\frac{-\hbar^2}{2m}}{7a1e284aa30551298d3492e8afab7d8eeaca7d01ece3037724b8ff3987c480ab}{f2415ce63eca429eab46b34ec9f8c6898c92e660288661c7958e8c517520e8d0}
\begin{equation}
\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)
\label{eq:f2415ce63eca429eab46b34ec9f8c6898c92e660288661c7958e8c517520e8d0}
\end{equation}

% step ID = 0009394834
\LHSofexprequalsLHSofexpr{f2415ce63eca429eab46b34ec9f8c6898c92e660288661c7958e8c517520e8d0}{67dc31740772af75abbee14e0431bcd127e608f0de63bc1f3b51a468cb7ed9bb}{8b03059cf5c11d71957c64242e9cda7cf1a87505dc956661ddfb03e9caf6c184}
\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:8b03059cf5c11d71957c64242e9cda7cf1a87505dc956661ddfb03e9caf6c184}
\end{equation}

% step ID = 0002455452
\declareinitialexpression{ee90bbda1906044d164dfa43ccae0cb060f0dd45af67fb42313a259c7e6ce386}
\begin{equation}
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
\label{eq:ee90bbda1906044d164dfa43ccae0cb060f0dd45af67fb42313a259c7e6ce386}
\end{equation}

% step ID = 0002954835
\substituteLHSofexprintoexpr{8b03059cf5c11d71957c64242e9cda7cf1a87505dc956661ddfb03e9caf6c184}{ee90bbda1906044d164dfa43ccae0cb060f0dd45af67fb42313a259c7e6ce386}{b0e11fa02f266de4f10e2251b9a6c663c1428df2271ee339c82d45aceec00e65}
\begin{equation}
{\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)
\label{eq:b0e11fa02f266de4f10e2251b9a6c663c1428df2271ee339c82d45aceec00e65}
\end{equation}

% step ID = 0006756574
\declarefinalexpression{b0e11fa02f266de4f10e2251b9a6c663c1428df2271ee339c82d45aceec00e65}
\bibliographystyle{plain}
\bibliography{pdg_derivation_citations.bib}
\end{document}
% EOF
