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\begin{document}

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\section*{Schrödinger equation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{abstract}{Abstract}\dotfill \pageref*{abstract} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{decomposition_into_phase_and_amplitude}{Decomposition into phase and amplitude}\dotfill \pageref*{decomposition_into_phase_and_amplitude} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The Schr\"o{}dinger equation (named after [[Erwin Schrödinger]]) is the [[evolution equation]] of [[quantum mechanics]] in the [[Schrödinger picture]]. Its simplest version results from replacing the classical expressions in the nonrelativistic, mechanical equation for the energy of a point particle, by [[operators]] on a [[Hilbert space]]:

We start with a point particle with mass $m$, impulse $p$ moving in the space $\mathbb{R}^3$ with a given potential function $V$, the energy of it is the sum of kinetic and potential energy:

\begin{displaymath}
E = \frac{p^2}{2 m} + V
\end{displaymath}
Quantizing this equation means replacing the coordinate $x \in \mathbb{R}^3$ with the [[Hilbert space]] $L^2(\mathbb{R}^3)$ and

\begin{displaymath}
E \to i \hbar  \frac{\partial}{\partial t}
\end{displaymath}
\begin{displaymath}
p \to -i  \hbar \nabla
\end{displaymath}
with the [[Planck constant]] $h$ and

\begin{displaymath}
\hbar = \frac{h}{2 \pi}
\end{displaymath}
the reduced Planck constant.

This results in the Schr\"o{}dinger equation for a single particle in a potential:

\begin{displaymath}
i \hbar  \frac{\partial}{\partial t} \psi(t, x) = - \frac{\hbar^2}{2 m} \nabla^2 \psi(t, x) + V(t, x) \psi(t, x)
\end{displaymath}
The last term is the multiplication of the functions $V$ and $\psi$.

The right hand side is called the [[Hamilton operator]] $H$, the Schr\"o{}dinger equation is therefore mostly stated in this form:

\begin{displaymath}
i \hbar \psi_t = H \psi
\end{displaymath}
\hypertarget{abstract}{}\subsection*{{Abstract}}\label{abstract}

\ldots{}

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\ldots{}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{decomposition_into_phase_and_amplitude}{}\subsubsection*{{Decomposition into phase and amplitude}}\label{decomposition_into_phase_and_amplitude}

Consider for simplicity, the [[mechanical system]] of a [[particle]] of [[mass]] $m$ propagating on the [[real line]] $\mathbb{R}$ and subject to a [[potential]] $V \in C^\infty(\mathbb{R})$, so that the Schr\"o{}dinger equation is the [[differential equation]] on [[complex number|complex]]-valued [[functions]] $\Psi \colon \mathbb{R}\times \mathbb{R} \to \mathbb{C}$ given by

\begin{displaymath}
i \hbar
  \frac{\partial}{\partial t}
  \Psi
  =
  \frac{\hbar^2}{2m}
  \frac{\partial^2}{\partial^2 x} \Psi
  + 
  V \Psi
  \,,
\end{displaymath}
where $\hbar$ denotes [[Planck's constant]].

By the nature of [[complex numbers]] and by the discussion at \emph{[[phase and phase space in physics]]}, it is natural to parameterize $\Psi$ -- away from its [[zero locus]] -- by a [[complex phase]] function

\begin{displaymath}
S \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}
\end{displaymath}
and an [[absolute value]] function $\sqrt{\rho}$

\begin{displaymath}
\sqrt{\rho} \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}
\end{displaymath}
which is positive, $\sqrt{\rho} \gt 0$, as

\begin{displaymath}
\Psi 
    \coloneqq 
  \exp\left(\frac{i}{\hbar} S / \hbar\right) \sqrt{\rho}
  \,.
\end{displaymath}
Entering this Ansatz into the above Schr\"o{}dinger equation, that [[complex number|complex]] equation becomes equivalent to the following two [[real number|real]] equations:

\begin{displaymath}
\frac{\partial}{\partial t} S
  = 
  - 
  \frac{1}{2m} \left(\frac{\partial}{\partial x}S\right)^2
  - 
  V
  + 
  \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}
\end{displaymath}
and

\begin{displaymath}
\frac{\partial}{\partial t} \rho
  = 
  -
  \frac{\partial}{\partial x}
  \left(
    \frac{1}{m} \left(\frac{\partial}{\partial x}S\right) \rho
  \right)
  \,.
\end{displaymath}
Now in this form one may notice a similarity of the form of these two equations with other equations from [[classical mechanics]] and [[statistical mechanics]]:

\begin{enumerate}%
\item The first equation is similar to the [[Hamilton-Jacobi equation]] that expresses the classical [[action functional]] $S$ and the [[canonical momentum]]

\begin{displaymath}
p \coloneqq \frac{\partial}{\partial x} S
\end{displaymath}
except that in addition to the ordinary [[potential energy]] $V$ there is an additional term

\begin{displaymath}
Q 
   \coloneq
 \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}
\end{displaymath}
which is unlike what may appar in an ordinary [[Hamilton-Jacobi equation]]. The perspective of Bohmian mechanics is to regard this as a correction of [[quantum physics]] to classical [[Hamilton-Jacobi theory]], it is then called the \emph{quantum potential}. Notice that unlike ordinary potentials, this ``quantum potential'' is a function of the density that is subject to the potential. (Notice that this works only away from the [[zero locus]] of $\rho$.)


\item The second equation has the form of the [[continuity equation]] of the [[flow]] expressed by $\frac{1}{m}p$.



\end{enumerate}
(In the context of \emph{[[Bohmian mechanics]]} one regard this equivalent rewriting of the [[Schrödinger equation]] as providing a [[hidden variable theory]] formulation of [[quantum mechanics]].)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Tomonaga-Schwinger equation]]


\item [[Schrödinger picture]]


\item [[heat equation]]


\item [[wave equation]], [[Klein-Gordon equation]]


\item [[Dirac equation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Any introductory textbook about [[quantum mechanics]] will explain the Schr\"o{}dinger equation (from the viewpoint of physicists mostly).

\begin{itemize}%
\item Wikipedia: \href{http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation}{Schr\"o{}dinger equation}

\end{itemize}
[[!redirects Schrödinger's equation]]

[[!redirects Schroedinger equation]] [[!redirects Schroedinger's equation]]



\end{document}