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\begin{document}

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\section*{semiclassical state}

\begin{quote}%
under construction


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

\hypertarget{geometric_quantum_theory}{}\paragraph*{{Geometric quantum theory}}\label{geometric_quantum_theory}

[[!include geometric quantization - contents]]

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

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{TraditionalInSchr&#246;dingerPicture}{Semiclassical state of the non-relativistic particle in a potential}\dotfill \pageref*{TraditionalInSchr&#246;dingerPicture} \linebreak
\noindent\hyperlink{as_a_wave_function}{As a wave function}\dotfill \pageref*{as_a_wave_function} \linebreak
\noindent\hyperlink{as_a_lagrangian_submanifold_of_phase_space_equipped_with_a_halfdensity}{As a Lagrangian submanifold of phase space equipped with a half-density}\dotfill \pageref*{as_a_lagrangian_submanifold_of_phase_space_equipped_with_a_halfdensity} \linebreak
\noindent\hyperlink{InSymplecticGeometry}{In symplectic geometry / geometric quantum theory}\dotfill \pageref*{InSymplecticGeometry} \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}

In [[physics]], a \emph{semiclassical state} is the approximation to a [[quantum state]] in [[semiclassical approximation]].

In the original sense of the [[WKB approximation]], in the [[Schrödinger picture]] a semiclassical state is a [[wave function]] which solves the [[Schrödinger equation]] to first order in [[Planck's constant]] $\hbar$.

In the broader formalization of quantum physics in [[symplectic geometry]]/[[geometric quantization]] one finds that such WKB semiclassical states are formalized as being [[Lagrangian submanifolds]] of the given [[phase space]] [[symplectic manifold]] equipped with with a [[half-density]].

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

We first give the traditional definition of semiclassical states according to the [[WKB method]] for a non-relativistic [[particle]] propagating on the [[Euclidean space]] $\mathbb{R}^n$ with its standard [[kinetic action]] and some arbitrary [[force]] potential

\begin{itemize}%
\item \hyperlink{TraditionalInSchr&#246;dingerPicture}{Of a non-relativistic particle in a potential}

\end{itemize}
Then we discuss the formalization of this in the broader context of [[symplectic geometry]]/[[geometric quantization]] in

\begin{itemize}%
\item \hyperlink{InSymplecticGeometry}{In symplectic geometry}

\end{itemize}
\hypertarget{TraditionalInSchr&#246;dingerPicture}{}\subsubsection*{{Semiclassical state of the non-relativistic particle in a potential}}\label{TraditionalInSchr&#246;dingerPicture}

Consider the [[physical system]] given by a non-relativistic [[particle]] of [[mass]] $m$ propagating on the [[Cartesian space]] $\mathbb{R}^n$ with standard [[kinetic action]] and sunbject to a [[force]] induced by a given potential [[smooth function]] $V \colon \mathbb{R}^n \to \mathbb{R}$.

\hypertarget{as_a_wave_function}{}\paragraph*{{As a wave function}}\label{as_a_wave_function}

The [[Hamilton operator]] for this system is the standard

\begin{displaymath}
\hat H \coloneqq 
  -
  \frac{\hbar^2}{2 m} \Delta  + V
  \,,
\end{displaymath}
where

\begin{displaymath}
\Delta = \mathbf{d}^\dagger \mathbf{d}  =   \sum_{i = 1}^n \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^i}
\end{displaymath}
is the [[Laplace operator]] on $\mathbb{R}^n$ regarded as a [[Riemannian manifold]] with its canonical flat [[metric]] ($\mathbf{d}$ is the [[de Rham differential]]).

Then for

\begin{displaymath}
\psi \colon \mathbb{R}^n \times \mathbb{R} \to \mathbb{C}
\end{displaymath}
a smooth 1-parameter collection of [[smooth functions]] (of [[wave functions]]), the [[Schrödinger equation]] is

\begin{displaymath}
i \hbar \frac{d}{ dt} \psi 
  = 
  \hat H \psi
  \,,
\end{displaymath}
where $\frac{d}{d t}$ is the [[differentiation]] with respect to the additional parameter ([[time]]).

We say that $\psi$ is a \emph{stationary solution} to the Schr\"o{}dinger equation if it is a solution of the form

\begin{displaymath}
\psi(x, t) = \phi(x)\exp(- i \omega t)
\end{displaymath}
for some $\omega \in \mathbb{R}$. For the following it is useful to decompose the remaining [[complex number|complex]]-valued [[smooth function]]

\begin{displaymath}
\phi \colon \mathbb{R}^n \to \mathbb{C}
\end{displaymath}
into its modulus and phase by writing it as

\begin{displaymath}
\phi(x)  = \exp(i S(x)/\hbar) a(x)
\end{displaymath}
for two smooth functions $S \colon \mathbb{R}^n \to \mathbb{R}$ and $a \colon \mathbb{R}^n \to \mathbb{R}_{\geq 0}$.

In fact it is often useful (such as in the symplecto-geometric interpretation that we turn to \hyperlink{InSymplecticGeometry}{below}) to restrict attention to non-vanishing solutions (or else to solutions restricted to their [[support]]) in which case we can regard $\phi$ as a function of the form

\begin{displaymath}
\phi \colon \mathbb{R}^n \to \mathbb{C}^\times \simeq U(1) \times \mathbb{R}_{\gt 0}
\end{displaymath}
and then this decomposition is unique up to a global global offset of $S$ by $2\pi i \cdot n$ for $n \in \mathbb{Z}$.

In terms of this decomposition the [[Schrödinger equation]] becomes

\begin{displaymath}
\begin{aligned}
    0
    &= 
    \left(i \hbar \frac{d}{dt} - \hat H\right) \psi
    \\
    & =
    \left(
      \left(
        \frac{{\vert \mathbf{d} S \vert}^2 }{2 m }
        + 
        (V - \hbar \omega)
      \right)
      -
      \frac{i \hbar}{ 2m a}
      \mathbf{d}^\dagger \left(a^2 \mathbf{d} S\right)
    \right)
    \exp( i S / \hbar ) a
    +
    \mathcal{O}(\hbar^2 )
  \end{aligned}
  \,,
\end{displaymath}
where $\mathbf{d} S$ is the [[gradient]] [[covector field]] of $S$, where $\mathbf{d}^\dagger ( a^2 \mathbf{d}S)$ is the [[divergence]] of $a ^2 \mathbf{d}S$, and where $\mathcal{O}(\hbar^2)$ denotes all further terms that are non-linear in $\hbar$.

This means that $\psi(-,t) = \exp(i S / \hbar) a \exp(- i \omega )$ is a \textbf{semiclassical stationary state} with [[energy]]

\begin{displaymath}
E \coloneqq \hbar \omega
\end{displaymath}
if the phase $S$ and the modulus $a$ satisfy the following two conditions:

\begin{enumerate}%
\item The phase function $S$ satisfies the [[Hamilton-Jacobi equation]] or [[eikonal]] equation

\begin{displaymath}
H(x, \nabla S(x)) = 
  \frac{\vert \mathbf{d} S\vert^2}{2 m } + V = 
  E
  \,,
\end{displaymath}

\item The modulus $a$ is such that $a^2 \mathbf{d} S$ satisfies the homogeneous [[transport equation]] in that it is a [[divergence]]-free [[vector field]].



\end{enumerate}
\hypertarget{as_a_lagrangian_submanifold_of_phase_space_equipped_with_a_halfdensity}{}\paragraph*{{As a Lagrangian submanifold of phase space equipped with a half-density}}\label{as_a_lagrangian_submanifold_of_phase_space_equipped_with_a_halfdensity}

The above characterization of semiclassical [[wave functions]] of the non-relativistic particle in a potential has a natural equivalent reformulation in terms of [[symplectic geometry]]/[[geometric quantization]].

The [[phase space]] is

\begin{displaymath}
T^* \mathbb{R}^n \simeq \mathbb{R}^{2 n}
  \,.
\end{displaymath}
Into this space is canonically embedded as the 0-section:

\begin{displaymath}
0
  = 
 (x \mapsto (x, p = 0))
  \;
  \colon 
  \;
  \mathbb{R}^n 
   \hookrightarrow
  T^* \mathbb{R}^n
\end{displaymath}
which is a [[Lagrangian submanifold]].

Now every phase function $S \colon \mathbb{R}^n \to \mathbb{R}$ as above induces a deformation of this by regarding the [[de Rham differential]] $\mathbf{d}S$ as a [[section]] of the [[cotangent bundle]]

\begin{displaymath}
\mathbf{d}S \colon \mathbb{R}^n \hookrightarrow T^* X
\end{displaymath}
(This is what related [[phase and phase space in physics]].)

This is again a [[Lagrangian submanifold]]. We write

\begin{displaymath}
\pi \colon im(\mathbf{d}S) \to \mathbb{R}^n
\end{displaymath}
for the restriction of the [[cotangent bundle]] projection to this Lagrangian submanifold.

The fact that $S$ satisfies the [[Hamilton-Jacobi equation]] means equivalently that this Lagrangian submanifold is the level-set of the [[Hamiltonian]] $H \colon \mathbb{R}^n \to \mathbb{R}$ at energy $E = \hbar \omega$

\begin{displaymath}
im(\mathbf{d}S) = H^{-1}(E)
  \,.
\end{displaymath}
For the interpretation of the modulus function $a$ in this reformulation, first notice that for $vol$ the canonical [[volume form]] on $\mathbb{R}^n$, the homogeneous transport equation

\begin{displaymath}
div(  a^2 \mathbf{d}S) = 0
\end{displaymath}
is equivalent to

\begin{displaymath}
\mathcal{L}_{\nabla S} ( a^2 vol ) = 0
\end{displaymath}
where on the left we have the [[Lie derivative]] along the [[gradient]] of $S$. Next observe that

\begin{displaymath}
\nabla S = \pi_* (v_H)|_{im(\mathbf{d}S)}
\end{displaymath}
where $v_{H}$ is the [[Hamiltonian vector field]] corresponding to $H$.

This means that the transport equation is equivalently

\begin{displaymath}
\mathcal{L}_{(v_H)} \pi^* (a^2 vol) = 0
  \,.
\end{displaymath}
Hence this says that $\pi^* a^2 vol$ is a [[volume form]] on $im(\mathbf{d}S)$ which is invariant with respect to the Hamiltonian flow of time evolution.

Finally, if instead of a [[volume form]] we choose a [[half-density]] $\sqrt{vol}$, then $a \sqrt{vol}$ is another half-density and the condition is that this be invariant under the Hamiltonian flow.

In summary then, the semiclassical wave fuction is equivalently

\begin{enumerate}%
\item a [[Lagrangian submanifold]]

\begin{itemize}%
\item which is a level-set of the [[Hamiltonian]] at the given [[energy]]

\end{itemize}

\item such that $\mathbf{a} \coloneqq a \sqrt{vol}$ is a [[half-density]] on the Lagranian submaifold

\begin{itemize}%
\item which in addition is invariant under the Hamiltonian flow.

\end{itemize}


\end{enumerate}
This formulation now suggests a more general definition of semiclassical states in [[symplectic geometry]]/[[geometric quantization]].

\hypertarget{InSymplecticGeometry}{}\subsubsection*{{In symplectic geometry / geometric quantum theory}}\label{InSymplecticGeometry}

(\ldots{})

abstracting the above we have that

(\ldots{})

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

[[!include classical-to-quantum notions - table]]

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

An introduction to the formulation of semiclassical states in [[symplectic geometry]] is in the first section of

\begin{itemize}%
\item Sean Bates, [[Alan Weinstein]], \emph{Lectures on the geometry of quantization} (\href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf})

\end{itemize}
[[!redirects semiclassical states]]



\end{document}