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\noindent
\begin{center}
\'{E}cole polytechnique de Bruxelles \hfill  PHYSH401/2019-2020 \\[1cm]
{\Large Quantum Mechanics II \\[0.3cm]
{\large Exercise 2: Wigner Representation}} \\

\vspace{5 mm}
\textrm{2 October 2019}
\date{2 October 2019}
\end{center}

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Wigner representation of quantum states is equivalent to the one by density operators.
\begin{enumerate}
\item The Wigner function for a system in the state $\hat\rho$ is defined in the \emph{phase space} $(x,p)$ as
\[
W(x,p) = \frac{1}{2\pi \hbar}\int_{-\infty}^\infty e^{ipy/\hbar}\langle x-y/2|\hat\rho|x+y/2\rangle\, dy.
\] 
%%
where kets $|x\rangle$ are the eigenstates of the position operator:
\[
\hat x |x\rangle =  x |x\rangle.
\]

\begin{itemize}
\item[a)] Using the identity
\[\frac{1}{2\pi}\int_{-\infty}^\infty  e^{ip(x-a)}  \, dp = \delta(x-a) 
\] 
show that the integral of the Wigner function over $p$ is the probability density for $x$ and \emph{vice versa}.

\item[b)]  Verify that the expectation value of the kinetic energy operator $\hat T = \hat p ^2/2m$ is given by
\[
\langle \hat T \rangle =  \int_{-\infty}^\infty\int_{-\infty}^\infty T(p)  W(x,p) \, dx \, dp,
\]
where $T (p)$ is a function of $p$.

%Verify that the expectation values of  the position and momentum operators are given by
%\begin{eqnarray*}
%\langle \hat x \rangle & = &  \int_{-\infty}^\infty\int_{-\infty}^\infty x\,  W(x,p)  \, dx\, dp ,\\
%\langle \hat p \rangle & = &  \int_{-\infty}^\infty\int_{-\infty}^\infty p\,  W(x,p)  \, dx\, dp.
%\end{eqnarray*}

\item[c)] Verify that the expectation value of the  potential energy  operator $\hat U = U(\hat x)$ is given by
\[
\langle \hat U \rangle  =   \int_{-\infty}^\infty\int_{-\infty}^\infty U(x)  W(x,p)  \, dx \, dp .
\]
\end{itemize}
%Utiliser la propri\'et\'e pr\'ec\'edente pour prouver que $S_{\pm} |\psi \rangle$ est un \'etat propre de $P_{21}$ avec valeur propre $\pm 1$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  Quantum superposition and mixture of two Gaussian states. Use here  $\hbar=1$.

Let two (non-normalized) Gaussian states be given by the wave functions: 
\begin{eqnarray*}
 \psi_1(x) & = & \exp \left[-(x-5)^2\right], \\ 
 \psi_2(x) & = & \exp \left[-(x+5)^2\right].
\end{eqnarray*}
\begin{itemize}
\item[a)] Find (up to a constant) the Wigner function of equiprobable statistical mixture of the two states.

\item[b)] Find (up to a constant) the Wigner function of the superposition of the two states given by equal amplitudes.

\item[c)] Compare the two Wigner functions. In which case does the Wigner function show non-classical features of the state? 
%Why is the Wigner function called \emph{quasi probability} distribution?
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item \emph{Coherent state} is an eigenstate of the annihilation operator. Use here  $\hbar=1$.

\[
\hat a |\alpha\rangle =\alpha |\alpha\rangle , \qquad \mathrm{where} \quad  \hat a = (\hat x + i \hat p)/\sqrt{2}.
\]

The (complex) eigenvalue $\alpha$ is related to the phase space variables $x$ et $p$ as
\begin{eqnarray*}
  x & = & \sqrt{2}\,\mathrm{Re}{(\alpha)},\\ 
  p & = & \sqrt{2}\,\mathrm{Im}{(\alpha)}.
\end{eqnarray*}


\begin{itemize}
\item[a)] Find the average number of particles in the coherent state.
\item[b)] Find the representation of the coherent state in the eigenbasis of the number operator.
What gives this representation for the coherent state with $\alpha=0$?
\item[c*)] Find up to a normalization constant the wave function of the coherent state in $x$-representation
$\varphi_\alpha(x)=\langle x|\alpha\rangle$ using the representation of 
the annihilation operator in terms of the position and momentum as given above.
Remember that in the position  representation we have
\begin{eqnarray*}
\hat x & = & \int_{-\infty}^\infty x |x\rangle\langle x|\, dx \\ [2ex]
\hat p & = & -i \int_{-\infty}^\infty \frac{d}{dx} |x\rangle\langle x|\, dx .
\end{eqnarray*}
\item[d*)] Find  the Wigner function of the coherent state using its wave function in the form
\[
\varphi_\alpha(x)= \frac{1}{\sqrt[4]{\pi}}e^{ -\frac{1}{2}(x-x_0)^2+ip_0x }=\frac{1}{\sqrt[4]{\pi}}e^{-\frac{1}{2}p_0^2}e^{ip_0x_0}e^{ -\frac{1}{2}(x-x_0-ip_0)^2}
 .
\]
What is the shape of this Wigner function in the phase space? 
\end{itemize}


\end{enumerate}


\newpage 

\section*{Reminder}

Integrals of Gaussian functions

\begin{eqnarray*}
\int_{-\infty}^{+\infty}e^{-a(x+b)^2}dx & =  & \sqrt{\frac{\pi}{a}}\, ,\\[2ex]
\int_{-\infty}^{+\infty}e^{-ax^2+bx+c}dx & = & \sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}+c}\, .
\end{eqnarray*}


\noindent Example

\makebox[1\textwidth]{
	\parbox[b]{0.7\textwidth}{ Wigner's quasiprobability distributions for Fock states :
		\begin{enumerate}
			\item [a)] $|0\rangle$ 
			\item[]
			\item[]
			\item [b)] $|1\rangle$
			\item[]
			\item[]
			\item [c)] $|5\rangle$.
			\item[]
		\end{enumerate}
			}
	\hspace{0.5cm}
	\includegraphics[width=0.25\textwidth]{Wigner_functions.jpg}
}

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