\input phyzzx

\def\a{\hat a^{\vphantom\dagger}}
\def\ad{\hat a^\dagger}
\def\b{\hat b^{\vphantom\dagger}}
\def\bd{\hat b^\dagger}
\def\bx{{\bf x}}
\def\bp{{\bf p}}
\def\bmp{{\bf-p}}
\def\eqdef{\buildrel\rm def\over=}

\hsize=7in	\hoffset=-0.35in
\vsize=9in	\voffset=-0.25in
\quad\noindent PHY--396~K.\qquad
Problem set \#3.\qquad
Due September 23, 2003.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
The first exercise is about first-quantized {\it v.} second-quantized
forms of one-body and two-body operators acting on identical bosons.
In class, we wrote the wave function of an $N$--particle state
$\ket{\alpha_1,\ldots,\alpha_N}=\ket{\{n_\beta\}}$ as
$$
\phi_{\alpha_1,\ldots,\alpha_N}(\bx_1,\ldots,\bx_N)\
=\,\sqrt{\prod_\beta n_\beta!\over N!}
\sum_{\textstyle{{\rm distinct\ permutations}\atop
(\tilde\alpha_1,\ldots,\tilde\alpha_N)\ {\rm of}\ (\alpha_1,\ldots,\alpha_N)}}
\phi_{\tilde\alpha_1}(\bx_1)\cdots\phi_{\tilde\alpha_{N}}(\bx_N),
\eqn\WaveFunction
$$
and we defined the annihilation operators $\a_\alpha$ according to
$$
\a_\alpha\ket{\{n_\beta\}}\
=\ \sqrt{n_\alpha}\ket{\{n'_\beta=n_\beta-\delta_{\alpha\beta}\}} .
\eqno\eq
$$
\spointbegin
Consider an $N$--particle state $\ket{N,\Psi}$ with a completely generic
totally-symmetric wave function $\Psi(\bx_1,\ldots,\bx_N)$.
Show that the $(N-1)$--particle state $\ket{(N-1),\Psi'}=\a_\gamma\ket{N,\Psi}$
has wave function
$$
\Psi'(\bx_1,\ldots,\bx_{N-1})\ =\ \sqrt{N}\int\!\!d^3\bx_N\,
\phi^*_\gamma(\bx_N)\,\Psi(\bx_1,\ldots,\bx_{N-1},\bx_N) .
\eqn\Annihilation
$$
Hint: First verify this formula for $\Psi$ of the form~\WaveFunction, and then
generalize to arbitrary (but totally-symmetric) $\Psi$ by linearity.

\pointcon
Now consider a one-body operator $\hat R_1$.
In the first-quantized formalism $\hat R_{\rm tot}$ acts on $N$--particle
states according to
$$
\hat R_{\rm tot}^{(1)}\ =\,\sum_{i=1}^N\hat R_1(i{\rm th\over{}}\ {\rm particle})
\eqno\eq
$$
while in the second-quantized formalism it becomes
$$
\hat R_{\rm tot}^{(2)}\
=\,\sum_{\alpha,\beta}\bra\alpha\hat R_1\ket\beta\,\ad_\alpha\a_\beta\,.
\eqn\OneBody
$$
\spoint %b
Use eq.~\Annihilation\ to verify that for any two $N$--particle states
$\bra{N,\Psi_1}$ and $\ket{N,\Psi_2}$
$$
\bra{N,\Psi_1}\hat R_{\rm tot}^{(1)}\ket{N,\Psi_2}\
=\ \bra{N,\Psi_1}\hat R_{\rm tot}^{(2)}\ket{N,\Psi_2} .
\eqno\eq
$$
Hint: Use $\hat R_1=
\sum_{\alpha,\beta}\ket\alpha\bra\alpha\hat R_1\ket\beta\bra\beta$.

\pointcon
Next, consider a two-body operator $\hat S_2$ which acts in the
first-quantized formalism according to
$$
\hat S^{(1)}_{\rm tot}\ =\ \half \sum_{i\neq j}
\hat S_2(i{\rm th\over{}}\ {\rm and}\ j{\rm th\over{}}\ {\rm particles})
\eqno\eq
$$
and in the second-quantized formalism according to
$$
\hat S^{(2)}_{\rm tot}\ =\,\sum_{\alpha,\beta,\gamma,\delta}
\bra\alpha\otimes\bra\beta \hat S_2 \ket\gamma\otimes\ket\delta
\ad_\alpha\ad_\beta\a_\gamma\a_\delta\,.
\eqn\TwoBody
$$
\spoint %c
Again, show that for any two $N$--particle states
$\bra{N,\Psi_1}$ and $\ket{N,\Psi_2}$
$$
\bra{N,\Psi_1}\hat S_{\rm tot}^{(1)}\ket{N,\Psi_2}\
=\ \bra{N,\Psi_1}\hat S_{\rm tot}^{(2)}\ket{N,\Psi_2} .
\eqno\eq
$$

\spoint %d
Finally, let $\hat A_1$ be a one-body operator,
let $\hat B_2$ and $\hat C_2$ be  two-body operators,
and let $\hat A$, $\hat B$ and $\hat C$ be the corresponding second-quantized
operators defined similar to eqs.~\OneBody\ and~\TwoBody.
\subpar
Show that if $\hat C_2=\left[\left(\hat A_1(1{\rm st\over{}})+
	\hat A_1(2{\rm nd\over{}})\right),\hat B_2\right]$
then $\hat C=[\hat A,\hat B]$
\subpar
Hint: First, calculate the commutator $[\ad_\alpha\ad_\beta\a_\gamma\a_\delta,
\ad_\mu\a_\nu]$.

\bigskip
\point
The second problem is about coherent states of harmonic oscillators
and free quantum fields.
Let us start with a harmonic oscillator $\hat H=\hbar\omega\ad\a$.
\spointbegin
For any complex number $\xi$ we define a {\it coherent state}
$\ket\xi\eqdef\exp\bigl(\xi\ad-\xi^*\a\bigr)\ket0$.
Show that
$$
\ket\xi\ =\ e^{-|\xi|^2/2}\,e^{\xi\ad}\ket0 \quad
{\rm and}\quad \a\ket\xi\ =\ \xi\ket\xi .
\eqn\Coherent
$$

\spoint %b
Calculate the uncertainties $\Delta q$ and $\Delta p$ for a coherent state
$\ket\xi$ and verify their minimality: $\Delta q\Delta p=\half\hbar$.
Also, verify $\delta n=\sqrt{\bar n}$ where $\bar n\eqdef\vev{\hat n}=|\xi|^2$.
\subpar
Hint: use $\a\ket\xi=\xi\ket\xi$ and $\bra\xi\ad=\xi^*\bra\xi$.

\spoint %c
Show that for $\xi(t)=\xi_0 e^{-e\omega t}$ the coherent state
$\ket{\xi(t)}$ satisfies the time-dependent Schr\"odinger equation
$i\hbar{d\over dt}\ket{\xi(t)}=\hat H\ket{\xi(t)}$.

\spoint %d
The coherent states are not quite orthogonal to each other.
Calculate their overlap $\vev{\eta|\xi}$.

\pointcon
Now consider coherent states of
 multi-oscillator systems and hence quantum fields.
In particular, let us focus on the creation and annihilation fields
$\hat\Psi^\dagger(\bx)$ and $\hat\Psi(\bx)$
for non-relativistic spinless bosons.
\spoint %e
Generalize (a) and construct coherent states $\ket\Phi$ which satisfy
$$
\hat\Psi(\bx)\ket\Phi\ =\ \Phi(\bx)\ket\Phi
\eqn\CoherentQFT
$$
for any given classical complex field $\Phi(\bx)$.

\spoint %f
Show that for any such coherent state, $\Delta N=\sqrt{\bar N}$ where
$$
\bar N\ \eqdef\ \bra\Phi\hat N\ket\Phi\ = \int\!\!d\bx\,|\Phi(\bx)|^2.
\eqno\eq
$$

\spoint %g
Let
$$
\hat H\ = \int\!\!d\bx\left(
{\hbar^2\over 2M}\nabla\hat\Psi^\dagger\cdot\nabla\hat\Psi\
+\ V(\bx)\hat\Psi^\dagger\hat\Psi\right) 
$$
and show that for any classical field configuration $\Phi(\bx,t)$
that satisfies the classical field equation
$$
i\hbar\partder{}{t}\Phi(\bx,t)\
=\ \left(-{\hbar^2\over 2M}\nabla^2+V(\bx)\right)\Phi(\bx,t) ,
$$
the time-dependent coherent state $\ket{\Phi}$ satisfies
the true Schr\"odinger equation
$$
i\hbar\partder{}{t}\ket{\Phi}\ =\ \hat H \ket{\Phi} .
\eqno\eq
$$

\spoint %h
Finally, show that the quantum overlap $|\vev{\Phi_1|\Phi_2}|^2$
between two different coherent states is exponentially small
for any {\sl macroscopic} difference $\delta\Phi(\bx)=\Phi_1(\bx)-\Phi_2(\bx)$
between the two field configurations.



\bye
