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1.  Consider the simple harmonic oscillator


\begin{displaymath}
\hat{H} =  {{p^2} \over {2m}} + {{ m \omega^2 x^2} \over {2}}
\end{displaymath}


Write it  in ``second quantized'' form , by expressing x and p in terms of creation
and annihilation operators.

\begin{displaymath}
\hat{H} = \hbar\omega(a^{\dagger} a + \frac{1}{2}).
\end{displaymath}

\bigskip

(i) Derive the equation of motion for $a$ and show that

\begin{displaymath}
a(t) = a(0)e^{- i \omega t}.
\end{displaymath}

Repeat this for $ a^{\dagger}$.

(ii) Using the relationship

\begin{displaymath}
a^{\dagger} = \frac{1}{\sqrt{2\hbar}} \left[ \frac{\hat{p}}{\sqrt{m \omega}} + i\hat{x} \sqrt{m \omega} \right].
\end{displaymath}

use the results of (i) to show that

%\begin{displaymath}
\begin{eqnarray*}\nonumber
\hat{p}(t) & =& \hat{p}(0) \cos \ \omega t - m\omega \hat{x}(o) sin \ \omega t \\
\hat{x}(t)&  =&  \hat{x}(0)\cos \ \omega t + \frac{\hat{p}(0)}{m \omega}sin \ \omega t.\\ \nonumber
\end{eqnarray*}
%\end{displaymath}

(iii) The above operator expressions appear identical to the classical equations of motion.

Why?

2.  Show that the Heisenberg spin operator S can be written in second quantized form as
\begin{eqnarray*}\nonumber
S_{x}& = &\phantom{+} \frac{1}{2}(c _{\uparrow}^{\dagger} c _{\downarrow} + c _{\downarrow}^{\dagger} c_{\uparrow}), \\
S_{y}& = &-\frac{i}{2}(c_{\uparrow}^{\dagger} c_{\downarrow} - c_{\downarrow}^{ \dagger} c_{\uparrow}), \\ \nonumber
S_{z}& = &\phantom{+} \frac{1}{2}(c_{\uparrow}^{\dagger} c_{\uparrow}   - c_{\downarrow}^{ \dagger} c_{\downarrow}). \\ \nonumber
\end{eqnarray*}

Check the commutation relations!

3.  Consider a system of fermions created by the field $\psi$ $^{\dag}$ $\vec{(r)}$ interacting under the Yukawa potential
\begin{displaymath}
V(r) = \frac{Ae^{-\lambda r}}{4 \pi r}.
\end{displaymath}

(i) Write the Hamiltonian in second quantized form, using the position basis.\\

(ii) Write the Hamiltonian
in  second quantized notation in the momentum basis, where
\begin{displaymath}
c_{\vec{k}} \ ^{\dagger} = \int d^{3}r \psi ^{\dagger}(\vec{r})e^{i \vec{k}. \vec{r}}.
\end{displaymath}


You will find it helpful to derive the Fourier representation
\begin{displaymath}
V(r) = \int \frac{d^{3}q} {(2\pi)^{3}}
e^{i\vec{q}.\vec{r} } \frac{A}{(q^{2} + \lambda^{2})}.
\end{displaymath}




4.  For the spin 1/2 fermions considered in question 3, prove the ``Hubbard Stratonovich'' identity
\begin{displaymath}
e^{-a (n_{\uparrow}-1/2)(n_{\downarrow} - 1/2)} = \frac{1}{2}e^{-a/4} \sum_{\sigma = \pm 1} e^{a\sigma (n_{\uparrow} -n_{\downarrow})}
\end{displaymath}

where $\mbox{cosh} a = e^{\frac{a}{4}}$ (Hint: consider the matrix representation of the operator
on the left hand side in the four dimensional Fock space of the up and down electrons.)
We shall later see how, considering $a \sim \delta t$ to be an infinitesimal time interval,
we can use this relation to treat interacting spins as spins moving in a fluctuating Ising
magnetic field.
\\

6.  Let $a^{\dagger}_{\alpha}$ , $a_{\alpha}$ be Fermion creation and annihilation operators, and assume
$$ H_{o} = \sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha}a_{\alpha}.$$

 (i) Compute $Tr[e^{\beta H_{o}}  a^{\dagger}_{\alpha} a_{\alpha}]$\\

(ii) Compute Tr [$e^{-\beta H_{o}} a_{\alpha}a^{\dagger}_{\alpha}$].\\

(iii) Compute $Tr [e^{-\beta H_{o}} a_{\alpha}a^{\dagger}_{\alpha}]$.  Using the identity $e^{-\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha} = -T \frac{\partial}{\partial \epsilon_{\alpha}} e^{-\beta H_{o}}$ check your answer to (i). \\

(iv) Confirm that the expectation
\begin{displaymath}
\langle n_{a} \rangle = Z^{-1} Tr[e^{-\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha} ] = f(\epsilon_{\alpha})
\end{displaymath}

where $f(x)$ is the Fermi function.\\

(v) Repeat the above procedure assuming that $a^{\dagger}_{a}$ and $a_{a}$ are boson operators.  What are the restrictions on the values of $\epsilon_{a}$ and what happens when one of these energies $\epsilon_{a}$ becomes zero?

(vi) Calculate


%\begin{equation}
$$
\left(
\begin{array}{l}
a^{\dagger}_{\alpha}(\tau) \\
a^{}_{\alpha}(\tau)
\end{array}
\right) =
e^{\tau H_o}
\left(
\begin{array}{l}
a^{\dagger}_{\alpha} \\
a^{}_{\alpha}
\end{array}
\right)
e^{-\tau H_o}
$$
%\end{equation}


4.  Let $\gamma$ be an oriented path from a to b in the complex plane.  If A(z) is an operator valued function of z, then define the line integral
\begin{displaymath}
\int_{\gamma} A(z)dz \equiv \sum_{i=1}^{N} A(z_{i})(z_{i+1}-z_{i})
\end{displaymath}


The time ordered product along the path $\gamma$ orders the operators along $\gamma$.  In the example
\begin{displaymath}
T_{\gamma}(A(z_{3})A(z_{2})A(z_{N})) = A(z_{N})A(z_{3})A(z_{1})
\end{displaymath}


Prove that

\begin{displaymath}
\left(
T_{\gamma} \exp \frac{1}{i} \int_{\gamma} H(\tau)d\tau
\right) ^{\dagger} =
\left(
T_{\gamma}\exp \frac{1}{i} \int_{\tilde\gamma} H(\tau)d\tau \right)
\end{displaymath}


where $\tilde \gamma$  is the path $\gamma$ traversed in the reverse direction (i.e. if $\gamma$ goes from $a$ to b, then $\tilde \gamma$ goes from b to $a$.)


6.  Define an antitime ordering operator
$\tilde{T}$

\begin{equation}
\tilde T [A(t_1)B(t_2)]=
\left\{
\begin{array}{ll}
A(t_1) B(t_2) & (t_1 < t_2 )\\
B(t_2) A(t_1) & (t_1 > t_2 )
\end{array}
\right .
\end{equation}


Define

\begin{displaymath}
\tilde U (t,t_{o}) = \tilde T \exp \frac{1}{i} \int_{t_{o}}^{t} H(\tau)d\tau
\end{displaymath}

What is the differential equation satisfied by $\tilde{U}$(t,0)?\\
10.  Consider two different representations of a Fermion Fock space defined by the creation operators
\begin{displaymath}
\psi^{\dagger} a = c^{\dagger}_{\beta} \langle \beta|\alpha \rangle
\end{displaymath}

(i) Show that $\psi^{\dagger}_{\alpha},c^{\dagger}_{\beta} = \langle \beta|\alpha \rangle $.\\

(ii) Show that $\langle 0|\psi^{\dagger} _{\alpha} c^{\dagger}_{ \beta} = \langle \beta | \alpha \rangle$.\\

(III) Prove that if
\begin{displaymath}
| \phi \rangle = c^{\dagger}_{ \beta_{N}} c^{\dagger}_{ \beta_{N-1}}
 \ldots c^{\dagger}_{ \beta_{1}} |0 \rangle
\end{displaymath}
is a state of $N$ fermions that
$\phi(\alpha_{1}, \alpha_{2} \ldots \alpha_{N}) =  \langle 0|\phi_{\alpha_{1}} \phi_{\alpha_{2}} \ldots \phi_{\alpha_N} |\phi \rangle$ is a Slater determinant

\begin{displaymath}
\phi (\alpha_{1}, \alpha_{2} \ldots \alpha_{N}) = det \langle \alpha_{i}| \beta_{j} \rangle
\end{displaymath}

(iv) If $a \equiv \vec{r}$ and  $\beta \equiv \vec{k}$ so that $\Psi^{\dagger} (\vec{r})$ creates an electron in the position eigenstate and $c^{\dagger}_{ \vec{k}}$ creates an electron in a momentum eigenstate, what is the overlap matrix element
$\langle \vec{r}| \vec{k} \rangle$, for states normalized in a box of volume $V$? Explicitly write out the Slater determinant in the position basis for a three particle state $\phi = |\vec{k_{3}}, \vec{k_{2}}, \vec{k_{1}} \rangle$.
\\

11.  (i) If $M = M^{\dagger}$ is an Hermitian $N$ dimensional matrix, $c^{\dagger}_{i}$ and $c_{i}$ $(i = 1 \ldots N)$  are creation and annihilation operators for Fermions (i.e.$ {c_{i},c^{\dagger}_{j}} = \delta_{ij}$, show that

\begin{displaymath}
c_{i} ^{\dagger} (t) = \left[ e^{-\hat M t}c^{\dagger}e^{\hat Mt} \right] _{j} = c^{\dagger}_{j} \left[e^{-M} \right]_{ji}\;\;\;\;
(M = c^{\dagger}_{j} {M}_{jk}c_{k})
\end{displaymath}
(Hint consider the differential equation satisfied by $c^{\dagger}(t)$).


7.  let U$_{I}$(t,t$_{0}$) be the evolution operator in the interaction picture and
U$_{S}$(t,t$_{0}$) the corresponding operator in the Schr\"{d}inger picture.  Prove that

\begin{displaymath*}
U_{I}(t,0) = e^{iH_{o}t}U_{S}(t,0)
\end{displaymath*}

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