Consider two different representations of a Fermion Fock space defined by the creation operators

$$\psi^{\dagger} a = c^{\dagger}_{\beta} \langle \beta\vert\alpha \rangle$$

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

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

(III) Prove that if

$$\vert \phi \rangle = c^{\dagger}_{ \beta_{N}} c^{\dagger}_{ \beta_{N-1}} \ldots c^{\dagger}_{ \beta_{1}} \vert \rangle$$

is a state of $N$ fermions that $\phi(\alpha_{1}, \alpha_{2} \ldots \alpha_{N}) = \langle 0\vert\phi_{\alpha_{1}} \phi_{\alpha_{2}} \ldots \phi_{\alpha_N} \vert\phi \rangle$ is a Slater determinant

$$\phi (\alpha_{1}, \alpha_{2} \ldots \alpha_{N}) = det \langle \alpha_{i}\vert \beta_{j} \rangle$$

(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 $\langle \vec{r}\vert \vec{k} \rangle$, for states normalized in a box of volume $V$? Explicitly write out the Slater determinant in the position basis for a three particle state $\phi = \vert\vec{k_{3}}, \vec{k_{2}}, \vec{k_{1}} \rangle$.