Homework Set 3

Problem 2: Pair correlation function of noninteracting spinless bosons

The pair correlation function gives the relative probability of finding a particle at position ${\mathbf{𝐫}}^{\prime }$ if we know that there is one at position $\mathbf{𝐫}$. It is defined by:

$G(\mathbf{𝐫}-{\mathbf{𝐫}}^{\prime })=⟨\mathrm{\Phi }(\mathbf{𝐫})|{\hat{\mathrm{\Psi }}}^{†}({\mathbf{𝐫}}^{\prime })\hat{\mathrm{\Psi }}({\mathbf{𝐫}}^{\prime })|\mathrm{\Phi }(\mathbf{𝐫})⟩=⟨{\mathrm{\Phi }}_0|{\hat{\mathrm{\Psi }}}^{†}(\mathbf{𝐫}){\hat{\mathrm{\Psi }}}^{†}({\mathbf{𝐫}}^{\prime })\hat{\mathrm{\Psi }}({\mathbf{𝐫}}^{\prime })\hat{\mathrm{\Psi }}(\mathbf{𝐫})|{\mathrm{\Phi }}_0⟩$

Here $|\mathrm{\Phi }(\mathbf{𝐫})⟩=\hat{\mathrm{\Psi }}(\mathbf{𝐫})|{\mathrm{\Phi }}_0⟩$ is a state with $N-1$ particles, obtained after removal of one particle at position $\mathbf{𝐫}$, so that pair correlation function is computed as the expectation value of the density operator in this new quantum state. Compute the pair correlation function for a system of translationally invariant noninteracting spinless bosons in many-body state

$|{\mathrm{\Phi }}_0⟩=|n_{{\mathbf{𝐤}}_0},n_{{\mathbf{𝐤}}_1},\dots ⟩$

by transforming field operators to creation and annihilation operators in momentum representation while assuming that bosons are enclosed in a box of volume $V$ with periodic boundary conditions. Your final result should be a function of $\mathbf{𝐫},{\mathbf{𝐫}}^{\prime }$ and sums over $n_{\mathbf{𝐤}}$, where you can also use that

$⟨{\mathrm{\Phi }}_0|{\hat{\mathrm{\Psi }}}^{†}(\mathbf{𝐫})\hat{\mathrm{\Psi }}(\mathbf{𝐫})|{\mathrm{\Phi }}_0⟩=\frac{1}{V}\sum _{\mathbf{𝐤}}{n}_{\mathbf{𝐤}}=N$.

==Problem 3: