Proving the Commutation Relation for Quantized Boson in a One-Dimensional Box

[Hill]

Homework Statement

Show for a boson particle in a box of volume $V$ that $$\frac 1 V \sum_{\mathbf{pq}} e^{i(\mathbf{px}-\mathbf{qy})}[\hat a_{\mathbf p},\hat a^\dagger_{\mathbf q}]=\delta^{(3)}(\mathbf x - \mathbf y)$$

Relevant Equations

$[\hat a_{\mathbf p},\hat a^\dagger_{\mathbf q}]=\delta_{\mathbf{pq}}$

To simplify, I consider a one-dimensional box of the size L. I need to show in this case that
$$\frac 1 L \sum_{pq} e^{i(px-qy)}[\hat a_p,\hat a^\dagger_q]=\delta(x -y)$$
With the commutation relation above, it becomes
$$\frac 1 L \sum_p e^{ip(x-y)}=\delta(x -y)$$
p is quantized: $p_m=\frac {2\pi m} L$

So I need to show that
$$\frac 1 L \sum_m e^{i \frac {2\pi m} L (x-y)}=\delta(x -y)$$
If $x \neq y$ the sum is $0$, but I don't know how to proceed otherwise.
A hint?

 

[Hill]

Got it.

$\langle x|p \rangle= \frac 1 {\sqrt L} e^{ipx}$
and
$\langle p|y \rangle= \frac 1 {\sqrt L} e^{-ipy}$

OOH,
$\langle x|y \rangle = \delta(x-y)$

OTOH, inserting the resolution of identity,
$\langle x|y \rangle = \sum_p \langle x|p \rangle \langle p|y \rangle=\frac 1 L \sum_p e^{ip(x-y)}$

Thus,
$\frac 1 L \sum_p e^{ip(x-y)}=\delta(x-y)$