I am trying to derive the following expression of the free-particle propagator in 2D, given by
$$ \rho_0(\mathbf{r},\mathbf{r'}, \beta) = \langle \mathbf{r'} \rvert e^{-\beta \hat{H}} \lvert \mathbf{r} \rangle = \frac{1}{4\pi \beta}e^{-\frac{(\mathbf{r}-\mathbf{r'})^2}{4\beta}}$$ where $\hat{H} = -\nabla^2_\mathbf{r}$ is the free-particle Hamiltonian in 2D, and where we assumed $\frac{\hbar^2}{2m} = 1$.
In a first attempt, I tried assuming rotational invariance of the wavefunction $\psi(r, \theta) = \frac{1}{\sqrt{2\pi}}R(r)$ (thus only considering s-wave solutions), which gives solutions of the Schroedinger equation in the form of Bessel functions of zero-th order. Unfortunately, I cannot reconcile these results with the equation above.