Show that this plane wave satisfies the Schrödinger Eqn

[Old_sm0key]

Homework Statement

I'm asked to show that the two dimensional plane wave (for constant C)
$$\psi \left ( \mathbf{r} \right )=Ce^{-i\mathbf{k}\cdot \mathbf{r}}$$
satisfies the Schrödinger equation:
$$-\frac{\hbar^{2}}{2m_e}\frac{\mathrm{d}^2 \psi\left ( \mathbf{r} \right )}{\mathrm{d} \mathbf{r}^2}=E\psi\left ( \mathbf{r} \right )$$
but I'm flummoxed by differentiating wrt a vector, $\mathbf{r}$. Can someone explain how I should understand this differential, so that I can produce the required equality?

Thanks in advance.

 

[Orodruin]

The intended meaning of $d^2/d\boldsymbol{r}^2$ is the Laplace operator $\nabla^2$.