What is the name of this equation:
$$\frac {d^2\psi}{dr^2}+k^2\psi=0?$$
(I want a Wikipedia link for this equation, but I don't know what its name is.)
Point: In this equation, the wave function is only function of space $\psi (r)$, for example $\psi=Ae^{\mathrm i kr}$.
This equation is known as Helmholtz equation and is widely used especially in optics.
As shubham notes, the Schroedinger equation is indeed a Helmholtz equation in 1D (your case), but the general form of the equation is:
$$ - \frac{\hbar^2}{2m} \nabla^2 \Psi \left( \mathbf{r}, t \right) + \operatorname{V \left( \mathbf{r},t \right)} \Psi = i \hbar \frac{\partial \Psi \left(\mathbf{r}, t\right)}{\partial t} $$
This is a diffusion equation in which the diffusion equation is a constant. For comparison, we have (if $V = 0$):
$$ D\nabla^2 \Psi = \frac{\partial \Psi}{\partial t} \quad \implies D = \frac{i \hbar}{2 m} $$
However, if we have a static case, then the term on the right hand side reduces to a Helmholtz equation (if we have $V = 0$ everywhere in our space):
$$ \nabla^2 \Psi + \frac{2mE}{\hbar^2} \Psi $$
where you can infer that
$$ \left|\mathbf{k}\right|^2 = \frac{2mE}{\hbar^2} $$
Which, as you see is a general result, which holds in 3D as well.
1) This is the equation you get on solving the Schrodinger equation for a particle in a box. Assume the particle to be in a box of length a under a potential $V(x)=0$ within the box and $\infty$ when outside the box Then applying separation of variables to the Schrodinger equation we get $\hat{H}\psi =E\psi $ which is simply \begin{equation} \frac{d^2\psi}{dr^2}=-k^2\psi\end{equation} 2)The above general equation is also the equation to study vibrating membranes of 2 or 3 dimensions.
