I am currently reviewing this webpage, near the end it shows how the one-dimensional time independent Schrondinger equation can easily be extended to three dimensions.

My question is specifically with the change in notation used between the 1D case

$$ -\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} + V(x) \Psi(x) = E\Psi(x)$$

and the 3D case

$$ -\frac{\hbar^2}{2m} \nabla^2\Psi(r) + V(r) \Psi(r) = E\Psi(r)$$

Specifically, in mathematical "plain English" how is the change from $\frac{d^2 \Psi}{dx^2}$ to $ \nabla^2\Psi$ read and interpreted?

From how I understand it it is read as "the rate of change, of the rate of change" which is interpreted in a single dimension. Versus, "The gradient" (which to mean is essientially the same but on a manifold instead of a line) which is interpreted in 3 dimensions.

I also do want to point out that I am aware that $r$ is a vector of the $i,j,k$ basis in Euclidean space. (But I am not sure if I stated that exactly correctly)

I am currently reviewing this webpage, near the end it shows how the one-dimensional time independent Schrondinger equation can easily be extended to three dimensions.

My question is specifically with the change in notation used between the 1D case

$$ -\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} + V(x) \Psi(x) = E\Psi(x)$$

and the 3D case

$$ -\frac{\hbar^2}{2m} \nabla^2\Psi(\mathbf{r}) + V(\mathbf{r}) \Psi(\mathbf{r}) = E\Psi(\mathbf{r})$$

Specifically, in mathematical "plain English" how is the change from $\frac{d^2 \Psi}{dx^2}$ to $ \nabla^2\Psi$ read and interpreted?

From how I understand it it is read as "the rate of change, of the rate of change" which is interpreted in a single dimension. Versus, "The gradient" (which to mean is essientially the same but on a manifold instead of a line) which is interpreted in 3 dimensions.

I also do want to point out that I am aware that $\mathbf{r}$ is a vector of the $i,j,k$ basis in Euclidean space. (But I am not sure if I stated that exactly correctly)