The Laplacian is the following linear differential operator
d2/dx2 + d2/dy2 + d2/dz2
so it is the same thing as the divergence of the gradient. It can be generalized to dimensions other than 3 in the obvious way. The Laplacian occurs in many different situations in physics. For example, the following partial differential equation describes the electric potential generated by a distribution of electric charge :
(d2/dx2 + d2/dy2 + d2/dz2) V = q/e0
where V is the potential, e0 is a constant and q is the charge density. Another example where the Laplacian makes an appearance is the wave equation:
(d2/dt2 - v2(d2/dx2 + d2/dy2 + d2/dz2)) p = 0
where p(x,y,z,t) is for example the air pressure in a certain place at a certain time and v is the velocity of the waves in question. This partial differential equation can, with appropriate boundary condition s, be used to model
propagation of sound or radio waves, for example.
Here is the heat equation, which describes the propagation of heat:
(d/dt - k (d2/dx2 + d2/dy2 + d2/dz2)) T = 0
where k is a constant describing heat conduction and T is the temperature. Here is a
Schrödinger equation, which describes the quantum-mechanical behavior of a free particle if relativity is neglected:
(i d/dt + hbar/2m (d2/dx2 + d2/dy2 + d2/dz2))psi = 0
, where psi is the wave function, i is the imaginary unit, hbar is the Planck's constant divided by 2 pi, and m is the mass of the particle. As you see, the Schrödinger equation looks almost like the heat equation, except for the imaginary unit in front of the time derivative, which makes it behave quite differently!