This question started as an attempt to better understand the origin of the Schrödinger operator.
Consider a real valued scalar field $f(x,y,z)$
and the squared absolute value of its gradient $E=|\nabla f(x,y,z)|^2$.
In topics related to thermal physics, electrostatics, elasticity, acoustics or similar,
which scalar fields $f$ appear such that $E$ is interpreted
as the density of the internal energy of $f$?
References will be appreciated.