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=|∇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.
Best regards,
Daniel