Legendre structure in quantum mechanics

Thermodynamics relationships can be expressed in different fashion,
choosing as independent variables different collections of
intensive-extensive variables combinations. For instance, the natural
variables of the internal energy E are S, V, N, all of them extensive.
For the Helmholtz free energy F one has, instead, T, V, N. However, F
and E contain exactly the same amount of information. This is due to
the fact that thermodynamic potentials are interconnected via Legendre
transforms. We show in this communication that a similar Legendre
transform structure underlies the celebrated Schrödinger equation.