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.