In the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrödinger equation in the form
can be obtained as the Euler-Lagrange equation corresponding to a Lagrangian density of the form
I have a discomfort with this derivation. As far as I know a Lagrangian is a classical object. Is it justified in constructing a Lagrangian that has $\hbar$ built into it?