quantum mechanics - The Schrödinger equation as an Euler-Lagrange equation

In the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrodinger equation in the form $$i\hbar\frac{\partial\psi}{\partial t}=\Big[-\frac{\hbar^2\nabla^2}{2m}+U(\textbf{r})\Big]\psi(\textbf{r},t)$$ can be obtained as the Euler-Lagrange equation corresponding to a Lagrangian of the form $$L=i\hbar\psi^*\dot{\psi}-\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi-U(\textbf{r})\psi^*\psi.$$

I have a discomfort with this derivation. As fas as I know a Lagrangian is a classical object. Is it justified in constructing a Lagrangian that has $\hbar$ built into it?

Answer

Firstly, one may think of this as a mathematical rather than physical procedure. In the end one is simply constructing a functional,

$$S = \int \mathrm dt \, L$$

whose extremisation, $\delta S = 0$ leads to the Schrodinger equation. However, Lagrangians containing $\hbar$ are not uncommon. In quantum field theory, one can construct effective actions from computing Feynman diagrams, which may have factors of $\hbar$, outside of natural units.