The Schrödinger equation as an Euler-Lagrange equation

Question

In section 1.2 on p. 14 in the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrödinger equation in the form

$$i\hbar\frac{\partial\psi}{\partial t}~=~\Big[-\frac{\hbar^2\nabla^2}{2m}+U(\textbf{r})\Big]\psi(\textbf{r},t)\tag{1.93}$$

can be obtained as the Euler-Lagrange equation corresponding to a Lagrangian density of the form

$$L~=~i\hbar\psi^*\dot{\psi}-\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi-U(\textbf{r})\psi^*\psi.\tag{1.94}$$

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?

Answer 1

1. As JamalS correctly points out in his answer:

   • Quantum actions in QFT are allowed to have $\hbar$-dependence.

   • If we merely want a stationary action principle for the TDSE, and view the action functional as just a mathematical tool without physical consequences beyond the EL equations, then the $\hbar$-dependence doesn't matter.

2. However, perhaps OP's discomfort with Mahan's TDSE derivation is spurred by the following deeper question:

   How we can get the correct semiclassical limit$^1$ and loop expansion$^2$ of a second-quantized path integral$^3$ $$Z~=~\int\! {\cal D}\frac{\psi_2}{\sqrt{\hbar}}{\cal D}\frac{\psi_2^{\ast}}{\sqrt{\hbar}} ~\exp\left(\frac{i}{\hbar} S\right),\tag{1}$$ if the Schroedinger action $S$ depends on $\hbar$, so that various parts of the actions $S$ scales/are suppressed inhomogeneously in the semiclassical limit $\hbar\to 0$?

   That's a good question. The answer is that there are implicit hidden $\hbar$-dependence, i.e. one should rescale the variables $$\psi~=~\frac{\psi_2}{\sqrt{\hbar}},\qquad m~=~\hbar m_2,\qquad U~=~\hbar U_2,\tag{2}$$ to obtain a classical ($\hbar$-independent) action$^4$ $$\begin{align} S~=~&\int \! \mathrm{d}t ~\mathrm{d}^3r \left( i\hbar \psi^{\ast}\dot{\psi}-\frac{\hbar^2}{2m} |\nabla\psi|^2 -U|\psi|^2 \right)\cr ~\stackrel{(2)}{=}~&\int \! \mathrm{d}t ~\mathrm{d}^3r \left( i \psi_2^{\ast}\dot{\psi}_2-\frac{1}{2m_2} |\nabla\psi_2|^2 -U_2|\psi_2|^2 \right) ,\end{align}\tag{3}$$ and to restore a correction loop expansion.

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$^1$ For the semiclassical limit, see e.g. this Phys.SE post.

$^2$ For the $\hbar$/loop-expansion, see e.g. this Phys.SE post.

$^3$ Here the subscript 2 refers to a properly normalized second-quantized formulation.

$^4$ Schwartz, section 22.1, p. 395, points out that the coupling constant $\frac{1}{m}$ has negative mass dimension, and hence corresponds to a non-renormalizable coupling.

Answer 2

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.