# The Schrödinger equation as an Euler-Lagrange equation

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?

• This really seems to be deriving a classical field equation which happens to look like the Schrödinger equation for a quantum particle. The interpretations are completely different. Feb 10, 2018 at 10:28
• Sep 24, 2021 at 13:17

1. As user 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 \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.

--

$$^1$$ 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.

• Notes for later: The coupling constant $\frac{1}{m}$ has negative mass dimension, and hence corresponds to a non-renormalizable coupling, cf. Schwartz, section 22.1, p. 395. Feb 26 at 18:22

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.