# Lagrangian of Schrodinger field

The usual Schrodinger Lagrangian is $$\tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi,$$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ vanishing. This Lagrangian density is not real but differs from a real Lagrangian density $$\tag 2 \frac{i}{2}(\psi^{*}\partial_{t}\psi -\psi \partial_{t}\psi^{*} ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi$$ by a total derivative.

My trouble is that these two lagrangian densities lead to different conjugate momenta and hence when setting equal time commutation relations, I am getting different results (a factor of 2 is causing the problem). I can rescale the fields but then my Hamiltonian also changes. Then applying Heisenberg equation of motion, I don't get the operator Schrodinger equation.

Is it possible to work with the real Lagrangian density and somehow get the correct commutation relations? I would have expected two Lagrangians differing by total derivative terms to give identical commutation relations (since canonical transformations preserve them). But perhaps I am making some very simple error. Unless all conjugate momenta are equivalent for two Lagrangians differing by total derivatives, how does one choose the correct one?

I guess the same thing happens for other first order systems like Dirac Lagrangian also.

• I don't have time for a detailed reply to your question, but it may help to have a look at the end of Sec. 7.2 in Weinberg's textbook (vol. 1). He discusses the effect of adding a total time derivative to the Lagrangian and shows that while modifying the canonical momentum, it does not affect the commutation relations of the theory. – Tomáš Brauner Sep 30 '11 at 10:17

Here we will for simplicity only consider the Schrödinger system. We will assume that

$$\phi~=~(\phi^1+i\phi^2)/\sqrt{2} \tag{A}$$

is a bosonic complex field, and that

$$\phi^*~=~(\phi^1-i\phi^2)/\sqrt{2} \tag{B}$$

is the complex conjugate, where $\phi^a$ are the two real component fields, $a=1,2$. [Note the change in notation $\psi\longrightarrow\phi$ as compared with the OP's question (v1).]

1) The Lagrangian density

$${\cal L}~:=~ i\phi^{*}\dot{\phi} + \frac{1}{2m} \phi^* \nabla^2\phi \tag{C}$$

for the Schrödinger field $\phi$ is already on the Hamiltonian form

$${\cal L}~=~ \pi\dot{\phi} - {\cal H}. \tag{D}$$

Simply define complex momentum

$$\pi~:=~i \phi^{\ast}, \tag{E}$$

and Hamiltonian density

$${\cal H}~:=~-\frac{1}{2m} \phi^{\ast} \nabla^2\phi. \tag{F}$$

More generally, this identification is a simple example of the Faddeev-Jackiw method.

2) Recall that the Euler-Lagrange equations do not change by adding a $4$-divergence $d_{\mu}\Lambda^{\mu}$ to the Lagrangian density

$${\cal L} ~~\longrightarrow~~ {\cal L}^{\prime}~:=~{\cal L} + d_{\mu}\Lambda^{\mu},\tag{G}$$

cf. e.g. this Phys.SE post. [We use the symbol $d_{\mu}$ (rather than $\partial_{\mu}$) to stress the fact that the derivative $d_{\mu}$ is a total derivative, which involves both implicit differentiation through the field variables $\phi^a(x)$, and explicit differentiation wrt. $x^{\mu}$.] Therefore, we can (via spatial integration by parts) choose an equivalent Hamiltonian density

$${\cal H} ~~\longrightarrow~~ {\cal H}^{\prime}~:=~\frac{1}{2m}|\nabla\phi|^2 ~=~\frac{1}{4m}(\nabla\phi^1)^2 +\frac{1}{4m}(\nabla\phi^2)^2,\tag{H}$$

and we can (via temporal integrations by part) choose an equivalent kinetic term

$$i\phi^*\dot{\phi}~=~ \pi\dot{\phi} ~~\longrightarrow~~$$ $$\frac{1}{2}(\pi\dot{\phi}-\phi\dot{\pi}) ~=~ \frac{i}{2}(\phi^*\dot{\phi}-\phi\dot{\phi}^*)~=~\frac{1}{2}(\phi^2\dot{\phi}^1-\phi^1\dot{\phi}^2)~~\longrightarrow~~\phi^2\dot{\phi}^1. \tag{I}$$

The last expression shows (in accordance with the Faddeev-Jackiw method) that

$$\text{The second component }\phi^2 \text{ is the momenta for the first component }\phi^1. \tag{J}$$

3) Alternatively, we can perform a Dirac-Bergmann analysis$^1$ directly. Consider for instance the Lagrangian density

$${\cal L}^{\prime}~=~ (\alpha+\frac{1}{2})\phi^2\dot{\phi}^1+(\alpha-\frac{1}{2})\phi^1\dot{\phi}^2 - {\cal H}^{\prime},\tag{K}$$

where $\alpha$ is an arbitrary real number. [The term $d(\phi^1\phi^2)/ dt$, which is multiplied by $\alpha$ in ${\cal L}^{\prime}$, is a total time derivative.] Let us check that the quantization procedure does not depend on this parameter $\alpha$. We introduce canonical Poisson brackets

$$\{\phi^a({\bf x},t),\phi^b({\bf y},t)\}_{PB} ~=~0,$$ $$\{\phi^a({\bf x},t),\pi_b({\bf y},t)\}_{PB} ~=~\delta^a_b ~ \delta^3 ({\bf x}-{\bf y}),$$ $$\{\pi_a({\bf x},t),\pi_b({\bf y},t)\}_{PB} ~=~0, \tag{L}$$

in the standard way. The canonical momenta $\pi_a$ are defined as

$$\pi_1~:=~\frac{\partial {\cal L}^{\prime}}{\partial \dot{\phi}^1} ~=~(\alpha+\frac{1}{2})\phi^2,$$ $$\pi_2~:=~\frac{\partial {\cal L}^{\prime}}{\partial \dot{\phi}^2} ~=~(\alpha-\frac{1}{2})\phi^1.\tag{M}$$

These two definitions produce two primary constraints

$$\chi_1~:=~\pi_1-(\alpha+\frac{1}{2})\phi^2~\approx~0,$$ $$\chi_2~:=~\pi_2-(\alpha-\frac{1}{2})\phi^1~\approx~0,\tag{N}$$

where the $\approx$ sign means equal modulo constraints. The two constraints are of second-class, because

$$\{\chi_2({\bf x},t),\chi_1({\bf y},t)\}_{PB}~=~\delta^3 ({\bf x}-{\bf y})~\neq~0. \tag{O}$$

Thus the Poisson bracket should be replaced by the Dirac bracket. [There are no secondary constraints, because

$$\dot{\chi}_a({\bf x},t) ~=~\{\chi_a({\bf x},t), H^{\prime}(t)\}_{DB} ~=~ 0, \qquad H^{\prime}(t)~:=~ \int d^3y \ {\cal H}^{\prime}({\bf y},t), \tag{P}$$

are automatically satisfied.] The Dirac bracket between the two $\phi^a$'s is

$$\{\phi^1({\bf x},t),\phi^2({\bf y},t)\}_{DB}~=~\delta^3 ({\bf x}-{\bf y}), \tag{Q}$$

leading to the same conclusion (J) as the Faddeev-Jackiw method. Note that the eqs. (O) and (Q) are independent of the parameter $\alpha$.

4) In all cases, the canonical equal-time commutator relations for the corresponding operators become

$$[\hat{\phi}^1({\bf x},t), \hat{\phi}^2({\bf y},t)] ~=~ i\hbar {\bf 1}~\delta^3 ({\bf x}-{\bf y}),$$ $$[\hat{\phi}({\bf x},t), \hat{\phi}^{\dagger}({\bf y},t)] ~=~ \hbar {\bf 1}~\delta^3 ({\bf x}-{\bf y}),$$ $$[\hat{\phi}({\bf x},t), \hat{\pi}({\bf y},t)] ~=~ i\hbar {\bf 1}~\delta^3 ({\bf x}-{\bf y}). \tag{R}$$

--

$^1$ See, e.g., M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1992.

• Thanks a lot to Qmechanic for very detailed answer. I really appreciate the help. – user5468 Oct 8 '11 at 17:39
• Notes for later: The Schrödinger action for multiple particles is $$I[\psi]~=~\int\! dt~\left[ \prod_{j=1}^N d^3{\bf x}_j \right] {\cal L}.$$ – Qmechanic Jun 26 '18 at 14:02
• Notes for later: The Schrödinger Lagrangian density is $${\cal L} ~=~\frac{i\hbar}{2}(\psi^{\ast}\dot{\psi}-\dot{\psi}^{\ast}\psi) - \sum_{j=1}^N\frac{\hbar^2}{2m_j}|\nabla_j\psi|^2 -V |\psi|^2$$ $$~=~ -\rho\dot{S}-\sum_{j=1}^N\frac{\hbar^2}{2m_j}(\nabla_j\sqrt{\rho})^2-\sum_{j=1}^N\frac{\rho}{2m_j}(\nabla_j S)^2 -\rho V,$$ where we rewrote the wavefunction $\psi=\sqrt{\rho}\exp\left(\frac{i}{\hbar}S\right)$ in "polar" coordinates $\rho$ and $S$. – Qmechanic Jun 26 '18 at 14:12
• Notes for later: Case $N=1$: Note that $\rho$ and $S$ are canonical variables $\{\rho({\bf x}),S({\bf y})\}=\delta^3({\bf x}-{\bf y})$, and that ${\cal L}$ is already on Hamiltonian first-order form. The Schrödinger picture suggests second quantization, cf. e.g. this Phys.SE post. – Qmechanic Jun 26 '18 at 14:59
• Notes for later: The EL eqs. wrt. $\rho$ and $S$ are $$\dot{S} - \sum_{j=1}^N\frac{\hbar^2}{2m_j\sqrt{\rho}}\nabla_j^2\sqrt{\rho} +\sum_{j=1}^N\frac{1}{m_j}(\nabla_jS)^2 + V ~\approx~ 0\qquad\text{and}\qquad \dot{\rho} + \sum_{j=1}^N\frac{1}{m_j}\nabla_j \cdot (\rho \nabla_jS) ~\approx~ 0,$$ respectively. The Madelung equations with ${\bf u}_j :=\frac{1}{m_j}\nabla_jS$ are straightforward consequences (and analogues of Navier–Stokes eqs). See also de Broglie–Bohm pilot-wave theory. – Qmechanic Jun 29 '18 at 16:55