How can the Euler-Lagrangian equation be applied to the Schroedinger Lagrangian of several fields?

Question

In a Lagrangian like the one that follows:

$$\mathcal{L} = {i} \Psi^*\dot{\Psi} - \frac{1}{2m} \nabla{\Psi} ^* \nabla \Psi.\tag{1}$$

How can I apply the Euler-Lagrangian equation, shown in $(2)$, to obtain an equation like $(3)$. I don't understand how I can deal with all the different fields?

$$ \frac{d}{dt}\left( \frac{\partial \mathcal{L}}{\partial \dot{\Psi}}\right)= \frac{\partial \mathcal{L}}{\partial \Psi} \tag{2}$$

$$i \frac{\partial \Psi}{\partial t} = - \frac{\nabla^2}{2m} \Psi \tag{3}$$

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I have seen in some notes that I must treat $\Psi$ and $\Psi^*$ as different fields and must first calculate the different derivatives, shown in $(4)$:

$$\frac{\partial \mathcal{L}}{\partial \dot{\Psi^*}}= i\Psi \hspace{5mm}; \hspace{5mm} \frac{\partial \mathcal{L}}{\partial \nabla{\Psi^*}}= - \frac{1}{2m} \nabla \Psi \hspace{5mm} ; \hspace{5mm} \frac{\partial \mathcal{L}}{\partial {\Psi^*}}= i\dot{\Psi} \tag{4}$$

But how I can apply $(2)$ to these to obtain $(3)$? And do I not need the $\frac{\partial \mathcal{L}}{\partial \nabla\Psi^*}$ ?

How can I use $(2)$ if I do not have a $\Psi$ variable or a $\dot\Psi^*$ to complete the equation?

Answer 1

I'll try to give a slightly more precise view which might be confusing at first but will pay off later.

You are minimizing the action $S[\psi,\psi^*]$ where the square brackets indicate that $S$ is a functional: an object which takes a function as argument and spits out a number. Here $\psi$ and $\psi^*$ are still considered completely independent functions. In the following equations I will drop the dependence on $\psi^*$ since otherwise it would be too long but you can just as easily put them back in. The action then looks like $$S[\psi]=\int\mathrm dx\,\mathcal L(\psi,\dot\psi,\nabla\psi,\nabla\dot\psi)\tag{1}$$ where $x$ is four vector so it includes time as well. Here I emphasize that $\mathcal L$ is just a regular function: it takes in 4 arguments and spits out a number. I could also fill in $a,b,c,d$ instead of $\psi,\dot\psi,\nabla\psi,\nabla\dot\psi$ in the Lagrangian (I will use $\psi$ instead of $\psi^*$ for brevity). $$\mathcal L(a,b,c,d)=\frac i 2ab-\frac 1{2m}dc\tag{2}$$ Now to minimize the action we will have to calculate the functional derivative. The functional derivative, $\frac{\delta S}{\delta\psi}( x)$, is formally defined as $$\int\mathrm d x\frac{\delta S}{\delta \psi}( x)\eta(x)=\lim_{\epsilon\rightarrow\infty}\frac{S[\psi+\epsilon\eta]-S[\psi]}{\epsilon}\tag{3}$$ where $\eta( x)$ is an arbitrary test function that satisfies the right boundary conditions, in this case the condition that $\eta$ goes to zero as $ x\rightarrow\infty$. $\eta$ is often written as $\delta\psi$. We can now calculate $S[\psi+\epsilon\eta]$ by Taylor expanding the Lagrangian: \begin{align} S[\psi+\epsilon\eta]=&\int\mathrm d x\,\mathcal L(\psi+\epsilon\eta,\dot\psi+\epsilon\dot\eta,\nabla\psi+\epsilon\nabla\eta,\nabla\dot\psi+\epsilon\nabla\dot\eta)\\ &=\int\mathrm dx\left[\mathcal L(\psi,\dot\psi,\nabla\psi,\nabla\dot\psi)+\\ \epsilon\,\eta(x)\frac{\partial}{\partial\psi}\mathcal L(\psi,\dot\psi,\nabla\psi,\nabla\dot\psi)+\\ \epsilon\,\dot\eta(x)\frac{\partial}{\partial\dot\psi}\mathcal L(\psi,\dot\psi,\nabla\psi,\nabla\dot\psi)+\\ \epsilon\nabla\eta(x)\frac{\partial}{\partial\nabla\psi}\mathcal L(\psi,\dot\psi,\nabla\psi,\nabla\dot\psi)+\\ \epsilon\nabla\dot\eta(x)\frac{\partial}{\partial\nabla\dot\psi}\mathcal L(\psi,\dot\psi,\nabla\psi,\nabla\dot\psi) \right]\tag{4} \end{align} This almost has the form of the LHS of (2) but $\eta$ still has derivatives acting on it. We can fix this by integrating by parts to move the derivative from $\eta$ to the other factor at the cost of introducing a minus sign. Plugging this into (3) gives \begin{align}\int\mathrm d x\frac{\delta S}{\delta \psi}( x)\eta(\mathbf x)&=\int\mathrm dx\left[\frac{\partial\mathcal L}{\partial\psi}-\frac{\partial}{\partial t}\frac{\partial\mathcal L}{\partial\dot\psi}-\nabla\cdot\left(\frac{\partial\mathcal L}{\partial\nabla\psi}\right)\pm\frac{\partial}{\partial t}\nabla\cdot\left(\frac{\partial\mathcal L}{\partial\nabla\dot\psi}\right)\right]\eta(x) \end{align} We recognize the term between brackets as the functional derivative and setting it to zero will give us the EL equations. I'm not so sure if the $\pm$ should be plus or minus but I think it's $+$. If we had included $\psi^*$ as a separate field from the start we would have got a similar equation but with $\psi^*$ instead of $\psi$.

I believe there's an error in your Lagrangian and that $\nabla\dot\psi$ should be $\nabla\psi$. The factor $\frac i 2\psi^*\dot\psi$ should either be $i\psi^*\dot\psi$ or $\frac i 2(\psi^*\dot\psi-\psi\dot\psi^*)$. See this question or this question. If you now calculate the EL you get the desired formula.

Answer 2

I'm not 100% confident in my answer as I am new to this topic, but I hope I am of help.

Edit: Your lagrangian is a slightly strange, as pointed out by AccidentalTaylorExpansion, so you may want to double check that it is correct. The below should hold for the lagrangian you have given however.

The lagrangian you have is a 'lagrangian density'. If the lagrangian is by L and the lagrangian density denoted by $\mathcal L$, then:

$$ \int L \ dt = \int \mathcal L \ dt dx^3 $$

The Euler Lagrange equation associated with extremising the action for the lagrangian density $\mathcal L(\psi, \dot \psi, \partial_{\mu} \psi, \partial_{\mu} \dot \psi)$ looks a bit different (Einstein summation in implied):

$$ \frac{\partial}{\partial t} \left ( \frac{\partial \mathcal L}{ \dot \psi } \right) + \frac{\partial}{\partial x^\mu} \left ( \frac{\partial \mathcal L}{ \partial (\partial_{\mu} \psi) } \right) + \frac{\partial^2}{\partial x^\mu \partial t} \left ( \frac{\partial \mathcal L}{ \partial (\partial_{\mu} \dot \psi) } \right) = \frac{\partial \mathcal L}{\partial \psi}$$

Where $\mu$ runs from 1 to 3 for the three spatial dimensions.

For complex fields, you treat the complex and complex conjugate as two separate fields. This is because there are two degrees of freedom (see: Complex Scalar Field - Euler Lagrange equation).

Meaning that you also get the equation: $$ \frac{\partial}{\partial t} \left ( \frac{\partial \mathcal L}{ \partial \dot \psi^* } \right) + \frac{\partial}{\partial x^\mu} \left ( \frac{\partial \mathcal L}{ \partial (\partial_{\mu} \psi^*) } \right) + \frac{\partial^2}{\partial x^\mu \partial t} \left ( \frac{\partial \mathcal L}{ \partial (\partial_{\mu} \dot \psi^*) } \right)= \frac{\partial \mathcal L}{\partial \psi^*}$$

Thus: $$ \frac{\partial \mathcal L}{\partial \dot \psi^* } = 0 $$ $$ \frac{\partial \mathcal L}{ \partial_{\mu} \psi^* } = -\frac{i}{2m} \partial^{\mu} \dot \psi $$ $$ \frac{\partial \mathcal L}{\partial \psi^*} = \frac{1}{2} \dot \psi$$ $$ \frac{\partial \mathcal L}{ \partial (\partial_{\mu} \dot \psi^*) } = 0 $$

So: $$ \dot \psi = -\frac{i}{m} \partial_{\mu} \partial^{\mu} \dot \psi $$

Additionally, from the EL equation with the real $\psi$:

Thus: $$ \frac{\partial \mathcal L}{\partial \dot \psi } = \frac{i}{2} \psi^* $$ $$ \frac{\partial \mathcal L}{ \partial_{\mu} \psi } = 0 $$ $$ \frac{\partial \mathcal L}{\partial \psi} = 0 $$ $$ \frac{\partial \mathcal L}{ \partial (\partial_{\mu} \dot \psi) } = - \frac{1}{2m} \partial_{\mu} \psi^*$$

So: $$ \frac{i}{2} \dot \psi^* = \frac{1}{2m} \partial_{\mu} \partial^{\mu} \dot \psi^*$$