Time dependent Schrodinger equation through variation principle - questions about derivation

Question

I'm reading a text which discusses time dependent variation principle (Geometry of the Time-Dependent Variational Principle in Quantum Mechanics by Kramer and Saraceno), and there is some part of a derivation which I cannot understand. I'm trying to do a similar derivation in a slightly different context, so if anyone here could shed some light on what I don't understand, that would be really helpful.

This is the relevant part:

Up to the integration by parts I understand everything, so everything that happens up until then you can take as given. They then write "therefore, up to some total derivatives, we obtain...".

But I do not understand why this is what they obtain. When I plug in the integration by parts (and ignore the total derivatives, since they won't affect the solution), I get something slightly different:

Note that I know the origin for one difference - for my case I am taking $\langle\psi | \psi\rangle=1$, and so the term which depends on $\langle\delta\psi | \psi\rangle$ vanishes (and that makes sense, I'm fine with it). But the Hamiltonian term seems to be different than what they get and I am not sure why.

Moreover, I am not sure how they made the step that leads to equation $(2.3)$ (with this treatment of the bra and ket as independent variables). Does anyone understand how they did it?

By the way, just for completeness, this is the relevant Lagrangian:

If anything here needs further clarification, let me know.

Answer 1

Well, let's see.

1. The Lagrangian is $$ L~=~ \frac{\langle \psi | i \partial_t -H |\psi\rangle}{2||\psi ||^2} ~+~ c.c.,\qquad ||\psi ||^2~\equiv~\langle \psi | \psi\rangle. \tag{A}$$ An infinitesimal variation is $$\begin{align} \delta L~=~&\frac{\langle \delta \psi | i \partial_t -H |\psi\rangle + \langle \psi | i \partial_t -H |\delta \psi\rangle}{2||\psi ||^2} \cr &+ \langle \psi | i \partial_t -H |\psi\rangle\delta \frac{1}{2||\psi ||^2} ~+~ c.c.\cr ~\stackrel{IBP}{\sim}~&\frac{\langle \delta \psi | i \partial_t -H |\psi\rangle + \langle \psi |- i \stackrel{\leftarrow}{\partial_t} -H |\delta \psi\rangle}{2||\psi ||^2} \cr &-\langle \psi | \delta \psi\rangle \partial_t \frac{i}{2||\psi ||^2} \cr & -\langle \psi | i \partial_t -H |\psi\rangle \frac{\langle \delta\psi | \psi\rangle+\langle \psi | \delta \psi\rangle }{2||\psi ||^4} ~+~ c.c.\cr ~=~&\frac{\langle \delta \psi | i \partial_t -H |\psi\rangle}{||\psi ||^2} + \langle \delta\psi | \psi\rangle \partial_t \frac{i}{2||\psi ||^2} \cr & -\langle\delta\psi |\psi\rangle \frac{\langle \psi | i \partial_t -H |\psi\rangle + \overline{\langle \psi | i \partial_t -H |\psi\rangle}}{2||\psi ||^4} ~+~ c.c. \end{align}\tag{B}$$ In the 2nd equality of eq. (B) we dropped a total time derivative term. In the 3rd equality of eq. (B) we strategically swopped terms with the $c.c.$ part, and we used that $H$ is self-adjoint.

2. We can treat $\langle \delta \psi |$ and $| \delta \psi \rangle$ as independent in eq. (B), cf. e.g. this Phys.SE post. This leads to$^1$ $$\begin{align} 0~\approx~& ( i \partial_t -H )|\psi\rangle + | \psi\rangle ||\psi ||^2 \partial_t \frac{i}{2||\psi ||^2} \cr & -| \psi\rangle \frac{\langle \psi | i \partial_t -H |\psi\rangle + \overline{\langle \psi | i \partial_t -H |\psi\rangle}}{2||\psi ||^2}. \end{align}\tag{C}$$ Contracting with $\langle \psi |$ yields $$ 0~\approx~ \partial_t \frac{i}{2||\psi ||^2} + \frac{\langle \psi | i \partial_t -H |\psi\rangle - \overline{\langle \psi | i \partial_t -H |\psi\rangle}}{2||\psi ||^4}. \tag{D}$$ Eqs. (C) & (D) indeed imply the projected TDSE in Ref. 1 $$ 0~\approx~ ( i \partial_t -H )|\psi\rangle -|\psi\rangle \frac{\langle \psi | i \partial_t -H |\psi\rangle}{||\psi ||^2}.\tag{2.3} $$

References:

1. P. Kramer & M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics 140, 1981.

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$^1$ The $\approx$ symbol means equality modulo EL equations.