goldstein schodinger's equation Lagragian problem.

[Peeter]

Problem 3 in the continuous systems and fields chapter of (the first edition, 1956 printing) of Goldstein's classical mechanics has the following Lagrangian:


L = \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*}
+ V \psi \psi^{*}
+ \frac{h}{2\pi i}
( \psi^{*} \dot{\psi}
- \psi \dot{\psi}^{*} )


The problem is to treat \psi and its conjugate as independent field variables and show that this generates Schodinger's equation and its conjugate.

Doing the problem I find I need h/4\pi i in this last term to make it work out. Could somebody with a newer edition of this text see if this is a corrected typo?

[malawi_glenn]

Still the same eq for the Lagrangian density.

I have third edition.

[Ben Niehoff]

See if it's here? (errata page)

http://astro.physics.sc.edu/goldstein/

[Peeter]

Does anybody see where I went wrong:


L
= \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*} + V \psi \psi^{*} + \frac{h}{2 \pi i} ( \psi^{*} \partial_t \psi - \psi \partial_t \psi^{*} )


= \frac{h^2}{8 \pi^2 m} \partial_k \psi \partial_k \psi^{*} + V \psi \psi^{*} +
\frac{h}{2 \pi i} ( \psi^{*} \partial_t \psi - \psi \partial_t \psi^{*} )


We have

\frac{\partial L}{\partial \psi^{*} } = V\psi + \frac{h}{2 \pi i} \partial_t \psi


and canonical momenta

\frac{\partial L}{\partial{(\partial_k \psi^{*})}} = \frac{h^2}{8 \pi^2 m} \partial_{k} \psi


\frac{\partial L}{\partial{(\partial_t \psi^{*})}} = -\frac{h}{2 \pi i} {\psi}



\frac{\partial L}{\partial \psi^{*}} = \sum_k \partial_k \frac{\partial L}{\partial{(\partial_k \psi^{*})}} + \partial_t \frac{\partial L}{\partial{(\partial_t \psi^{*})}}


V\psi + \frac{h}{2 \pi i} \partial_t \psi = \frac{h^2}{8 \pi^2 m} \sum_k \partial_{kk} \psi -\frac{h}{2 \pi i} \frac{\partial \psi}{\partial{t}}


which is off by a factor of two in the time term

-\frac{h^2}{8 \pi^2 m} \nabla^2 \psi + V\psi = \frac{h i}{\pi} \frac{\partial \psi}{\partial{t}}

[samalkhaiat]

[QUOTE=Peeter;1912727]Problem 3 in the continuous systems and fields chapter of (the first edition, 1956 printing) of Goldstein's classical mechanics has the following Lagrangian:


L = \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*}
+ V \psi \psi^{*}
+ \frac{h}{2\pi i}
( \psi^{*} \dot{\psi}
- \psi \dot{\psi}^{*} )


The problem is to treat \psi and its conjugate as independent field variables and show that this generates Schodinger's equation and its conjugate.

Doing the problem I find I need h/4\pi i in this last term to make it work out. Could somebody with a newer edition of this text see if this is a corrected typo?

You are right. I remember this "typo"! I wonder why it has not been corrected after all these years?
The 1/2 factor is necessary when working with Hermitian Lagrangians like the one you wrote. Non-hermitian Lagrangian on the other hand does not need the 1/2, but does the same job;

\mathcal{L} = i \hbar \psi^{*}\partial_{t}\psi - \frac{\hbar^{2}}{2m} \partial_{i}\psi^{*}\partial_{i}\psi - V(x) \psi^{*}\psi

regards

sam

[Peeter]

Thanks Sam.

Malawi,

What page and chapter is this problem in, in the third edition?

EDIT: fyi. I reported the problem and got the following response:

> I looked in the 1st, 4th and 6th printings of the 3rd edition. The problem appears on page 599 (#4). The correction you sent was made before the 1st printing since the 1/4 appears on page 599 of all printings.