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:

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

The problem is to treat $\psi$ and its conjugate as independent field variables and show that this generates Schrödinger'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:

$$L <br /> = \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*} + V \psi \psi^{*} + \frac{h}{2 \pi i} ( \psi^{*} \partial_t \psi - \psi \partial_t \psi^{*} )$$
$$= \frac{h^2}{8 \pi^2 m} \partial_k \psi \partial_k \psi^{*} + V \psi \psi^{*} + <br /> \frac{h}{2 \pi i} ( \psi^{*} \partial_t \psi - \psi \partial_t \psi^{*} )$$

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]

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

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

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