Lagrangians in Quantum Mechanics

[lugita15]

In classical mechanics the Lagrangian depends only on time, position, and velocity. It is not allowed to depend on any higher order derivatives of position. Does this principle remain true for Lagrangians in non-relativistic quantum mechanics? What about relativistic quantum field theory?

Any help would be greatly appreciated.
Thank You in Advance.

 

[Fredrik]

I don't think I've ever seen Lagrangian for a quantum mechanical theory that isn't a quantum field theory (except for string Lagrangians, which I haven't studied). Quantum field theory Lagrangians don't contain higher order derivatives because they would make the theory non-renormalizable. (It would probably be more correct to include those terms but they don't contribute much to physics at low energies, so it's safe to ignore them. This is fortunate since they are difficult to deal with mathematically).

 

[per.sundqvist]

In classical mechanics the Lagrangian depends only on time, position, and velocity. It is not allowed to depend on any higher order derivatives of position. Does this principle remain true for Lagrangians in non-relativistic quantum mechanics? What about relativistic quantum field theory?

Any help would be greatly appreciated.
Thank You in Advance.

The lagrangian in QM is the same as in classical mechanics. You could derive Schrödingers equ from the Lagrangian using Feynmans path integral formulation as described in http://en.wikipedia.org/wiki/Path_integral_formulation" .

In principle the Lagrangian could contain higher order derivatives. Using the variational derivatives to get the equation of motion you get for instance something like this for a second order time-derivative:

$$\frac{d^2}{dt^2}\left(\frac{\partial L}{\partial x_{tt}}\right)$$

But if it is physical meaningful is another question... If you could transform your Lagrangian into a Hamiltonian, containing the canonic momentum p, then you could use it into an effective Schrödinger equation as well.

The relativistic non-quantum Lagrangian is not linear in the momentum either, but that's another story.