# Motion equation in quantum mechanics?

ndung200790
Please teach me this:
Why the motion equation must be:dA/dt=[H,A] where H-Hamintonian,A-operator of any observation,because with a local flow t(time) of vector X in a manifold we can write the Lie derivative:dA/dt=[X,A].(Where we consider time t as one-parameter group and as local flow of some operator X)
Thank you very much in advanced.

## Answers and Replies

Deric Boyle
It is not equation of motion it states time evolution of a state under operator A. After studying it, it comes out to be analogus(or same) to classical equations of motion (Ehernfest theorem).
You need to thoroughly study foundations of mathematical aspects applied to quantum mechanics. von Neumann's work might help.

Why the motion equation must be:dA/dt=[H,A] where H-Hamintonian,A-operator of any observation

An essentially trivial derivation is in any decent textbook on quantum mechanics.
What is it that you don't get?

ndung200790
I would like to derive all equations:Schrodinger,Klein-Gordon,Dirac equation from this"motion" equation.It seem to me that dynamics of time is energy,the translation symmetry in time relate with Hamintonian.So if time t is local flow of vector H,then following Lie derivative we have the motion equation dA/dt=[H,A].

I would like to derive all equations:Schrodinger,Klein-Gordon,Dirac equation from this"motion" equation.It seem to me that dynamics of time is energy,the translation symmetry in time relate with Hamintonian.So if time t is local flow of vector H,then following Lie derivative we have the motion equation dA/dt=[H,A].

For these equation, the relevant dynamics is i hbar psidot = H psi.
H = p^2/2m +V(x) gives Schroedinger,
H=sqrt(p^2+m^2) gives the physical (positive energy) part of Klein-Gordon,
H=gamma_0(gamma dot p + m) gives Dirac (with both physical and unphysical part; the physical part is obtained by multiplying with a projector to E>=0).

dA/dt=[H,A] holds in each of these cases for arbitrary A if one changes from the Schroedinger picture to the Heisenberg picture according to the standard recipe.

I'd like to suggest that you get more practice in standard QM before you explore your own conjectures about how to structure the standard material.

ndung200790
Thank you very much for your kind helping and kind advice