Time dependent transformations

[tavi_boada]

Hey people,

I just finished reading a chapter in a book on quantum mechanics that has deeply disturbed me. The chapter was about symmetry in quantum mechanics. It was divided in two basic parts: time dependent and time independent transformations.

Time independent transformations were quite tractable in the sense that nothing special or surprising happened. The Schrödinger equation was left unchanged.

For time dependent transformations a new term to the Hamiltonian had to be added in order to preserve the form of the equation. The new Hamiltonian was the same as the old one transformed plus a term involving a firs order time derivative of the transformation operator. How can that be? Aren't all inertial frames of reference equivalent? Is Schrödinger's equation not covariant?

I know nothing of relativistic QM, so sorry if this question offends those who do know something about it.

 

[selfAdjoint]

No Schroedinger's equation is not covariant. Schroedinger originally tried for a covariant equation and wound up with the equation now called Klein Gordon; this describes a boson and had, in the understanding of the 1920s, a problem of being unstable to negative energy solutions. So Schroedinger abandoned it and developed his non-relativistic equation.

The problem of a covariant equation that describes fermions like the electron was famously solved by Dirac.

 

[tavi_boada]

I'm a litle confused here. According to Jackson (Classical Electrodynamics, ed Wiley) the Schrödinger equation is invariant under the Galilean group of transformations (!??). However, according to Sakurai (Modern Quantum Mechanics) the new hamiltonian is the old transformed plus a term involving a first order derivative of the transformation operator (which is first order in time, and thus not zero).

Furthermore, I have read in other books comments on the new hamiltonian like, "this additional term ( the one involing the first order time derivative) plays the same role in QM as the forces of inertia do in classical physics"(?!).

How can that be!That something that depends linearly on time affects the equations of motion!No principle of relativity!?

Thanks for replying and sorry for the stubborness!ciao

 

[selfAdjoint]

Sure Schroedinger is Galilean invariant. So is Newton's mechanics. Schroedinger splits time and space; space is an operator but time is just a parameter. This is non-covariant from the git-go.

Special relativity covariance is a symmetry of the Poincare group.

I'm not prepared to discuss your other issues off the top of my head. I'll study Sakurai a little first.

 

[reilly]

Time dependent xforms are 1. used a lot in classical and quantum physics, and 2. often are mathematically difficult and opaque.

In classical mechanics, TDX are used, think of body centered coordinates and the theory of the spinning top; think of transforming away Corolius forces by appropriate rotational coordinates -- the most general form are the equations of motions in GR, identical to the most general equations derived from a Lagrangian in generalized coordinates. Necessarily, such xforms change energy/Hamiltonian if for no other reason than the KE must change.

In QM time dependent xforms are often used in QFT -- Heisenberg Picture, InteractionPicture -- and in magnetic resonance work --the so called Bloch Eq.s in rotating frames. These, by the way, are quite analogous to time dependent Contact Transformations in classical mechanics.(See Goldstein's Classical Mechanics, for example.

These topics are scattered all over the literature, probably can be found in advanced QM texts. Do a Google, and you will find more than you want to know.

BTW, Dirac's major insight in deriving the Dirac Eq. was to realize relativistic form invariance required first order spatial derivatives if the equation was to be first order in the time derivative.
Regards,
Reilly Atkinson