Re: The Heisenberg picture vs. the Schrodinger picture

On May 28, 11:03 pm, Edward Green <spamspamsp...@xxxxxxxxxxx> wrote:
If two things are mathematically equivalent, well then, they are
equivalent, and there is no point is wondering if one is more
"physical" than the other.  That said, the Schrodinger picture at
first glance seems to fit my prejudices of being more physical.

A "system" is generally a mathematical object -- an element in a
mathematical space -- which evolves according to some laws, which laws
in turn my include some of the character of "constraints".

E.g.: a (model) box of gas molecules may be taken to evolve according
to Newtonian dynamics, with the added conditions of collisions with
the walls, and possibly each other.

Such a system evolves in time even when its laws are stationary, but
we also may evolve the laws -- e.g., we may move in one wall of the
box as a piston.

The Schrodinger picture seems to follow this scheme pretty well.  A
system state evolves in time according to some laws (the Hamiltonian),
which itself may be stationary, or else also depend on time.  The
Heisenberg picture does not seem to follow this scheme: the "system
state" is now stationary, and the "operator" (the laws?) themselves
evolve in time -- according to some law.

This immediately seems to blur the distinction between evolution of
the system according to stationary laws, and evolution of the laws,
lumping it all together in the evolution of the operator.  I also have
a hard time seeing why -- in the box of gas example -- if we take the
initial configuration of the gas to be highly non-equilibrium, and the
contraints to be fixed, we would want to regard the "state" of the
system as invariant.  This just seems perverse, no matter how we
justify it mathematically.

Please comment.

The operator is not a "law" but represents an "observation".
So, Schroedinger and Heisenberg picture are equivalent ONLY if the
operator used in the Heisenberg picture has a complete spectrum and
does not project onto an true subspace. In the latter case, a
reconstruction of the full state evolution in the Schroedinger picture
from the Heisenberg picture would not be possible anymore!
.

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