Understanding Environment-Induced Superselection in Quantum Systems

[fanieh]

What exactly is the meaning of environment being slow?

Maximillian Schlosshauer book Decoherence mentions this:

"Let us now consider the situation in which the modes of the environment are "slow" in comparison with the evolution of the system. That is, we assume that the highest frequencies (i.e. energies) available in the environment are smaller than the separation between the energy eigenstates of the system. In this situation, the environment will be able to monitor only quantities which are constants of motion. In the case of nondegeneracy , this quantity will be the energy of the system, thus leading to the environment-induced superselection of the energy eigenstates for the system (i.e., eigenstates of the self-Hamiltonian of the system)."

1. If the energies in the environment is smaller than the separation between the energy eigenstates of the system, does it mean it is accepted or not (as quantum as I know only unit can be accepted in QM)?

2. Why did it mention "constants of motion".. it is talking about position? Or is momentum consider constants of motion, why?

3. What does "nondegeneracy" mean?

4. Can you give an example where the environment is "slow" and energy of the system is chosen?

Help appreciated much.

 

[PeterDonis]

What exactly is the meaning of environment being slow?

The quote you give answers this: "[W]e assume that the highest frequencies (i.e. energies) available in the environment are smaller than the separation between the energy eigenstates of the system..."

If the energies in the environment is smaller than the separation between the energy eigenstates of the system, does it mean it is accepted or not

What do you mean by "accepted"?

Why did it mention "constants of motion".. it is talking about position?

No. A constant of the motion is an observable that does not change with time; i.e., it commutes with the Hamiltonian. The Hamiltonian itself, i.e., energy, is always one such observable. There may be others, depending on the specific system.

What does "nondegeneracy" mean?

It means each eigenstate of the observable (energy in this case) has a different, unique eigenvalue. So in the case of energy, it means every energy eigenstate has a different energy. Some systems have multiple energy eigenstates with the same energy; that complicates the analysis.

Can you give an example where the environment is "slow" and energy of the system is chosen?

Does the book give any?

 

[fanieh]

The quote you give answers this: "[W]e assume that the highest frequencies (i.e. energies) available in the environment are smaller than the separation between the energy eigenstates of the system..."

But in what sense are "frequencies" related to "energies".. is it talking about frequency of photons?

What do you mean by "accepted"?

Photons emit by quanta (whole numbers) and not like 1.5 or 1.6888... is this not related to the "... are smaller than the separation between the energy eigenstates of the system"... so I understood bigger than the separation as no longer whole number (quantum)... is it like the concept where only certain wavelength can fit the orbital of atoms?

No. A constant of the motion is an observable that does not change with time; i.e., it commutes with the Hamiltonian. The Hamiltonian itself, i.e., energy, is always one such observable. There may be others, depending on the specific system.
It means each eigenstate of the observable (energy in this case) has a different, unique eigenvalue. So in the case of energy, it means every energy eigenstate has a different energy. Some systems have multiple energy eigenstates with the same energy; that complicates the analysis.
Does the book give any?

I think it mentioned about sugar handedness or the fact atoms are hidden inside molecules hence can't interact (line of sight) with the environment preferred basis and so position not chosen... but I just want other or actual examples from other people.

 

[PeterDonis]

in what sense are "frequencies" related to "energies"

$E = \hbar \omega$. This applies to anything, not just a photon.

Photons emit by quanta (whole numbers) and not like 1.5 or 1.6888... is this not related to the "... are smaller than the separation between the energy eigenstates of the system"...

The quote you give doesn't say anything specifically about photons, so I don't know why you are thinking of them specifically. The point is that for the system to go from one energy eigenstate to another, the energy has to come from somewhere, and the only place available is the environment. If the highest energies available in the environment are much smaller than the separation between the system's energy eigenstates, then the environment can't supply enough energy for the system to go from one energy eigenstate to another; so the system will just stay in the eigenstate it starts out in (which will usually be the ground state).