# Quantum measurement limit of decoherence and scattering: why are position pointer said classical?

In decoherence theory, we explain the decoherence by hamiltonian evolutions between a system and its environment. Calling $$H$$ the total hamiltonian, I have:

$$H=H_S + H_E + H_{SE}$$

A pointer state $$|s\rangle$$ is a state of the system $$S$$ for which the associated observable $$|s\rangle \langle s |$$ will commute with the total Hamiltonian. Then, coherences between pointer states will be killed under the Hamiltonian evolution.

One particular limit is the quantum-measurement limit for which $$H \approx H_{SE}$$. As typical interactions between a system and its environment are of the form:

$$H_{SE} = X \otimes E$$

Where $$X$$ is the position operator of the system, we figure out that the pointer states are the states $$|x\rangle$$, and thus the coherence between position eigenstates will be killed.

My question:

Now, a particular example of this situation if I understood is the quantum scattering. Basically our system $$S$$ (a molecule for example) will interact with environmental molecules under $$H_{SE}$$. And this interaction will kill superposition in the position basis as I explained. And from this we say that it matches the classical behavior in which the molecule doesn't have position superposition.

However I am puzzled by this. Indeed for me the classical limit should be a packet that is narrow both in position and momentum, like a coherent state.

Thus I am not sure to really understand why quantum scattering, as it is a particular case of quantum measurement limit, would fit as a good quantum to classical correspondance. If the particle is localized in space it is spread in momentum and thus highly non classical.

Does that mean that experimentally, we really see that molecules in a gaz for example are well defined in position, but very poorly in momentum (and then this explanation would match the decoherence theory) ?

A paper in which is explicitly said that collisional decoherence is a particular case of the quantum measurement limit (i.e hamiltonian dominated by interacting part) is The quantum-to-classical transition and decoherence on page 5, beginning of second column.

That's correct. The approximation $$H\approx H_{SE}$$ can be used to show the tendency for pointer states to be localized in position, but that's just an approximation. The kinetic term $$H_S$$ tends to make a particle's wavefunction spread out, which competes with the tendency of $$H_{SE}$$ to make it localized. The balance of these two trends results in the particle being somewhat localized in position, but still spread out enough that it can have a somewhat well-defined momentum, too, so its location can move like a classical particle's location does.
The "measurement limit" is the limit in which the decoherence in one observable's eigenbasis is so rapid and so complete that competing terms don't have a chance. This is the defining feature of a perfect measurement. Natural measurements, like the mutual "measurements" incurred by the molecules in a macroscopic sample of gas, are far enough away from being perfect that both position and momentum can be approximately well-defined (without violating the uncertainty principle $$\Delta x\Delta p\gtrsim\hbar$$, of course).
• Thank you. Allright then $H \approx H_{SE}$ is a "strong measurement" limit that in a way corresponds to what we expect from projection postulate (the state of the system is brutally projected somewhere). It corresponds to an apparatus super strongly interacting with a system. And it can only occur if the system is small enough (for a too big system $H_S$ would be too big and this limit couldn't be reached). Would you agree ? Jun 8, 2020 at 9:22
• Then, $H=H_S+H_{SE} (+H_E)$ where all term are comparable is something more natural in which spread will both be in momentum and space. It is a more natural situation in which the system is big (so $H_S$ is big), and it interacts with other systems around big as well. But $H_{SE}$ and $H_S$ are then somewhat comparable in size. Then does that mean the author makes a little mistake of interpretation by saying that $H \approx H_{SE}$ would match the scattering of molecules ? Jun 8, 2020 at 9:24
• @StarBucK The conditions under which the approximation $H\approx H_{SE}$ holds depend on the state as well as on things like the size of the system. I've mainly seen it used in a heuristic way: solving the equations with $H_S+H_{SE}$ is too hard, so we look at what would happen if only $H_S$ or only $H_{SE}$ were present and then use intuition to interpolate. Even in a scattering situation, $H\approx H_{SE}$ is only an approximation, and the approximation leaves room for the scattered molecule to still have an approximately-well-defined momentum (and an approximately well-defined position). Jun 8, 2020 at 13:06
• Hello again, sorry for my late comment but do you have in mind a scattering model I could look at in which we see that indeed the bare Hamiltonian plays a role into localizing in momentum as well. I searched for this for a long time unsuccessfully. In the book "Decoherence and the Quantum-To-Classical Transition", they add the bare Hamiltonian in the dynamic but the pointer states are still exactly position states. I could only see the competition between $H_S$ and $H_{SE}$ that leads to localization both in momentum and position for quantum Brownian motion. I am interested for the same for Jun 27, 2020 at 14:44