Partial quantum / partial classical?

[exmachina]

I came up with the following thought experiment:

Let's assume we have a long carbon polymer (see below) in vacuum somewhere, and there's some wavefunction (nuclear/electron, let's assume born-oppenheimer applies) governing the behaviour of this molecule.

/\/\/\/\/\/\/\/\/\/\/\/\

I shine a highly focused beam so it hits only the three connected carbons on the left side of the polymer.

So, would we expect that:

1) the left 3 carbon's electronic wavefunction collapses?

2) the other carbons (where light doesn't shine on) still exhibits wave-like behaviour?

 

[alxm]

Well, if you view the total electronic wave function for the entire molecule in terms of single-electron wave functions (Slater determinant), then the single-electron wave functions for whichever electrons were 'measured', will 'collapse'.

 

[exmachina]

so it then does become possible for parts of the molecule to behave "classically" while other parts quantum mechanically? what about the nuclear wavefunction? what would cause that to collapse?

 

[f95toli]

You have to be careful when you talk about "collapse" here. Shining a beam of light onto something does not in itself constitute a measurement. it would certainly be possible (albeit complicated) to describe the situation described here using only quantum mechanics (well, quantum electrodynamics).

Look up e.g. the Jaynes-Cummings Hamiltonian to see an example of a simpler (singe atom) but conceptually similar system.

 

[exmachina]

is the QED formulation applicable primarily to systems where there is light? what about in a closed container where no light can penetrate?

 

[f95toli]

Well, QED deals -as the name implies- with electromagnetics, so I guess the answer is yes.
QED is bascally the "quantum version" of classical electromagnetics and is used to describe things like lasers etc

If you were able to put an object in a "perfect box" with no dissipation, there would not be any decoherence (i.e. no "collapse", which btw. is a bit of a misleading word)

 

[fermi]

I came up with the following thought experiment: ... I shine a highly focused beam so it hits only the three connected carbons on the left side of the polymer.
So, would we expect that: 1) the left 3 carbon's electronic wavefunction collapses? ...

I believe your experimental set-up plus the questions that follow are not consistent with the Uncertainty Principle. If you need a beam that can be so tightly focused that you make sure that it hits the leftmost three carbon atoms but not the other ones, then the wavelength of your photons should be in the order of three times the diameter of carbon atoms. That's about 10 Angstoms, give or take a few. So the wavelength must be 10A^o or smaller. If it is 10A^o, then the energy of the photons must be 20 kEV or larger. These are very energetic x-rays, of which energy is 20000 larger than the molecular binding energies. This is even 40 times larger than the binding energy of the innermost electrons in the Carbon atoms.

Therefore, it is most likely that the light will pass through the molecule with no effect. Occasionally however, it will interact with the molecule so violently that it will destroy it completely. Typically the photon will knock off one or more electrons (it may even knock off an inner shell electron which starts a cascade of transitions and generates many more photons and many more electrons flying all over the place and doing their own secondary damage.) What I am saying therefore is that the molecule cannot react coherently with a very energetic photon. The fact that the carbon atoms happen to be bound together in a molecule is almost irrelevant when you get whacked by a 20kEV photon.

If on the other hand, you reduce the photon energy down to 1.0 eV, then you avoid this problem I explained above, and you can start talking about the "collapse". But in this latter case, you have no right to say that the photons hit only the leftmost three carbon atoms. You are hitting the whole molecule coherently, even if it has a thousand carbon atoms in a chain.

 

[exmachina]

if we take the case that a 1 eV beam of photon IS hitting every part of the molecule,

1. how would one begin to model collapse?
2. how would this collapse change if we increase the energy of the photon?

 

[f95toli]

Again, if all you have is a beam of light hitting a molecule there is NO "collapse" of any sort. Remember that e.g. all photo-induced chemical reactions are "quantum" in some sense.