Incomplete collapse of wavefunction

[hilbert2]

I'm otherwise pretty comfortable with the postulates of quantum mechanics, but I find it difficult to understand situations where a measurement causes only incomplete collapse of the wave function...

Suppose we have an electron in a state described by some wave function. Then we measure its position. If the QM postulates were taken literally, the measurement should collapse the wavefunction into a position eigenstate, a Dirac delta function, that is. But that would be against energy conservation, because an electron in a position eigenstate can have *any* value of kinetic energy with equal probability.

I think the position measurement collapses the wavefunction into some kind of an almost-position-eigenstate, like a gaussian spike that is narrow if the position was measured very accurately and wide if the measurement was inaccurate. To measure the position accurately, we should scatter something with very short De Broglie wavelength and very high energy off the electron, which explains the uncertainty in the electrons momentum after the measurement.

Probably an inaccurate measurement of the total energy of a molecule could also collapse the molecular wavefunction into a state that is still a superposition of several eigenstates of the Hamiltonian.

I don't really understand this, because the postulates don't tell how to handle these kinds of situations. I think this has something to do with the concept of a 'weak' or 'nondemolition' measurement.

 

[kith]

The Born rule can be understood as relating the probability of an experimental outcome to the expectation value of the corresponding projection operator: P(a_i) = E_ψ(|a_i><a_i|) = |<a_i|ψ>|². I don't think there's a problem to apply this to situations of incomplete collapse / higher dimensional projectors.

 

[hilbert2]

^ But is the state of the electron after a position measurement even a projection of the original state? If the complex phase of the new wavefunction behaves differently, it would not be one.

 

[DrChinese]

I'm otherwise pretty comfortable with the postulates of quantum mechanics, but I find it difficult to understand situations where a measurement causes only incomplete collapse of the wave function...

Suppose we have an electron in a state described by some wave function. Then we measure its position. If the QM postulates were taken literally, the measurement should collapse the wavefunction into a position eigenstate, a Dirac delta function, that is. But that would be against energy conservation, because an electron in a position eigenstate can have *any* value of kinetic energy with equal probability.

I think the position measurement collapses the wavefunction into some kind of an almost-position-eigenstate, like a gaussian spike that is narrow if the position was measured very accurately and wide if the measurement was inaccurate. To measure the position accurately, we should scatter something with very short De Broglie wavelength and very high energy off the electron, which explains the uncertainty in the electrons momentum after the measurement.

Probably an inaccurate measurement of the total energy of a molecule could also collapse the molecular wavefunction into a state that is still a superposition of several eigenstates of the Hamiltonian.

I don't really understand this, because the postulates don't tell how to handle these kinds of situations. I think this has something to do with the concept of a 'weak' or 'nondemolition' measurement.

I don't know if I would use the term "incomplete collapse" by itself. It is always relative to some basis. This is much easier to see with something like spin. If you know an electron's x-spin, obviously its z-spin is completely uncertain. But if you measure at 45 degrees towards z, you learn something about the z spin. Is that full or partial collapse? Depends on what basis you are referring to. Obviously it is full collapse in 1 basis and partial in many others.

So you could say that the issue is referring to position as a "real" observable versus other observables that are combinations of position and momentum. In a sense, they are all equally real.

 

[hilbert2]

^ According to the postulates of QM, collapse should happen relative to the eigenbasis of the observable that was measured.

Nice idea making operators that are combinations of position and momentum... One just has to make sure they are hermitian (not just any combination is).

EDIT: I suppose you were trying to say that when an incompletely accurate position measurement is made, one is actually measuring some observable that gives some information about both position and momentum.

 

[kith]

But is the state of the electron after a position measurement even a projection of the original state?

According to the postulates of QM, collapse should happen relative to the eigenbasis of the observable that was measured.

Doesn't this answer your question? You expand your initial state in the eigenbasis of the observable and the final state is one of these eigenstates. That's a projection.

An easy way to picture a position measurement with finite resolution is a one-dimensional CCD. The ordinary position operator is X = ∫dx x|x><x|. The CCD operator would be something like X_Δ = Ʃ_i x_i|x_i><x_i| where the x_i correspond to the CCD cells, Δx is the resolution and |x_i><x_i| is the projection operator for the cell x_i. It could be written as |x_i><x_i| = ∫dx ^x_i+Δx_xi-Δx|x><x|.

 

[hilbert2]

^ Yes I understand that. It's like a 'smoothed' position operator and its eigenstates are only approximations of a delta function. I guess that is the real observable that is being measured in a finite-resolution position measurement.