Measurement as a unitary process

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Discussion Overview

The discussion revolves around the nature of measurement in quantum mechanics, specifically addressing whether the measurement process leads to a true eigenstate or if the system remains in a state where one coefficient is significantly larger than others after measurement. The scope includes theoretical implications of measurement, unitary evolution, and entanglement.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that after measurement, the system could be in a state where one coefficient is much larger than others, suggesting a form of unitary evolution rather than a true eigenstate.
  • Others argue that the unitary process involved in measurement does not change the relative amplitudes of the wave function's terms, maintaining that entanglement does not affect these amplitudes.
  • A participant questions the consistency of the measurement postulate, suggesting that new interactions imply a changing Hamiltonian, which complicates the understanding of measurement outcomes.
  • Another participant provides an example involving a Stern-Gerlach apparatus to illustrate that the entanglement created during measurement does not alter the initial superposition of states.
  • There is a suggestion that alternative approaches may be necessary to explain how a state can achieve a higher amplitude for one of the pointer states, but this is met with resistance and labeled as personal speculation.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of measurement on the wave function's coefficients, with some supporting the idea of significant amplitude differences and others rejecting this notion. The discussion remains unresolved, with competing views on the nature of measurement and its effects.

Contextual Notes

Limitations include the dependence on interpretations of quantum mechanics and the unresolved nature of the measurement postulate's implications. The discussion highlights the complexities involved in reconciling unitary evolution with measurement outcomes.

DaTario
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TL;DR
Hi All,

would it be reasonable to think that after the measurement the system has only experienced unitary evolutions and is in a state such that one of the coefficients is just much larger than the others?
Hi All,

We say that the Schroedinger equation stipulates a smooth and unitary evolution for the wave function, and that the measurement causes the wave function to collapse into one of the eigenstates of the operator that represents the observable parameter being measured by the apparatus. My question is the following: would it be reasonable to suppose that the apparatus introduced an interaction with the particle (system) that would lead the unitary evolution of the wave vector to be directed by multifurcations to a state such that the coefficient of a certain eigenstate would be many orders magnitude greater than the coefficients corresponding to the other eigenstates? In other words, instead of a true eigenstate, would it be reasonable to think that after the measurement the system has only experienced unitary evolutions and is in a state such that one of its coefficients is just much larger than the others?

Best regards,

DaTario
 
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DaTario said:
TL;DR Summary: Hi All,

would it be reasonable to think that after the measurement the system has only experienced unitary evolutions and is in a state such that one of the coefficients is just much larger than the others?

In other words, instead of a true eigenstate, would it be reasonable to think that after the measurement the system has only experienced unitary evolutions and is in a state such that one of its coefficients is just much larger than the others?
A true eigenstate which is a result of measurement undertakes unitary evolution by Hamiltonian of the system. A little time after measurement the state would become as you describe.
 
DaTario said:
would it be reasonable to think that after the measurement the system has only experienced unitary evolutions and is in a state such that one of its coefficients is just much larger than the others?
No.

The unitary process involved with measurement is an interaction that entangles the measured system with the measuring device. But this process does not change the relative amplitudes of any of the terms in the wave function of the measured system.
 
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PeterDonis said:
No.

The unitary process involved with measurement is an interaction that entangles the measured system with the measuring device. But this process does not change the relative amplitudes of any of the terms in the wave function of the measured system.
Ok, the entanglement has to do with the interaction between the apparatus and system. After this it seems somewhat contradictory (at least in most cases) to say that there is no change in amplitudes of the state of the particle, since a new interaction usually means that the hamiltonian is changing with time. It seems that the application of the measurement postulate is becoming more and more an inacceptable feature of the theory. So we have to find another way to arrive at a state with one of the 'pointer states' amplitude much higher than the others. Do you think it is well said?
 
DaTario said:
the entanglement has to do with the interaction between the apparatus and system.
That interaction creates the entanglement.

DaTario said:
After this it seems somewhat contradictory (at least in most cases) to say that there is no change in amplitudes of the state of the particle
The relative amplitudes of the terms in the expansion of the particle's state in some particular basis. The interaction between the particle and the measuring device does not affect those at all.

For a simple example, suppose we pass a spin-1/2 particle through a Stern-Gerlach apparatus. Suppose that, in the basis corresponding to the orientation of the apparatus, the particle's state is an equal superposition of spin up and spin down. The apparatus entangles the particle's spin with its momentum, so that there are two separate output beams, the "up" beam and the "down" beam, and the final state is an equal superposition of "spin up, momentum in the up beam" and "spin down, momentum in the down beam". The "equal superposition" part has not changed at all.

DaTario said:
a new interaction usually means that the hamiltonian is changing with time
Yes, during the measurement the Hamiltonian includes an interaction term between the measured system and the measuring device that is not present before or after the measurement.

In the Stern-Gerlach example above, the interaction term in the Hamiltonian that entangles the particle's spin with its momentum is only present while the particle is passing through the apparatus, not before or after.

DaTario said:
It seems that the application of the measurement postulate is becoming more and more an inacceptable feature of the theory
No, your understanding of how measurement is modeled in QM as it currently stands is flawed.

DaTario said:
So we have to find another way to arrive at a state with one of the 'pointer states' amplitude much higher than the others.
This is personal speculation and is off limits here.

DaTario said:
Do you think it is well said?
No. See above.
 
Thank you PeterDonis, I understand my discussion is off limits here.
 
DaTario said:
I understand my discussion is off limits here.
And with that, this thread is closed.
 

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