Measurement as a unitary process

[DaTario]

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

 

[anuttarasammyak]

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.

 

[PeterDonis]

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.

 

[DaTario]

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?

 

[PeterDonis]

the entanglement has to do with the interaction between the apparatus and system.

That interaction creates the entanglement.

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.

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.

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.

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.

Do you think it is well said?

No. See above.

 

[DaTario]

Thank you PeterDonis, I understand my discussion is off limits here.

 

[PeterDonis]

I understand my discussion is off limits here.

And with that, this thread is closed.