Measurement and the Conservation of Momentum

[Bernard Brault]

If you prepare a particle with a “relatively precise” momentum by the act of filtering or measuring its momentum. It’s state will collapse into a momentum eigenstate and the measured momentum will be the corresponding eigenvalue.
The position state will now be nearly uniformly spread out and the position unknown. But you can do a position measurement and the particle will register at a specific location but now it is the momentum that becomes unknown.

That seems to violate the principle of conservation of momentum. Prior to measuring position, we knew it’s monentum.

Possibly in both cases, it is the expected value of momentum that does not change.

But after measuring position, you could measure momentum again and get a different value then what you started with.
Where did the original momentum go?

 

[PeroK]

If you prepare a particle with a “relatively precise” momentum by the act of filtering or measuring its momentum. It’s state will collapse into a momentum eigenstate and the measured momentum will be the corresponding eigenvalue.
The position state will now be nearly uniformly spread out and the position unknown. But you can do a position measurement and the particle will register at a specific location but now it is the momentum that becomes unknown.

That seems to violate the principle of conservation of momentum. Prior to measuring position, we knew it’s monentum.

Possibly in both cases, it is the expected value of momentum that does not change.

But after measuring position, you could measure momentum again and get a different value then what you started with.
Where did the original momentum go?

The process of measuring position affects the momentum of the particle.

 

[Nugatory]

Where did the original momentum go?

The measurement involves an exchange of momentum between the particle and the measuring device, so the total momentum of the combined system is conserved.

 

[Bernard Brault]

The measurement involves an exchange of momentum between the particle and the measuring device, so the total momentum of the combined system is conserved.

In delayed measurements, where does the missing momentum goes?
It appears that cheating is allowed as long as you cannot catch the cheater.

 

[Nugatory]

In delayed measurements, where does the missing momentum goes?
It appears that cheating is allowed as long as you cannot catch the cheater.

You'll have to describe the exact measurement scenario before we can properly resolve the paradox that you're seeing. Nonetheless, we know how it has to turn out: the momentum operator commutes with the complete Hamiltonian of the quantum system, so one way or another momentum is going to be conserved.

You do want to be aware that popular descriptions of how the uncertainty principle allows for unobserved violations of classical conservation laws (quantum fluctuations of energy, virtual particles popping in and out of existence, that we can cheat on momentum conservation as long as the cheating is not detected) are terribly misleading.

 

[name123]

You'll have to describe the exact measurement scenario before we can properly resolve the paradox that you're seeing. Nonetheless, we know how it has to turn out: the momentum operator commutes with the complete Hamiltonian of the quantum system, so one way or another momentum is going to be conserved.

You do want to be aware that popular descriptions of how the uncertainty principle allows for unobserved violations of classical conservation laws (quantum fluctuations of energy, virtual particles popping in and out of existence, that we can cheat on momentum conservation as long as the cheating is not detected) are terribly misleading.

With regards to a single particle, I am not clear how under certain interpretations it can be thought to have a momentum value until measured. What I am thinking about is the Copenhagen Interpretation for example and the double slit experiment. I thought the "particle" was supposed to pass through all slits at the same time, and since getting to the slits involves velocities in different directions (or so I would assume) I was not clear on how prior to being measured the particle could be thought to have a momentum. So using the Copenhagen Interpretation is the "particle" in the double slit experiment thought to have a momentum prior to measurement?

 

[atyy]

https://arxiv.org/abs/1005.0569
Position Measurements Obeying Momentum Conservation
Paul Busch, Leon Loveridge