Ballentine on the quantum Zeno paradox

[bhobba]

TBH, I'm a bit disappointed that you're being so short with me. I've tried to help plenty of other people on PF in the past but I hardly ever request assistance for myself.

Sorry mate - accept my apologies.

Its just I seem to go through the same stuff over and over - and you do indeed help me and others quite a bit.

So fire away and I will do my best.

Thanks
Bill

 

[bhobba]

What do you mean by "physical continuitity". (A link to a paper or previous thread is fine if you can't be bothered explaining.)

Its associated with the idea of 'filtering' measurements from Ballentine - page 246. These are measurements that don't destroy the state but rather change it as a result of the measurement.

I thought he used my argument but can't find it there so it may be my imagination.

Here it is. From physical continuity we expect the same measurement just after such a measurement will give the same result. Also we expect the state to change insignificantly. Let the observable of the observation be ∑ yi bi><bi|. Suppose yi is the outcome. If P is the state after the observation then Trace (P |bi><bi|) = 1. A little math shows P must be|bi><bi| (I will post the detail of you like). This is the projection postulate ie the state after the observation is the corresponding eigenvector of the outcome.

Thanks
Bill

 

[strangerep]

Its associated with the idea of 'filtering' measurements from Ballentine - page 246. These are measurements that don't destroy the state but rather change it as a result of the measurement.

[...] From physical continuity we expect the same measurement just after such a measurement will give the same result. Also we expect the state to change insignificantly. Let the observable of the observation be ∑ yi bi><bi|. Suppose yi is the outcome. If P is the state after the observation then Trace (P |bi><bi|) = 1. A little math shows P must be|bi><bi| (I will post the detail of you like). This is the projection postulate ie the state after the observation is the corresponding eigenvector of the outcome.

OK, I think I understand what you're talking about, though perhaps "idempotency of filtering operators" may be a more specific phrase than "physical continuity" in this context.

It's curious that Ballentine [p246, bottom] cites the Stern-Gerlach setup as example of "measurement of the filtering type, in which the ensemble of systems generated by the ρ-state preparation is separated into subensembles according to the value of the dynamical variable R". He apparently considers that "measurement" has occurred immediately after the beam has passed the non-uniform magnetic field and divided into 2 beams. However, I maintain that "measurement" has not occurred until a silver atom (or whatever) has impacted a final detector and caused a count to increment. Before that, one has merely applied an operator (implemented by interaction with the magnetic field), obtaining a new state but not yet a number.

 

[atyy]

It's curious that Ballentine [p246, bottom] cites the Stern-Gerlach setup as example of "measurement of the filtering type, in which the ensemble of systems generated by the ρ-state preparation is separated into subensembles according to the value of the dynamical variable R". He apparently considers that "measurement" has occurred immediately after the beam has passed the non-uniform magnetic field and divided into 2 beams. However, I maintain that "measurement" has not occurred until a silver atom (or whatever) has impacted a final detector and caused a count to increment. Before that, one has merely applied an operator (implemented by interaction with the magnetic field), obtaining a new state but not yet a number.

I'm not sure if this is what Ballentine had in mind, but I pictured here something like Fig 2 of http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/zurek%20phys%20today.pdf , where he puts a detector in the path, and then has decoherence and collapse so that the detector reads a definite outcome. Ballentine mentions decoherence on p245, so perhaps he's thinking of something similar on p246.