Zeno Effect: Quantifying the Impact of Observation on Events

[Dadface]

[this thread was forked from https://www.physicsforums.com/threads/zeno-effect-standard-derivation.951929/; the "equation above" can be found there]

If the quantum Zeno effect is real and if it can be quantified shouldn't the maths be specific to the event being observed? For example radioactive decay has been discussed recently on this forum and a relevant quantification is based on the assumption that the decay rate is proportional to N the number of radioactive nuclei present at a particular instant and also depends on the nature of the particular isotope being considered (expressed by a constant λ).

We can write: dN/dt = - λN. It's a quantification that seems to make sense and most importantly it seems to be well borne out by observations. But I can't believe that the decay rate can be altered by making observations that don't somehow change or disrupt the system and I would expect the equation above and any resulting variations of it to be expressed by some relevant mathematics. The maths used above seems to be of a general nature and non specific.

 

[Nugatory]

We can write: dN/dt = - λN. It's a quantification that seems to make sense and most importantly it seems to be well borne out by observations. But I can't believe that the decay rate can be altered by making observations that don't somehow change or disrupt the system

It doesn't. If we had a large number of independent particles, each being separately and individually observed in its own independent experimental apparatus, then the quantum Zeno effect would apply to each one (the quantum state of the entire collection would be the tensor product of the state's). But that's not the situation here; it's obviously unrealistic to prepare that quantum state and the $N$ that we're measuring isn't a Hermitian operator projecting a pure quantum state into an eigenstate of anything. Thus, the quantum Zeno effect isn't expected to and doesn't apply, just as it I can't arrest my childrens' growth by periodically marking their height against the door jamb.

 

[Dadface]

Thank you Nugatory. My present feeling is that there are no systems where the Zeno effect applies and the whole concept seems daft to me. I will probably look into it further when I get some spare moments.

 

[Nugatory]

My present feeling is that there are no systems where the Zeno effect applies

There are many. Google for "quantum Zeno effect observed"

and the whole concept seems daft to me

It might seem more sensible if you consider that all measurements are interactions between the system being measured and the measuring device, and that an interaction generally changes the system state. Arrange your experimental setup so that the measurement is likely to put the system in a particular state, and it won't be surprising that repeated measurements will tend to leave the system in that state.

 

[Dadface]

It makes sense but the changes needed to put a system into a particular state depends on what that system is and I'm surprised that this doesn't seem to be accounted for in in the maths. Anyway, I will do some searches and find out more of what it's all about. Thank's again.

 

[Nugatory]

It makes sense but the changes needed to put a system into a particular state depends on what that system is and I'm surprised that this doesn't seem to be accounted for in in the maths.

It is accounted for in the math, which shows a state that is a sum of different eigenfunctions before the measurement being projected by the measurement into a single eigenfunction.

 

[Dadface]

As I said in my opening post the maths seems to be of a general nature and not specific to different events. I don't yet know enough about events that display Zeno effects but I think there are different events that are reputed to do so. And although some different events may share common features they are still different and as such may need different physics to describe them. But the maths doesn't seem to take these differences into account.
Perhaps the general treatment boils down to a case of making simplifying assumptions? I'm going to leave this for now and possibly come back to it after a bit more googling. Thanks for your feedback

 

[Agrippa]

If the quantum Zeno effect is real and if it can be quantified shouldn't the maths be specific to the event being observed?

If the effect applies to all quantum mechanical observables, then no, there's no reason why the math should be specific to some particular observable.

But I can't believe that the decay rate can be altered by making observations that don't somehow change or disrupt the system

Observations (measurements) do disrupt systems in quantum mechanics (how exactly this comes about is a controversial issue in the foundations of quantum mechanics, but it does come about).

There's a simpler way of understanding the QZE if you assume that measurement causes collapse (i.e. the von Neumann interpretation):

Assume that a system begins in an eigenstate |1> of some observable O, and will evolve to eigenstate |2> of that same observable O, under Schroedinger evolution. Since |1> and |2> are orthogonal, and since Schroedinger evolution rotates the state vector continuously and without jumps, it follows that Schroedinger evolution must rotate the state vector through superpositions of |1> and |2> before it can put the system into eigenstate |2>.

The first superposition attributes effectively all of the squared amplitude to the initial eigenstate (c₁|1> + c₂|2> ; |c₁|²>>|c₂|²). As time passes, the squared amplitude is passed over to the |2> eigenstate. Hence, the faster one does a measurement of observable O, the more likely the system will collapse back to initial state |1>.

Plug in any quantum mechanical observable for O.