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Warning: This is a highly hypothetical question.

I am bothered with Dirac's description of the system when making a measurement. Without quoting his statement (from The Principles of Quantum Mechanics, Dirac, 1930), he simply states that one of the axioms of quantum mechanics is that regardless of the current state of the system, once we make an observation, the system forcefully "jumps" to an eigenstate of the observable. It, particularly, jumps to the eigenstate associated with that specific eigenvalue that we measured.

Now, my question goes as follows, let us assume a hypothetical continuous observer; that is, an observer that observers (measures) the system continuously without a pause. Let us assume, furthermore, that we observed the system in one state, and hence ensured that it jumped to an eigenstate associated with the measurements (eigenvalue), according to Dirac. Let us assume, additionally, that we start applying external effects to the system (probably forces) so as to change the system's state, which is admissible by quantum mechanics. Let us assume, now, that we let the continuous observer observe the system while applying the external forces (agents or any physical effect). We have two possibilities at this point: It is either the case that the state will NOT change, or that it will ACTUALLY change.

If the state changes while continuously observing, then Dirac's axiom fails miserably and this postulate of quantum mechanics collapses.

On the other hand, if the state remains the same while observing, then by making the external applied forces arbitrary, we can generate any possible state of the system. But then, this means that regardless of the state of the system, the measurement will be always the same with this specific observer (measurer). However, since we know that the system can actually have different measurements when measured at different times and in different states (then by different observers), we can say that the measurement has nothing to do with the state of the system. Rather, it is associated with the observer. Hence, by studying the observer itself, we can get the measurement without bothering to measure the system.

To save you some time, my photonics professor said that there is nothing like a continuous observer. His answer did not really convince me for I am creating a hypothetical situation.

Norbert Schuch
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Con
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    You might be interested in https://en.wikipedia.org/wiki/Quantum_Zeno_effect – Ozzy Feb 08 '24 at 13:39
  • Related - Does the collapse of the wave function happen immediately everywhere? – mmesser314 Feb 08 '24 at 15:38

  • "Dirac's axiom fails miserably and this postulate of quantum mechanics collapses." Please elaborate. How does the state changing in time imply Dirac's axiom fails miserably? The axiom does not necessarily require that eigenvalue and its eigenstates are constant in time. E.g. when external static/magnetic electric field varies in time, atom Hamiltonian eigenvalues change in time.

     – Ján Lalinský Feb 08 '24 at 23:09
  • @JánLalinský The state changing in time according to quantum theory is deterministic. What I mentioned in my question is "arbitrary" external forces. Think about it as a limit case. If I observe the value of the system then it "jumps" to some state. Then if I look again immediately at it, it jumps to that state (the same in this case). If I take this process to the limit (since the observer is continuous), then I will keep generating that specific state although I am applying an external force that is supposed to change the state. If the value changes, then my observation did not "force"... – Con Feb 11 '24 at 17:37
  • @JánLalinský ... the system to "jump" to that specific state again and again, as Dirac's axiom suggests. The nuance here is the word "continuous." I know that this hypothetical case is not easy to imagine, but it brings an interesting case. – Con Feb 11 '24 at 17:38

  • Then if I look again immediately at it, it jumps to that state (the same in this case) This is so only at the same time, but not after some time elapses. The continuous observation only implies that the state will be an eigenstate at all times, but it does not imply that this eigenstate does not change in time. External field changing in time means the eigenstate will change in time, and the actual state will follow that eigenstate in time.

     – Ján Lalinský Feb 11 '24 at 22:57
  • Note that despite the frequent (continuous) measurement, the Hamiltonian eigenstate does change in time, because it does not depend on any measurement, only on the Hamiltonian, which changes in time due to external field changing. Thus it is not possible to keep the system in the old eigenstate of quantity $H$ by measuring $H$, because after non-zero time, the eigenstate changes, and after the measurement at time $t_0+\Delta t$, we can only obtain eigenstate of $H_{t_0+\Delta t}$, which differs from the eigenstate of $H_{t_0}$. – Ján Lalinský Feb 11 '24 at 23:05

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What happens after a measurement is controversial and different accounts of what is happening in reality will give different answers. My answer presupposes that quantum mechanical equations of motion such as the Schrödinger or Dirac equation are accurate descriptions of reality.

A measurement is an interaction between the measurement device $M$ and a system $S$ that produces a record: a piece of information about the system that can be copied, discussed, criticised etc. A measurement interaction takes place suppresses interference between the different values of the measured observable:

https://arxiv.org/abs/0707.2832

As a result the different values of the measured observable evolve autonomously and to reflect this you update the relative state of the system concerned:

https://arxiv.org/abs/2008.02328

There are other theories that provide different equations of motion and so different accounts of measurement such as spontaneous collapse or pilot wave theories:

https://arxiv.org/abs/2310.14969

https://arxiv.org/abs/2212.12175

alanf
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