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Let's say I have a wavefunction $\Psi$ which can be decomposed into a sum of it's energy eigenstates:

$$ \Psi = a|1\rangle + b|3\rangle + c|8\rangle + d|10\rangle$$

Where, of course, $|a|^2 + |b|^2 + |c|^2 + |d|^2 =1 $.

And let's say I have a device which can measure the energy of this wavefunction. Unfortunately, the device has an inherent uncertainty of $\pm3$.

I measure $\Psi$ and find it to have an energy of $7\pm3$. After my measurement the wavefunction has "collapsed" (to some extent?). I can think of a few possibilities for post-measurement $\Psi$:

1) $\Psi$ really is in either $|8\rangle$ or $|10\rangle$. The problem statement is wrong: any uncertainly in energy measured is a laboratory issue. It must have an exact physical answer.

2) $\Psi = c'|8\rangle + d'|10\rangle$

Where, $|c'|^2 + |d'|^2 =1 $.

I might even go so far as to say $c' = c/(|c|^2 + |d|^2)^{\frac{1}{2}}$

My immediate answer: the measurement simply eliminated the possibility of the $|1\rangle$ and $|3\rangle$ eigenstates.

3) $\Psi = e|4\rangle + f|5\rangle + g|6\rangle + h|7\rangle +k|8\rangle + m|9\rangle + n|10\rangle$

The "measurement" isn't really a measurement; just a disruption to the wavefunction.

Would any of my answers be correct? Is what I've described not a 'measurement' according to QM? What would it be then?

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    $\begingroup$ Search term: 'weak measurement'. $\endgroup$ – dmckee Sep 26 '17 at 19:39
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    $\begingroup$ or POVM, which were invented for the purpose you describe. $\endgroup$ – ZeroTheHero Sep 26 '17 at 19:54
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    $\begingroup$ i.e. positive-operator valued measure. Come on! Question authors systematically get scold when they don't spell out their abbreviations ;-) $\endgroup$ – user154997 Sep 26 '17 at 20:27
  • $\begingroup$ @LucJ.Bourhis oh well... shame on me then, although a G-search of POVM immediately pulls up the wiki page: en.m.wikipedia.org/wiki/POVM $\endgroup$ – ZeroTheHero Sep 26 '17 at 20:45
  • $\begingroup$ True indeed! Surprising… $\endgroup$ – user154997 Sep 26 '17 at 21:00
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A Positive Operator Valued Measurement (POVM) to describe your measurement could be given by elements $$M_{i}=\frac{1}{3+\min\{i,4\}}\sum_{|j-i|\leq 3} |j\rangle\langle j|$$ for $i=1,2,3,...$, which are positive semidefinite and sum up to the identity. Maybe in practice not all outcomes within your $\pm 3$ uncertainty have the same probability, but I assume that's the measurement you describe in your question.
The post-measurement state following outcome '7' is $$|\Psi'\rangle\propto M_7|\Psi\rangle=c|8\rangle + d |10\rangle,$$ exactly as you suggest in 2).

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