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Going through David J. Griffiths' Quantum Mechanics book today, I read the following:

If the operator $Q(\hat x,-i\hbar\ \partial_x)$ has a continous spectrum with eigenvector $f_y(x)$ corresponding to the eigenvalue $y$, the probability that the found value after making a measurement is between $z$ and $z+dz$ is $|c(z)|^2dz$, where $$\Psi(x)=\int_{-\infty}^{+\infty}dy\ c(y) \ f_y(x)$$ Thus in the measurement of an operator with a continuous spectrum, the wave function collapses to a spike in the neighbourhood of the measured value, depending on the precision of the measuring apparatus.

And hence my question: I interpreted the last statement to mean that when the measuring apparatus is really precise, it perturbs the original system very heavily, and thus causes what we would call more of a spike, leading to a very low spread in the wave function instantly after the measurement.

Is this correct?

Physiker
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2 Answers2

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Yes, that's correct — at least that's the correct interpretation of what Griffiths is saying. I described another perspective on the same idea in another answer. It's a good enough model for many practical purposes.

A more realistic model would include the measurement equipment as part of the quantum system, so that the observable $Q$ becomes entangled with the measurement equipment as a result of the measurement. We eventually still need to invoke the projection postulate to account for whatever outcome we actually experience, but instead of projecting directly onto an interval of eigenstates of $Q$, we project onto an eigenstate of a downstream observable, like a digital readout on the measurement equipment. This accounts for the measurement's finite resolution in a more natural way, because it gives quantum theory a chance to describe the upstream part of the measurement (the direct interaction with $Q$) as a physical/unitary process instead of as an artificial/nonunitary projection.

But that's usually mathematically too difficult. The simplified approach described by Griffiths is much more practical, even though it's not as natural.

Chiral Anomaly
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You end with

", leading to a very low spread in the wave function instantly after the measurement"

Τhe wave function after the measurement is a different wave function, as the measurement changes the boundary conditions . That is why one talks of "collapse".

If the measurement method has a small error, assume it a statistical gaussian, one gets a spike for the value. The smaller the error the sharper the spike in the measurements.

anna v
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