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There is a picture usually depicted in introductory quantum mechanics about how the probability pattern appears in the classical limit, that the oscillation is too fast and that it cannot be resolved by the experimental apparatus, hence the detector will detect only the average value.

Image of oscillation probability

I do not see how the detector "decides" to give the average value if the observed value oscillate too fast between 2 numbers!

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What kind of detector is it? – David Z Feb 11 '12 at 1:42
David's question is absolutely key here. Many detectors are moderately complicated even in their idealized form, and as realized they can be even more complicated. – dmckee Feb 11 '12 at 2:04
@DavidZaslavsky Please see this familiar picture – Revo Feb 11 '12 at 5:23
The context of this picture is explained in this link. It's a picture of a quantum SHO state. The text explains the reason (uncertainty principle) why the detector can't resolve the peaks. – twistor59 Feb 11 '12 at 8:23
If you superpose a few neighboring states, the oscillations wash out even without a detector. The goal of a detector is to set up a collapsed state which is not a pure energy eigenstate, but a superposition of several nearby ones. – Ron Maimon Apr 9 '12 at 0:04

An analog pointer will simply be too slow to follow a high frequency motion. Thus it follows variations in the driving force only when there is a consistent trend during the response time.

No decision is needed, just as a dumb student doesn't decide not to follow a lecture - he can't!

But a digital detector (e.g., a digital LED display) may well oscillate between two numbers when the mean isn't close to one of the representable numbers. I have seen it happen....

On a formal level, a book by Grabert (Projection operator techniques in nonequilibrium statistical mechanics) discusses how to get reduced (classical or quantum) descriptions from a quantum description involving high frequencies.

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The question is more complicated than it appears. Take the example of a light beam and a photodetector.It is instructive to look at how energy is distributed among the various modes (frequency components) of the radiation. The detector responds only to a certain specific subset of frequency and this response is conditioned by the response time of the detector.

In other words, a photodetector is an intensity correlator. Hence, faster detectors can resolve finer details or conversely, slower detectors wash out the details and you get some mean value of intensity measurement.

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This is complete nonsense. The probability amplitude is not oscillating at all in the classical limit, rather it is a bump at a certain position with a certain momentum.

The example picture you give is of a moderately large n energy eigenstate. In this energy eigenstate, the particle is bouncing back and forth smeared out over the whole line, so that it continuously interferes with itself going in opposite directions, giving rise to a real valued wavefunction with maxima and minima.

For a real particle whose position is seen by continuous measurement by photons, like a classical mass on a spring, the energy is indefinite. The known position and momentum make a smeared out wavepacket when you measure all the outgoing photons (otherwise it's a density matrix). The wavefunction appropriate to such a thing is a localized wave-packet of width comparable to the wavelength of light, which has a complex amplitude that varies as the momentum. A model for such a state is

$$ \psi(x) = e^{-a x^2 + ip\cdot x}$$

You can write such a state as a superposition of nearby energy eigenstates, and it doesn't have any minima. The minima appear only when a wave diffracts off a reflecting barrier and interferes with itself going the other way, something which is only relevant near the classical turning points.

If you actually had a particle carefully prepared in the state you pictured, a detector which looked to see if the particle was at a minimum would never find the particle. If it looked near the maxima, it would find the particle at a higher rate than the classical density. The average density of the particle is still given by

$$ \langle|\psi(x)|^2 \rangle = {T\over v(x)}$$

Where T is the classical period and v is the classical velocity at position x. But the interpretation that this is a classical motion is not correct, and is being misused by the author of the book.

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