A Thought Experiment in the HUP

1. Nov 15, 2012

hollowsolid

A number of threads in this forum oscillate between pure mathematical descriptions of the
HUP and QM generally, while others *try* to provide a physical, visualisable causal model to explain them. Heisenberg's original quantum microscope explanation for the HUP was such an example which concluded that any measurement of a system disturbed it and affected subsequent measurements. However the relevance and accuracy of this explanation are doubted in many quarters.

The following thought experiment offers a slightly different explanation which a number of forum readers would recognise and have offered in various forms. However I think there is
a subtlety to this one that does make it different.

Consider a particle in free motion which collides with a device that has the following components:

(a) a known mass which the particle collides with it at a perpendicular angle such that
the particle does not glance off but transfers all of its momentum to the target mass.

(b) a scale next to the target mass in our units of choice and which serves to record
the distance the target mass moves after the free particle collides.

(c) The units of distance and of mass are roughly commensurate i.e. we are not taking
about very different scales here such as X ray wavelengths of distance vs Kg of mass.

Consider the following extremes:

(a) the particle impacts and stops dead without moving the target mass. In such a
case we know with great certainty where the particle is at point of impact but we know
nothing about its momentum since the scale has not budged i.e. the measurement device
has not registered because the particle has not engage it.

(b) The particle moves the target mass a considerable distance and in the presence of
various resistive forces (distant gravitational sources etc) finally comes to rest relative
the scale. In this situation we would have gained considerable accuracy about the
particle's momentum but lost accuracy on the exact position at which it had this momentum.

In general, the longer the period that the target mass is moved, the more that resolution of
position is lost at the expense of gains in measurement of momentum and the shorter
the period the target mass is moved the greater the resolution of position at the expense of accuracy in momentum.

Or stated differently,

(a) if the target mass is relatively small relative to the particle, then the scale will return a more accurate momentum measure because
the unit measure of the target is smaller and momentum is "bled" out of the target over time. The price to be paid for this accurate measurement is due to the time differential
effect on position.

(b) However if the measurement period is contracted because the target is more massive relative to the particle then the unit of mass measure will incur greater uncertainty in itself
in terms of momentum. Furthermore its "bleeding out" of momentum over time will be much shorter leading to greater certainty in position.

Now consider that the apparatus and its units are at the planck scale such that the
target mass and distance scale are on the order of subatomic particles or less. Any given
measurement will be in multiples of plancks constant.

In QM, measurements of single particles tell us little. Typically we measure ensembles of particles i.e. a statistical population. Intuitively we would expect that the distribution
of momentum and position measurements of such an ensemble of identical particles would be gaussian and that this would represent their accuracy of measurement about some mean
value across many particle measurements.

Less intuitively we would expect that as the scale of measurement approached the
planck scale then the standard deviation of measurement error would also approach that
constant. Given that the smallest real scale possible could only be some multiple of planck's
constant then this limits the certainty that is possible in terms of real measurement.

In this situation measurements converge on planck's constant not just as a standard
deviation of error about a true mean but in fact the standard deviation becomes the
best estimate of the true value. For example, if my only measurement device was gloves
with a diameter of 1 meter then repeated measurements would return an error of
1m and my best estimate of the location of an object would be the radius i.e. 1m/2.

In short, all real measurements in a real universe take time and the notion of simultaneous
measurement does not really exist any more than it does in relativity- but clearly for different reasons.

Applying this notion to the simplest example of the HUP- simple diffraction through a
slit- leads to the following interpretation:

A free particle passes through a wide slid unimpeded, does not interact with the slit and
therefore its momentum as measured by impact on a screen yields small variations from a target point. It has low uncertainty in momentum but high uncertainty in position.

As the width of the slit diminishes, the particle increasingly interacts with the objects
that comprise it (particles, fields, waves or whatever your preference). The smaller the
slit the greater amount of TIME involved in its transit (I would welcome any experimental
evidence that suggests that particle transit times within a diffraction grating vary as
width decreases- intuitively it should) since the particle in engaging a measurement
system.

The smaller the slit, the greater the engagement with the components of the slit and the greater the variation in the momenta of particles as they exist the slit. The variation in momenta is revealed as a dispersion of impact points across the screen,.

I think this interpretation is different from Heisenberg's microscope and other analogies
which attempt to localise an object using electromagnetic radiation. In the case of the heisenberg microscope the use of higher energy light to accurately
locate the particle adds energy to the system which increasingly changes it from T1. But
I think this is a flawed explanation.

What is different
about my interpretation is that it rejects the notion of a simultaneous measurement in
the real world while most physical analogies of the HUP represent measurement as a
disturbance of a system that degrades the accuracy of subsequent measurements. i.e.
that measurement1 changes the system so that subsequent measurements of variables
(measurement 2) lose accuracy on their value at T1.

In my interpretation it is not the disturbance of the system by the act of measurement
such as adding energy, momentum etc. but the simple fact that measurement takes time
and that more accurate measurement takes more time. Simultaneity of measurement just does not exist and ultimately our limit to simultaneity lies at the Planck scale.

I have no idea how this interpetation could be applied to the case of EPR/Bell inequality experiments. However I would be interested in thinking that suggests that simultaneity
in EPR contexts is not as clear cut as we think including notions of decoherence in the
Bell inequality set up and explanation.

Anyway it's all fun.