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MeanSquare Fluctuation
One physical quantity of great interest is the variance in the position of atoms at equilibrium,
.
For our model oscillator,
; so
.
This meansquare fluctuation about the average position is related to the B factors
of crystallography and is also measurable by neutron scattering [3] and by Mössbauer spectroscopy [4].
It is one of the most important quantities to keep an eye on in molecular dynamics simulations as well.
What is this fluctuation for the harmonic oscillator in equilibrium at constant
according to classical and quantum mechanics? We again use equations 2, 4, and 5,
now with , and consider the
same three frequencies of proton vibration.
Because and
for harmonic oscillation,
the quantum and classical results are proportional to those obtained above for the average energy.
That is,
. Again, m is taken as the mass of a proton.
(To plot the three frequencies on one scale, results
have been scaled to the f = 100/ps values (green): Results for f = 10/ps (blue) were scaled by 0.01 ; results for f = 1/ps (cyan)
were scaled by 0.0001. Consequently, the three classical curves (red),
, coincide.)

Meansquare fluctuation of H atom undergoing harmonic oscillation. 
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Steinbach
20190201