On 08-07-2021 01:51, Bruce Kellett wrote:
On Wed, Jul 7, 2021 at 5:26 PM smitra <[email protected]> wrote:
On 07-07-2021 01:17, Bruce Kellett wrote:
Your idea of QM is sadly flawed. The real professional quantum
analysis given in the quoted paper shows how the observed effects
are
completely consistent with quantum mechanics. The emission of
thermal
radiation by the heated balls leads to a clear and evident loss of
coherence. Your pseudo-analysis has nothing to do with either
quantum
mechanics or the actual set-up of this buckyball experiment.
As I said, what I wrote is 100% consistent with their results.
Your analysis, as I understand it, suggests that if the IR photons are
not observed, no interference pattern is seen. This seems to overlook
the fact that the photons are emitted only probabilistically -- there
need not be any photons at all, and yet the experiment sees a
buckyball interference pattern at the lower temperatures. This pattern
is gradually washed out as the temperature of the balls is increased.
No photons are ever detected in that experiment.
Do you dispute that that is what the paper by Hornberger et al. says?
Bruce
I don't dispute these results. The buckyballs are coming from a thermal
reservoir at some finite temperature. We can avoid working with mixed
states by simply considering the interference pattern for each pure
state separately and then summing over the probability distribution over
the pure states. But for this discussion we want to focus on what the
interference pattern will be if all the buckyballs are in the same
exited state. If we put the entire system ina finite volume then we have
a countable set of allowed k-values for the photon momenta. We then have
a set of allowed states for the photons defined by the allowed momenta
and a polarization. We can then label these photon states using a number
and then specify an arbitrary state for the photons by specifying how
many photons we have in each state, and these numbers can then be equal
to zero. If they are all zero then no photons are present.
If only the right slit is open, the state of the buckyball and the
photons just before the screen is hit can be denoted as:
|Right> = sum over n1, n2,n3,...|R(n1,n2,n3,...)>|n1,n2,n3,......>
where |R(n1,n2,...)> denotes the quantum state of the buckyball if it
emits n1 photons in state 1, n2 photons in state 2 etc. The state of the
photons is then denoted as |n1,n2,n3,......>
If only the left slit is open, the state of the buckyball and the
photons just before the screen is hit can be denoted as:
|Left> = sum over n1, n2,n3,...|L(n1,n2,n3,...)>|n1,n2,n3,......>
where |R(n1,n2,...)> denotes the quantum state of the buckyball. Here we
note that the state of the photons will pick up a phase factor relative
to the case of only the right slit being open, but we can then absorb
this phase factor in |L(n1,n2,n3,...)>.
With both slits open, we'll then have a state of the form:
|psi> = 1/sqrt(2) [|Right> + |Left>]
The inner product of |psi> with some position eigenstate |x>, <x|psi> is
then a state vector for the photon states, the squared norm of that
state vector is the probability of finding the buckyball at position x,
because this is the sum over the probabilities for photons over all
possible photon states. So, the probability is then:
P(x) = ||<x|psi>||^2 = 1/2 [||<x|Right>||^2 + ||<x|Left>||^2] +
Re[<Left|x><x|Right>]
We can then evaluate the interference term as follows:
Re[<Left|x><x|Right>] = Re sum over n1, n2,n3,...m1,m2,m3
<x|L(m1,m2,m3,...)>*<x|R(n1,n2,n3,...)><m1,m2,m3,...|n1,n2,n3,......>
Using that <m1,m2,m3,...|n1,n2,n3,......> = 0 unless m_j = n_j for all
j, in which case this inner product equals 1, we then have:
Re[<Left|x><x|Right>] = Re sum over n1, n2,n3
<x|L(n1,n2,n3,...)>*<x|R(n1,n2,n3,...)> =
Re[<x|L(0,0,0...)>*<x|R(0,0,0,...>] +
Re[<x|L(1,0,0...)>*<x|R(1,0,0,...>] + ..... (1)
As explained above when there are photons present then we've absorbed
the phase factor due to translation of the photon states in the states
|L(n1,n2,n3...)>. For each wave vector k there is a factor for each
photon with that wavevector of exp(i k dot r) where r is the position of
the left slit w.r.t. the right slit. So, the total phase factor will be:
Product over j of exp(i kj nj dot r)
In the experiment there is then an additional summation over the pure
states of the buckyballs. If the temperature is low then the summation
will consist of states for which |R(0,0,0,..> and |L(0,0,0,..> are the
dominant terms, as most of the time no photons will be emitted. At
higher temperatures the typical states there will be contributions from
different numbers pf photons, so the interference pattern will be a sum
of many different terms in (1) with comparable norms, they come with
different phase factors due to the different numbers of photons with
different momenta. So, the interference pattern will be washed out.
Saibal
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