On 5/12/2025 1:58 PM, Alan Grayson wrote:
On Friday, May 9, 2025 at 10:40:42 PM UTC-6 Brent Meeker wrote:
On 5/9/2025 7:08 PM, Alan Grayson wrote:
*I can see that the measurement spreads due to instrument
limitations are usually immensely larger than the much smaller
spreads accounted for by the UP, but what causes these much
smaller spreads? Is this a quantum effect? AG*
Yes. Quantum evolution is unitary, i.e. the state vector just
rotates in a complex Hilbert space so that probability is
preserved. Consequently the infinitesimal time translation
operator is U=1+e6/6t or in common notation 1-i(e/h)H where
H=ih6/6t and h is just conversion factor because we measure energy
in different units than inverse time. It's not mathematics, but an
empirical fact that h is a universal constant.
Brent
*If one wants to prepare a system in some momentum state to be
measured, doesn't this imply a pre-measurement measurement, *
Right, given that it's an ideal measurement. Most measurements don't
leave the system in the eigenstate that is the measurement result. An
ideal measurement is one that leaves the system in the state that the
measurement yielded.
*and the observable to be measured remains in that state on subsequent
measurements? *
Only if they're ideal measurements of that same variable or of other
variables that commute with it.
*If so, how can the unitary operator, which just changes the state of
the system's wf, create the quantum spread? *
You don't need a change in the wf to "create the quantum spread".
Having prepared in an eigenstate of A just measure some other variable B
that doesn't commute with A. In general A will be a superposition of
other variables, say A=xC+yD; that's just a change of coordinates. But
the system is not in an eigenstate of C or D.
Brent
*TY, AG*
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