70 From Classical Mechanics to Quantum Field Theory. A Tutorial
measurements on identical systems all prepared in the stateψa, or we can perform
different subsequent measurements on the same system in the stateψa.Inthe
second case, these measurements have to be performed very close to each other
in time to prevent the state of the system from evolving in view of Schr ̈odinger
evolution as said in (3) below.) Such states, where observable take definite values,
cannot be prepared forallobservables simultaneously as discussed in (2) below.
(2) Compatible and Incompatible Observables.The second noticeable fea-
ture of QM is the existence ofincompatible observables. Differently from CM,
there are physical quantities which cannot be measured simultaneously. There is
no physical instrument capable to do it. If an observableAisdefinedin a given
stateψ– i.e. it attains a precise valueawith probability 1 in case of a measure-
ment – an observableBincompatiblewithAturns out to benot definedin the
stateψ– i.e., it generally attains several different valuesb, b′,b′′...,none with
probability1, in case of measurement. So, if we perform a measurement ofB,we
generally obtain a spectrum of values described by a probabilistic distribution as
preannounced in (1) above.
Incompatibility is asymmetricproperty:Ais incompatible withBif and only
ifBis incompatible withA. However it is nottransitive.
There are alsocompatible observableswhich, by definition, can be measured
simultaneously. An example is the componentxof the position of a particle and
the componentyof the momentum of that particle, referring to a given inertial
reference frame. A popular case of incompatible observables is a pair ofcanonically
conjugated observables(see the first part) like the positionXand the momentumP
of a particle both along the same fixed axis of a reference frame. In this case, there
is a lower bound for the product of the standard deviations, resp. ΔXψ,ΔPψ,of
the outcomes of the measurements ofthese observables in a given stateψ(these
measurement has to be performed on different identical systems all prepared in
the same stateψ). This lower bound does not depend on the state and is encoded
in the celebrated mathematical formula of theHeisenberg principle(a theorem in
the modern formulations):
ΔXψΔPψ≥/ 2 , (2.1)where Planck constant shows up.
(3) Post Measurement Collapse of the State. In QM, measurementsgen-
erally change the state of the systemand produce a post-measurement state from
the state on which the measurement is performed. (We are here referring to ide-
alized measurement procedures, sincemeasurement procedures are very often