1549380323-Statistical Mechanics Theory and Molecular Simulation

372 Quantum mechanics

and the corresponding probability isPk=|αk|^2. Interestingly,{αk}are just the coef-
ficients of an expansion of|Ψ〉in the eigenvectors ofAˆ:

|Ψ〉=

∑

k

αk|ak〉. (9.2.26)

Thus, the more aligned the state vector is with a given eigenvector ofAˆ, the greater
is the probability of obtaining the corresponding eigenvalue in a given measurement.
Clearly, if|Ψ〉is one of the eigenvectors ofAˆ, then the corresponding eigenvalue must
be obtained with 100% probability, since no other result is possible in this state.
Although we have not yet discussed the time evolution of the state vector, one
aspect of this evolution can be established immediately. According toour discussion,
when a measurement is made and yields a particular eigenvalue ofAˆ, then immedi-
ately following the measurement, the state vector must somehow “collapse” onto the
corresponding eigenvector, for at that moment, we know with 100% certainty that a
particular eigenvalue was obtained as the result. Therefore, the act of measurement
changes the state of the system and its subsequent time development. Moreover, this
change is abrupt and discontinuous.^2
Finally, suppose a measurement ofAˆis performed many times, with each repe-
tition carried out on the same state|Ψ〉. If we average over the outcomes of these
measurements, what is the result? We know that each measurement yields a resultak
with probability|αk|^2. The average over these trials yields theexpectation valueofAˆ
defined by
〈Aˆ〉=〈Ψ|Aˆ|Ψ〉. (9.2.27)

In order to verify this definition, consider, again, the expansion in eqn. (9.2.26). Sub-
stituting eqn. (9.2.26) into eqn. (9.2.27) gives

〈Aˆ〉=

∑

j,k

α∗jαk〈aj|Aˆ|ak〉

=

∑

j,k

α∗jαkak〈aj|ak〉

=

∑

j,k

α∗jαkakδjk

=

∑

j

aj|αj|^2. (9.2.28)

The last line shows that the expectation value is determined by summing the possible
outcomes of a measurement ofAˆ(the eigenvaluesaj) times the probability|αj|^2 that

(^2) It is important to note that the notion of a “collapsing” wavefunction belongs to one of sev-
eral interpretations of quantum mechanics and the measurement process known as theCopenhagen
Interpretation. Another interpretation, the so-called “many-worlds” interpretation, states that our
universe is part of an essentially infinite “multiverse”; whenAˆis measured, a different outcome is
obtained in each member of the multiverse. It has been suggested that a more fundamental theory
of the universe’s origin (e.g. string theory or loop quantumgravity) will encode a more fundamental
interpretation. Many interesting articles and books existon this subject for curious readers who wish
to explore the subject further.