QMGreensite_merged

In obtaining [2.21], the orthonormality condition [2.16] has been used.
The resulting equation forhiA has the following interpretation. WhenA
is measured for a single member of the ensemble, the result obtained is
one of the eigenvalues ofA; however, which eigenvalue is obtained cannot
be specified in advance of the measurement. For the ensemble as a whole,
the resultjis obtained in a proportioncjcj; that is,cjcjis interpreted as
the probability that the resultjis obtained in a single measurement.
Consequently, although the allowed values ofAmust be members of the
discrete set of eigenvalues ofA, the observable expectation valuehiA can
have any (continuous) value consistent with [2.21].
The time-independent Schro ̈dinger equation is an eigenvalue
equation for the Hamiltonian operator. Substitution of [2.8] into [2.2]
yields

@
ðtÞ

@t

¼
iH
ðtÞ,

ðÞ

dexp½
iEtŠ

dt

¼
iH ðÞexp½
iEtŠ,

E ðÞ¼H ðÞ:

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The eigenvalues of this equation are the energies of the system and the
eigenfunctions are termed thestationary statesof the system.

2.1.3 SIMULTANEOUSEIGENFUNCTIONS
Next, quantum mechanical restrictions on measurement of different
observable quantities are presented. Two operators, A and B,
corresponding to observable propertiesAandB, are considered as an
example. The eigenfunctions and eigenvalues ofAwill be designated
anda; the eigenfunctions and eigenvalues ofBwill be designated’and
b. The operators are called compatible if the result of measuringA(orB)
does not depend upon whetherB(orA) is measured first. Compatible, or
simultaneous, measurements ofAandBare possible only ifAandB
have the same eigenfunctions (but not necessarily the same eigenvalues).
An important theorem states that ifAB¼BA, thenAandBhave the
same complete set of eigenfunctions. The proof of this statement is
straightforward:

AB¼BA,

AB (^) i¼BA (^) i,
AB (^) i¼aiB (^) i:
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34 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY