Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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is completely
described by a state function Ø(q, t) or ket jØi, which depends on spatial
coordinates q and the time t. This function is sometimes also called a state
vector or a wave function. The coordinate vector q has components q1, q2, . . . ,
so that the state function may also be written as Ø(q1, q2, . . . , t). For a particle
or system that moves in only one dimension (say along the x-axis), the vector q
has only one component and the state vector Ø is a function of x and
t: Ø(x, t). For a particle or system in three dimensions, the components of q
are x, y, z and Ø is a function of the position vector r and t: Ø(r, t). The state
function is single-valued, a continuous function of each of its variables, and
square or quadratically integrable.
For a one-dimensional system, the quantity Ø\ufffd(x, t)Ø(x, t) is the probabil-
ity density for finding the system at position x at time t. In three dimensions,
the quantity Ø\ufffd(r, t)Ø(r, t) is the probability density for finding the system at
point r at time t. For a multi-variable system, the product Ø\ufffd(q1, q2,
. . . , t)Ø(q1, q2, . . . , t) is the probability density that the system has coordi-
nates q1, q2, . . . at time t. We show below that this interpretation of Ø
\ufffdØ
follows from postulate 3. We usually assume that the state function is normal-
ized …
Ø\ufffd(q1, q2, . . . , t)Ø(q1, q2, . . . , t)w(q1, q2, . . .) dq1 dq2 . . . ˆ 1
or in Dirac notation
hØjØi ˆ 1
where the limits of integration are over all allowed values of q1, q2, . . .
Physical quantities or observables
The second postulate states that a physical quantity or observable is represented
in quantum mechanics by a hermitian operator. To every classically defined
function A(r, p) of position and momentum there corresponds a quantum-
mechanical linear hermitian operator A^(r, ("=i)=). Thus, to obtain the quan-
tum-mechanical operator, the momentum p in the classical function is replaced
by the operator p^
p^ ˆ "
i
= (3:44)
or, in terms of components
p^x ˆ "
i
@
@x
, p^y ˆ "
i
@
@ y
, p^z ˆ "
i
@
@z
86 General principles of quantum theory
For multi-particle systems with cartesian coordinates r1, r2, . . . , the classical
function A(r1, r2, . . . , p1, p2, . . .) possesses the corresponding operator
A^(r1, r2, . . . , ("=i)=1, ("=i)=2, . . .) where =k is the gradient with respect to
rk . For non-cartesian coordinates, the construction of the quantum-mechanical
operator A^ is more complex and is not presented here.
The classical function A is an observable, meaning that it is a physically
measurable property of the system. For example, for a one-particle system the
Hamiltonian operator H^ corresponding to the classical Hamiltonian function
H(r, p) ˆ p
2
2m
‡ V (r)
where p2 ˆ p : p ˆ p2x ‡ p2y ‡ p2z , is
H^ ˆ ÿ "
2
2m
=2 ‡ V (r)
The linear operator H^ is easily shown to be hermitian.
Measurement of observable properties
The third postulate relates to the measurement of observable properties. Every
individual measurement of a physical observable A yields an eigenvalue ºi of
the operator A^. The eigenvalues are given by
A^jii ˆ ºijii (3:45)
where jii are the orthonormal eigenkets of A^. Since A^ is hermitian, the
eigenvalues are all real. It is essential for the theory that A^ is hermitian because
any measured quantity must, of course, be a real number. If the spectrum of A^
is discrete, then the eigenvalues ºi are discrete and the measurements of A are
quantized. If, on the other hand, the eigenfunctions jii form a continuous,
infinite set, then the eigenvalues ºi are continuous and the measured values of
A are not quantized. The set of eigenkets jii of the dynamical operator A^ are
assumed to be complete. In some cases it is possible to show explicitly that jii
forms a complete set, but in other cases we must assume that property.
The expectation value or mean value hAi of the physical observable A at
time t for a system in a normalized state Ø is given by
hAi ˆ hØjA^jØi (3:46)
If Ø is not normalized, then the appropriate expression is
hAi ˆ hØjA^jØihØjØi
Some examples of expectation values are as follows
3.7 Postulates of quantum mechanics 87
hxi ˆ hØjxjØi
hpxi ˆ Ø
\ufffd\ufffd\ufffd\ufffd "i @@x
\ufffd\ufffd\ufffd\ufffdØ
* +
hri ˆ hØjrjØi
hpi ˆ Ø
\ufffd\ufffd\ufffd\ufffd "i =
\ufffd\ufffd\ufffd\ufffdØ
* +
E ˆ hHi ˆ Ø
\ufffd\ufffd\ufffd\ufffdÿ "22m =2 ‡ V (r)
\ufffd\ufffd\ufffd\ufffdØ
* +
The expectation value hAi is not the result of a single measurement of the
property A, but rather the average of a large number (in the limit, an infinite
number) of measurements of A on systems, each of which is in the same state
Ø. Each individual measurement yields one of the eigenvalues ºi, and hAi is
then the average of the observed array of eigenvalues. For example, if the
eigenvalue º1 is observed four times, the eigenvalue º2 three times, the
eigenvalue º3 once, and no other eigenvalues are observed, then the expectation
value hAi is given by
hAi ˆ 4º1 ‡ 3º2 ‡ º3
8
In practice, many more than eight observations would be required to obtain a
reliable value for hAi.
In general, the expectation value hAi of the observable A may be written for
a discrete set of eigenfunctions as
hAi ˆ
X
i
Piºi (3:47)
where Pi is the probability of obtaining the value ºi. If the state function Ø for
a system happens to coincide with one of the eigenstates jii, then only the
eigenvalue ºi would be observed each time a measurement of A is made and
therefore the expectation value hAi would equal ºi
hAi ˆ hijA^jii ˆ hijºijii ˆ ºi
It is important not to confuse the expectation value hAi with the time average
of A for a single system.
For an arbitrary state Ø at a fixed time t, the ket jØi may be expanded in
terms of the complete set of eigenkets of A^. In order to make the following
discussion clearer, we now introduce a slightly more complicated notation.
Each eigenvalue ºi will now be distinct, so that ºi 6ˆ º j for i 6ˆ j. We let gi be
88 General principles of quantum theory
the degeneracy of the eigenvalue ºi and let jiÆi, Æ ˆ 1, 2, . . . , gi, be the
orthonormal eigenkets of A^. We assume that the subset of kets corresponding
to each eigenvalue ºi has been made orthogonal by the Schmidt procedure
outlined in Section 3.3.
If the eigenkets jiÆi constitute a discrete set, we may expand the state vector
jØi as
jØi ˆ
X
i
Xgi
ƈ1
ciÆjiÆi (3:48)
where the expansion coefficients ciÆ are
ciÆ ˆ hiÆjØi (3:49)
The expansion of the bra vector hØj is, therefore, given by
hØj ˆ
X
j
Xg j
âˆ1
c\ufffdjâh jâj (3:50)
where the dummy indices i and Æ have been replaced by j and â.
The expectation value of A^ is obtained by substituting equations (3.48) and
(3.50) into (3.46)
hAi ˆ
X
j
Xg j
âˆ1
X
i
Xgi
ƈ1
c\ufffdjâciÆh jâjA^jiÆi ˆ
X
j
Xg j
âˆ1
X
i
Xgi
ƈ1
c\ufffdjâciƺih jâjiÆi
ˆ
X
i
Xgi
ƈ1
jciÆj2ºi (3:51)
where we have noted that the kets jiÆi are orthonormal, so that
h jâjiÆi ˆ äijäÆâ
A comparison of equations (3.47) and (3.51) relates the probability Pi to the
expansion coefficients ciÆ
Pi ˆ
Xgi
ƈ1
jciÆj2 ˆ
Xgi
ƈ1
jhiÆjØij2 (3:52)
where equation (3.49) has also been introduced. For the case where ºi is non-
degenerate, the index Æ is not needed and equation (3.52) reduces to
Pi ˆ jcij2 ˆ jhijØij2
For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it
is more convenient to write the eigenvalue equation (3.45) in the form
A^jºi ˆ ºjºi
where º is now a continuous variable and jºi is the eigenfunction whose
eigenvalue is º. The expansion of the state vector Ø becomes
3.7 Postulates of quantum mechanics 89
jØi ˆ
…
c(º)jºi dº
where
c(º) ˆ hºjØi
and the expectation value of A takes the form
hAi ˆ
…
jc(º)j2 º dº (3:53)
If dPº is the probability of obtaining a value of A between º and º‡ dº, then
equation