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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(3.47) is replaced by
hAi ˆ
…
º dPº
and we see that
dPº ˆ jc(º)j2 dº ˆ jhºjØij2 dº
The probability dPº is often written in the form
dPº ˆ r(º) dº
where r(º) is the probability density of obtaining the result º and is given by
r(º) ˆ jc(º)j2 ˆ jhºjØij2
In terms of the probability density, equation (3.53) becomes
hAi ˆ
…
ºr(º) dº (3:54)
In some applications to physical systems, the eigenkets of A^ possess a
partially discrete and a partially continuous spectrum, in which case equations
(3.51) and (3.53) must be combined.
The scalar product hØjØi may be evaluated from equations (3.48) and
(3.50) as
hØjØi ˆ
X
j
Xg j
âˆ1
X
i
Xgi
ƈ1
c\ufffdjâciÆh jâjiÆi ˆ
X
i
Xgi
ƈ1
jciÆj2
ˆ
X
i
Xgi
ƈ1
jhiÆjØij2 ˆ
X
i
Pi
Since the state vector Ø is normalized, this expression givesX
i
Pi ˆ 1
Thus, the sum of the probabilities Pi equals unity as it must from the definition
of probability. For a continuous set of eigenkets, this relationship is replaced by
90 General principles of quantum theory
…
dPº ˆ
…
r(º) dº ˆ 1
As an example, we consider a particle in a one-dimensional box as discussed
in Section 2.5. Suppose that the state function Ø(x) for this particle is time-
independent and is given by
Ø(x) ˆ C sin5 ðx
a
\ufffd \ufffd
, 0 < x < a
where C is a constant which normalizes Ø(x). The eigenfunctions jni and
eigenvalues En of the Hamiltonian operator H^ are
jni ˆ

2
a
r
sin
nðx
a
\ufffd \ufffd
, En ˆ n
2 h2
8ma2
, n ˆ 1, 2, . . .
Obviously, the state function Ø(x) is not an eigenfunction of H^ . Following the
general procedure described above, we expand Ø(x) in terms of the eigenfunc-
tions jni. This expansion is the same as an expansion in a Fourier series, as
described in Appendix B. As a shortcut we may use equations (A.39) and
(A.40) to obtain the identity
sin5\u141 ˆ 1
16
(10 sin \u141ÿ 5 sin 3\u141‡ sin 5\u141)
so that the expansion of Ø(x) is
Ø(x) ˆ C
16
10 sin
ðx
a
\ufffd \ufffd
ÿ 5 sin 3ðx
a
\ufffd \ufffd
‡ sin 5ðx
a
\ufffd \ufffd&quot; #
ˆ C
16

a
2
r
(10j1i ÿ 5j3i ‡ j5i)
A measurement of the energy of a particle in state Ø(x) yields one of three
values and no other value. The values and their probabilities are
E1 ˆ h
2
8ma2
, P1 ˆ 10
2
102 ‡ 52 ‡ 12 ˆ
100
126
ˆ 0:794
E3 ˆ 9h
2
8ma2
, P3 ˆ 5
2
126
ˆ 0:198
E5 ˆ 25h
2
8ma2
, P5 ˆ 1
2
126
ˆ 0:008
The sum of the probabilities is unity,
P1 ‡ P3 ‡ P5 ˆ 0:794‡ 0:198‡ 0:008 ˆ 1
The interpretation that the quantity Ø\ufffd(q1, q2, . . . , t)Ø(q1, q2, . . . , t) is
the probability density that the coordinates of the system at time t are
3.7 Postulates of quantum mechanics 91
q1, q2, . . . may be shown by comparing equations (3.46) and (3.54) for A equal
to the coordinate vector q
hqi ˆ hØjqjØi ˆ
…
Ø\ufffd(q)Ø(q)q dô
hqi ˆ
…
r(q)q dô
For these two expressions to be mutually consistent, we must have
r(q) ˆ Ø\ufffd(q)Ø(q)
Thus, this interpretation of Ø\ufffdØ follows from postulate 3 and for this reason
is not included in the statement of postulate 1.
Collapse of the state function
The measurement of a physical observable A gives one of the eigenvalues ºn of
the operator A^. As stated by the fourth postulate, a consequence of this
measurement is the sudden change in the state function of the system from its
original form Ø to an eigenfunction or linear combination of eigenfunctions of
A^ corresponding to ºn.
At a fixed time t just before the measurement takes place, the ket jØi may
be expanded in terms of the eigenkets jiÆi of A^, as shown in equation (3.48). If
the measurement gives a non-degenerate eigenvalue ºn, then immediately after
the measurement the system is in state jni. The state function Ø is said to
collapse to the function jni. A second measurement of A on this same system,
if taken immediately after the first, always yields the same result ºn. If the
eigenvalue ºn is degenerate, then right after the measurement the state function
is some linear combination of the eigenkets jnÆi, Æ ˆ 1, 2, . . . , gn. A second,
immediate measurement of A still yields ºn as the result.
From postulates 4 and 5, we see that the state function Ø can change with
time for two different reasons. A discontinuous change in Ø occurs when some
property of the system is measured. The state of the system changes suddenly
from Ø to an eigenfunction or linear combination of eigenfunctions associated
with the observed eigenvalue. An isolated system, on the other hand, undergoes
a continuous change with time in accordance with the time-dependent Schro¨-
dinger equation.
Time evolution of the state function
The fifth postulate stipulates that the time evolution of the state function Ø is
determined by the time-dependent Schro¨dinger equation
92 General principles of quantum theory
i&quot;
@Ø
@ t
ˆ H^Ø (3:55)
where H^ is the Hamiltonian operator of the system and, in general, changes
with time. However, in this book we only consider systems for which the
Hamiltonian operator is time-independent. To solve the time-dependent Schro¨-
dinger equation, we express the state function Ø(q, t) as the product of two
functions
Ø(q, t) ˆ \u142(q)÷(t) (3:56)
where \u142(q) depends only on the spatial variables and ÷(t) depends only on the
time. In Section 2.4 we discuss the procedure for separating the partial
differential equation (3.55) into two differential equations, one involving only
the spatial variables and the other only the time. The state function Ø(q, t) is
then shown to be
Ø(q, t) ˆ \u142(q)eÿiEt=&quot; (3:57)
where E is the separation constant. Since it follows from equation (3.57) that
jØ(q, t)j2 ˆ j\u142(q)j2
the probability density is independent of the time t and Ø(q, t) is a stationary
state.
The spatial differential equation, known as the time-independent Schro¨din-
ger equation, is
H^\u142(q) ˆ E\u142(q)
Thus, the spatial function \u142(q) is actually a set of eigenfunctions \u142n(q) of the
Hamiltonian operator H^ with eigenvalues En. The time-independent Schro¨din-
ger equation takes the form
H^\u142n(q) ˆ En\u142n(q) (3:58)
and the general solution of the time-dependent Schro¨dinger equation is
Ø(q, t) ˆ
X
n
cn\u142n(q)e
ÿiEn t=&quot; (3:59)
where cn are arbitrary complex constants.
The appearance of the Hamiltonian operator in equation (3.55) as stipulated
by postulate 5 gives that operator a special status in quantum mechanics.
Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator
for a given system is sufficient to determine the stationary states of the system
and the expectation values of any other dynamical variables.
We next address the question as to whether equation (3.59) is actually the
most general solution of the time-dependent Schro¨dinger equation. Are there
other solutions that are not expressible in the form of equation (3.59)? To
3.7 Postulates of quantum mechanics 93
answer that question, we assume that Ø(q, t) is any arbitrary solution of the
parital differential equation (3.55). We suppose further that the set of functions
\u142n(q) which satisfy the eigenvalue equation (3.58) is complete. Then we can,
in general, expand Ø(q, t) in terms of the complete set \u142n(q) and obtain
Ø(q, t) ˆ
X
n
an(t)\u142n(q) (3:60)
The coefficients an(t) in the expansion are given by
an(t) ˆ h\u142n(q)jØ(q, t)i (3:61)
and are functions of the time t, but not of the coordinates q. We substitute the
expansion (3.60) into the differential equation (3.55) to obtainX
n
&quot;
i
@an(t)
@ t
‡ an(t) H^
\ufffd \ufffd
\u142n(q) ˆ
X
n
&quot;
i
@an(t)
@ t
‡ Enan(t)
\ufffd \ufffd
\u142n(q) ˆ 0 (3:62)
where we have also noted that the functions \u142n(q) are eigenfunctions of H^ in
accordance with equation (3.58). We next multiply equation (3.62) by \u142\ufffdk (q),
the complex conjugate of one of the eigenfunctions of the orthogonal set, and