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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of solutions \u142n(x) given by equation (2.40). The first four wave
functions \u142n(x) for n ˆ 1, 2, 3, and 4 and their corresponding probability
densities j\u142n(x)j2 are shown in Figure 2.2. The wave function \u1421(x) corre-
sponding to the lowest energy level E1 is called the ground state. The other
wave functions are called excited states.
If we integrate the product of two different wave functions \u142 l(x) and \u142n(x),
we find that…a
0
\u142 l(x)\u142n(x) dx ˆ 2
a
…a
0
sin
lðx
a
\ufffd \ufffd
sin
nðx
a
\ufffd \ufffd
dx ˆ 2
ð
…ð
0
sin l\u141 sin n\u141 d\u141 ˆ 0
(2:41)
where equation (A.15) has been introduced. This result may be combined with
the normalization relation to give…a
0
\u142 l(x)\u142n(x) dx ˆ ä ln (2:42)
where ä ln is the Kronecker delta,
ä ln ˆ 1, l ˆ n
ˆ 0, l 6ˆ n (2:43)
Functions that obey equation (2.41) are called orthogonal functions. If the
orthogonal functions are also normalized, as in equation (2.42), then they are
2.5 Particle in a one-dimensional box 51
said to be orthonormal. The orthogonal property of wave functions in quantum
mechanics is discussed in a more general context in Section 3.3.
The stationary states Ø(x, t) for the particle in a one-dimensional box are
given by substitution of equations (2.39) and (2.40) into (2.31),
Ø(x, t) ˆ

2
a
r
sin
nðx
a
\ufffd \ufffd
eÿi(n
2ð2"=2ma2) t (2:44)
The most general solution (2.33) is, then,
Ø(x, t) ˆ

2
a
r X
n
cn sin
nðx
a
\ufffd \ufffd
eÿi(n
2ð2"=2ma2) t (2:45)
Figure 2.2 Wave functions \u142i and probability densities j\u142ij2 for a particle in a one-
dimensional box of length a.
\u1421
0 a
x
|\u1421|2
0 a
\u1422
0 a
x
|\u1422|2
0 a
\u1423
0 a
x
|\u1423|2
0 a
\u1424
0 a
x
|\u1424|2
0 a
52 Schro¨dinger wave mechanics
2.6 Tunneling
As a second example of the application of the Schro¨dinger equation, we
consider the behavior of a particle in the presence of a potential barrier. The
specific form that we choose for the potential energy V (x) is given by
V (x) ˆ V0, 0 < x < a
ˆ 0, x , 0, x . a
and is shown in Figure 2.3. The region where x , 0 is labeled I, where
0 < x < a is labeled II, and where x . a is labeled III.
Suppose a particle of mass m and energy E coming from the left approaches
the potential barrier. According to classical mechanics, if E is less than the
barrier height V0, the particle will be reflected by the barrier; it cannot pass
through the barrier and appear in region III. In quantum theory, as we shall see,
the particle can penetrate the barrier and appear on the other side. This effect is
called tunneling.
In regions I and III, where V (x) is zero, the Schro¨dinger equation (2.30) is
d2\u142(x)
dx2
ˆ ÿ 2mE
&quot;2
\u142(x) (2:46)
The general solutions to equation (2.46) for these regions are
\u142I ˆ AeiÆx ‡ BeÿiÆx (2:47 a)
\u142III ˆ FeiÆx ‡ GeÿiÆx (2:47 b)
where A, B, F, and G are arbitrary constants of integration and Æ is the
abbreviation
Æ ˆ

2mE
p
&quot;
(2:48)
In region II, where V (x) ˆ V0 . E, the Schro¨dinger equation (2.30) becomes
V(x)
x
0 a
V0
I II III
Figure 2.3 Potential energy barrier of height V0 and width a.
2.6 Tunneling 53
d2\u142(x)
dx2
ˆ 2m
&quot;2
(V0 ÿ E)\u142(x) (2:49)
for which the general solution is
\u142II ˆ Ceâx ‡ Deÿâx (2:50)
where C and D are integration constants and â is the abbreviation
⠈

2m(V0 ÿ E)
p
&quot;
(2:51)
The term exp[iÆx] in equations (2.47) indicates travel in the positive x-
direction, while exp[ÿiÆx] refers to travel in the opposite direction. The
coefficient A is, then, the amplitude of the incident wave, B is the amplitude of
the reflected wave, and F is the amplitude of the transmitted wave. In region
III, the particle moves in the positive x-direction, so that G is zero. The relative
probability of tunneling is given by the transmission coefficient T
T ˆ jFj
2
jAj2 (2:52)
and the relative probability of reflection is given by the reflection coefficient R
R ˆ jBj
2
jAj2 (2:53)
The wave function for the particle is obtained by joining the three parts \u142I,
\u142II, and \u142III such that the resulting wave function \u142(x) and its first derivative
\u1429(x) are continuous. Thus, the following boundary conditions apply
\u142I(0) ˆ \u142II(0), \u1429I(0) ˆ \u1429II(0) (2:54)
\u142II(a) ˆ \u142III(a), \u1429II(a) ˆ \u1429III(a) (2:55)
These four relations are sufficient to determine any four of the constants A, B,
C, D, F in terms of the fifth. If the particle were confined to a finite region of
space, then its wave function could be normalized, thereby determining the fifth
and final constant. However, in this example, the position of the particle may
range from ÿ1 to 1. Accordingly, the wave function cannot be normalized,
the remaining constant cannot be evaluated, and only relative probabilities such
as the transmission and reflection coefficients can be determined.
We first evaluate the transmission coefficient T in equation (2.52). Applying
equations (2.55) to (2.47 b) and (2.50), we obtain
Ceâa ‡ Deÿâa ˆ FeiÆa
â(Ceâa ÿ Deÿâa) ˆ iÆFeiÆa
from which it follows that
54 Schro¨dinger wave mechanics
C ˆ F ⇠iÆ
2â
\ufffd \ufffd
e(iÆÿâ)a
D ˆ F âÿ iÆ
2â
\ufffd \ufffd
e(iƇâ)a
(2:56)
Application of equation (2.54) to (2.47 a) and (2.50) gives
A‡ B ˆ C ‡ D
iÆ(Aÿ B) ˆ â(C ÿ D)
(2:57)
Elimination of B from the pair of equations (2.57) and substitution of equations
(2.56) for C and for D yield
A ˆ 1
2iÆ
[(⇠iÆ)C ÿ (âÿ iÆ)D]
ˆ F e
iÆa
4iÆâ
[(⇠iÆ)2eÿâa ÿ (âÿ iÆ)2eâa]
At this point it is easier to form jAj2 before any further algebraic simplifica-
tions
jAj2 ˆ A\ufffdA
ˆ jFj2 1
16Æ2â2
[(â2 ‡ Æ2)2eÿ2âa ‡ (â2 ‡ Æ2)2e2âa ÿ (âÿ iÆ)4 ÿ (⇠iÆ)4]
ˆ jFj2 1
16Æ2â2
[(Æ2 ‡ â2)2(eâa ÿ eÿâa)2 ‡ 16Æ2â2]
ˆ jFj2 1‡ (Æ
2 ‡ â2)2
4Æ2â2
sinh2 âa
&quot; #
where equation (A.46) has been used. Combining this result with equations
(2.48), (2.51), and (2.52), we obtain
T ˆ 1‡ V
2
0
4E(V0 ÿ E) sinh
2(

2m(V0 ÿ E)
p
a=&quot;)
&quot; #ÿ1
(2:58)
To find the reflection coefficient R, we eliminate A from the pair of
equations (2.57) and substitute equations (2.56) for C and for D to obtain
B ˆ 1
2iÆ
[ÿ(âÿ iÆ)C ‡ (⇠iÆ)D] ˆ F e
iÆa
4iÆâ
[(Æ2 ‡ â2)eâa ÿ (Æ2 ‡ â2)eÿâa]
ˆ F e
iÆa
2iÆâ
(Æ2 ‡ â2)sinh âa
2.6 Tunneling 55
where again equation (A.46) has been used. Combining this result with
equations (2.52) and (2.53), we find that
R ˆ T jBj
2
jFj2 ˆ T
(Æ2 ‡ â2)2
4Æ2â2
sinh2 âa
Substitution of equations (2.48), (2.51), and (2.58) yields
R ˆ
V 20
4E(V0 ÿ E) sinh
2(

2m(V0 ÿ E)
p
a=&quot;)
1‡ V
2
0
4E(V0 ÿ E) sinh
2(

2m(V0 ÿ E)
p
a=&quot;)
(2:59)
The transmission coefficient T in equation (2.58) is the relative probability
that a particle impinging on the potential barrier tunnels through the barrier.
The reflection coefficient R in equation (2.59) is the relative probability that
the particle bounces off the barrier and moves in the negative x-direction. Since
the particle must do one or the other of these two possibilities, the sum of T
and R should equal unity
T ‡ R ˆ 1
which we observe from equations (2.58) and (2.59) to be the case.
We also note that the (relative) probability for the particle being in the region
0 < x < a is not zero. In this region, the potential energy is greater than the
total particle energy, making the kinetic energy of the particle negative. This
concept is contrary to classical theory and does not have a physical signifi-
cance. For this reason we cannot observe the particle experimentally within the
potential barrier. Further, we note that because the particle is not confined to a
finite region, the boundary conditions on the wave function have not imposed
any restrictions