# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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```îÿ d
dî
\ufffd \ufffdn
eÿî
2=2 (4:37)
This equation may be somewhat simplified if we note that
116 Harmonic oscillator
îÿ d
dî
\ufffd \ufffd
eÿî
2=2  îÿ d
dî
\ufffd \ufffd
eî
2=2eÿî
2  îeÿî2=2 ÿ îeî2=2eÿî2 ÿ eî2=2 d
dî
eÿî
2
 ÿeî2=2 d
dî
eÿî
2
so that
îÿ d
dî
\ufffd \ufffdn
eÿî
2=2  (ÿ1)neî2=2 d
n
dî n
eÿî
2
(4:38)
Substitution of equation (4.38) into (4.37) gives
jni  (ÿ1)n(2n n!)ÿ1=2ðÿ1=4eî2=2 d
n
dî n
eÿî
2
(4:39)
which may be used to obtain the entire set of eigenfunctions of N^ and H^ .
Eigenfunctions in terms of Hermite polynomials
It is customary to express the eigenfunctions for the stationary states of the
harmonic oscillator in terms of the Hermite polynomials. The infinite set of
Hermite polynomials Hn(î) is defined in Appendix D, which also derives many
of the properties of those polynomials. In particular, equation (D.3) relates the
Hermite polynomial of order n to the nth-order derivative which appears in
equation (4.39)
H n(î)  (ÿ1)neî2 d
n
dî n
eÿî
2
Therefore, we may express the eigenvector jni in terms of the Hermite
polynomial Hn(î) by the relation
jni  ön(î)  (2n n!)ÿ1=2ðÿ1=4 Hn(î)eÿî2=2 (4:40)
The eigenstates \u142n(x) are related to the functions ön(î) by equation (4.16),
so that we have
\u142n(x)  (2n n!)ÿ1=2 møð&quot;
\ufffd \ufffd1=4
Hn(î)e
ÿî2=2
î  mø
&quot;
\ufffd \ufffd1=2
x
(4:41)
For reference, the Hermite polynomials for n  0 to n  10 are listed in Table
4.1. When needed, higher-order Hermite polynomials are most easily obtained
from the recurrence relation (D.5). If only a single Hermite polynomial is
wanted and the neighboring polynomials are not available, then equation (D.4)
may be used.
4.3 Eigenfunctions 117
The functions ön(î) in equation (4.40) are identical to those defined by
equation (D.15) and, therefore, form a complete set as shown in equation
(D.19). Substituting equation (4.16) into (D.19) and applying the relation
(C.5b), we see that the functions \u142n(x) in equation (4.41) form a complete set,
so that X1
n0
\u142n(x)\u142n(x9)  ä(xÿ x9) (4:42)
Physical interpretation
The first four eigenfunctions \u142n(x) for n  0, 1, 2, 3 are plotted in Figure 4.2
and the corresponding functions [\u142n(x)]2 in Figure 4.3. These figures also
show the outline of the potential energy V (x) from equation (4.5) and the four
corresponding energy levels from equation (4.30). The function [\u142n(x)]2 is the
probability density as a function of x for the particle in the nth quantum state.
The quantity [\u142n(x)]2 dx at any point x gives the probability for finding the
particle between x and x dx.
We wish to compare the quantum probability distributions with those
obtained from the classical treatment of the harmonic oscillator at the same
energies. The classical probability density P(y) as a function of the reduced
distance y (ÿ1 < y < 1) is given by equation (4.10) and is shown in Figure
4.1. When equations (4.8), (4.14), and (4.30) are combined, we see that the
maximum displacement in terms of î for a classical oscillator with energy
(n 1
2
)&quot;ø is

2n 1p . For î,ÿ 2n 1p and î. 2n 1p , the classical
Table 4.1. Hermite polynomials
n Hn(î)
0 1
1 2î
2 4î2 ÿ 2
3 8î3 ÿ 12î
4 16î4 ÿ 48î2  12
5 32î5 ÿ 160î3  120î
6 64î6 ÿ 480î4  720î2 ÿ 120
7 128î7 ÿ 1344î5  3360î3 ÿ 1680î
8 256î8 ÿ 3584î6  13440î4 ÿ 13440î2  1680
9 512î9 ÿ 9216î7  48384î5 ÿ 80640î3  30240î
10 1024î10 ÿ 23040î8  161280î6 ÿ 403200î4  302400î2 ÿ 30240
118 Harmonic oscillator
probability for finding the particle is equal to zero. These regions are shaded in
Figures 4.2 and 4.3.
Each of the quantum probability distributions differs from the corresponding
classical distribution in one very significant respect. In the quantum solution
there is a non-vanishing probability of finding the particle outside the classi-
cally allowed region, i.e., in a region where the total energy is less than the
potential energy. Since the Hermite polynomial Hn(î) is of degree n, the wave
function \u142n(x) has n nodes, a node being a point where a function touches or
crosses the x-axis. The quantum probability density [\u142n(x)]2 is zero at a node.
Within the classically allowed region, the wave function and the probability
density oscillate with n nodes; outside that region the wave function and
probability density rapidly approach zero with no nodes.
E
n e
rg
y
7
2
hø
5
2
hø
3
2
hø
1
2
hø
n 5 3
n 5 2
n 5 1
n 5 0
x
0
\u1420
\u1421
\u1422
\u1423
V
Figure 4.2 Wave functions and energy levels for a particle in a harmonic potential well.
The outline of the potential energy is indicated by shading.
4.3 Eigenfunctions 119
While the classical particle is most likely to be found near its maximum
displacement, the probability density for the quantum particle in the ground
state is largest at the origin. However, as the value of n increases, the quantum
probability distribution begins to look more and more like the classical
probability distribution. In Figure 4.4 the function [\u14230(x)]2 is plotted along
with the classical result for an energy 30:5 &quot;ø. The average behavior of the
rapidly oscillating quantum curve agrees well with the classical curve. This
observation is an example of the Bohr correspondence principle, mentioned in
Section 2.3. According to the correspondence principle, classical mechanics
and quantum theory must give the same results in the limit of large quantum
numbers.
E
n e
rg
y
7
2
hø
5
2
hø
3
2
hø
1
2
hø
n 5 3
n 5 2
n 5 1
n 5 0
x
0
|\u1420|2
|\u1421|2
|\u1422|2
|\u1423|2
Figure 4.3 Probability densities and energy levels for a particle in a harmonic potential
well. The outline of the potential energy is indicated by shading.
120 Harmonic oscillator
4.4 Matrix elements
In the application to an oscillator of some quantum-mechanical procedures, the
matrix elements of xn and p^n for a harmonic oscillator are needed. In this
section we derive the matrix elements hn9jxjni, hn9jx2jni, hn9j p^jni, and
hn9j p^2jni, and show how other matrix elements may be determined.
The ladder operators a^ and a^y defined in equation (4.18) may be solved for x
and for p^ to give
x  &quot;
2mø
\ufffd \ufffd1=2
(a^y  a^) (4:43a)
p^  i m&quot;ø
2
\ufffd \ufffd1=2
(a^y ÿ a^) (4:43b)
From equations (4.34) and the orthonormality of the harmonic oscillator
eigenfunctions jni, we find that the matrix elements of a^ and a^y are
hn9ja^jni  np hn9jnÿ 1i  np än9,nÿ1 (4:44a)
hn9ja^yjni  n 1p hn9jn 1i  n 1p än9,n1 (4:44b)
The set of equations (4.43) and (4.44) may be used to evaluate the matrix
elements of any integral power of x and p^.
To find the matrix element hn9jxjni, we apply equations (4.43a) and (4.44) to
obtain
x
0
|\u14230(x)|2
Figure 4.4 The probability density j\u14230(x)j2 for an oscillating particle in state n  30.
The dotted curve is the classical probability density for a particle with the same
energy.
4.4 Matrix elements 121
hn9jxjni  &quot;
2mø
\ufffd \ufffd1=2
(hn9ja^yjni  hn9ja^jni)
 &quot;
2mø
\ufffd \ufffd1=2
(

n 1p än9,n1 

n
p
än9,nÿ1)
so that
hn 1jxjni 

(n 1)&quot;
2mø
r
(4:45a)
hnÿ 1jxjni 

n&quot;
2mø
r
(4:45b)
hn9jxjni  0 for n9 6 n 1, nÿ 1 (4:45c)
If we replace n by nÿ 1 in equation (4.45a), we obtain
hnjxjnÿ 1i 

n&quot;
2mø
r
From equation (4.45b) we see that
hnÿ 1jxjni  hnjxjnÿ 1i
Likewise, we can show that
hn 1jxjni  hnjxjn 1i
In general, then, we have
hn9jxjni  hnjxjn9i
To find the matrix element hn9j p^jni, we use equations (4.43b) and (4.44) to
give
hn9j p^jni  i m&quot;ø
2
\ufffd \ufffd1=2
hn9ja^y ÿ a^jni
 i m&quot;ø
2
\ufffd \ufffd1=2
(
```