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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n‡ 1p än9,n‡1 ÿ

n
p
än9,nÿ1)
so that
hn‡ 1j p^jni ˆ i

(n‡ 1)m"ø
2
r
(4:46a)
hnÿ 1j p^jni ˆ ÿi

nm"ø
2
r
(4:46b)
hn9j p^jni ˆ 0 for n9 6ˆ n‡ 1, nÿ 1 (4:46c)
We can easily show that
122 Harmonic oscillator
hn9j p^jni ˆ ÿhnj p^jn9i
The matrix element hn9jx2jni is
hn9jx2jni ˆ "
2mø
hn9j(a^y ‡ a^)2jni ˆ "
2mø
hn9j(a^y)2 ‡ a^y a^‡ a^a^y ‡ a^2jni
From equation (4.34) we have
(a^y)2jni ˆ n‡ 1p a^yjn‡ 1i ˆ (n‡ 1)(n‡ 2)p jn‡ 2i
a^y a^jni ˆ np a^yjnÿ 1i ˆ njni
a^a^yjni ˆ n‡ 1p a^jn‡ 1i ˆ (n‡ 1)jni
a^2jni ˆ np a^jnÿ 1i ˆ n(nÿ 1)p jnÿ 2i
(4:47)
so that
hn9jx2jni ˆ "
2mø
[

(n‡ 1)(n‡ 2)
p
än9,n‡2 ‡ (2n‡ 1)än9n
‡

n(nÿ 1)
p
än9,nÿ2]
We conclude that
hn‡ 2jx2jni ˆ hnjx2jn‡ 2i ˆ "
2mø

(n‡ 1)(n‡ 2)
p
(4:48a)
hnjx2jni ˆ "
mø
(n‡ 1
2
) (4:48b)
hnÿ 2jx2jni ˆ hnjx2jnÿ 2i ˆ "
2mø

n(nÿ 1)
p
(4:48c)
hn9jx2jni ˆ 0, n9 6ˆ n‡ 2, n, nÿ 2 (4:48d)
The matrix element hn9j p^2jni is obtained from equations (4.43b) and (4.47)
hn9j p^2jni ˆ ÿ m"ø
2
\ufffd \ufffd
hn9j(a^y ÿ a^)2jni ˆ ÿ m"ø
2
\ufffd \ufffd
hn9j(a^y)2 ÿ a^y a^
ÿ a^a^y ‡ a^2jni
ˆ ÿ m"ø
2
\ufffd \ufffd
[

(n‡ 1)(n‡ 2)
p
än9,n‡2 ÿ (2n‡ 1)än9n
‡

n(nÿ 1)
p
än9,nÿ2]
so that
4.4 Matrix elements 123
hn‡ 2j p^2jni ˆ hnj p^2jn‡ 2i ˆ ÿ m"ø
2
\ufffd \ufffd 
(n‡ 1)(n‡ 2)
p
(4:49a)
hnj p^2jni ˆ m"ø(n‡ 1
2
) (4:49b)
hnÿ 2j p^2jni ˆ hnj p^2jnÿ 2i ˆ ÿ m"ø
2
\ufffd \ufffd 
n(nÿ 1)
p
(4:49c)
hn9j p^2jni ˆ 0, n9 6ˆ n‡ 2, n, nÿ 2 (4:49d)
Following this same procedure using the operators (a^y \ufffd a^)k , we can find the
matrix elements of xk and of p^k for any positive integral power k. In Chapters
9 and 10, we need the matrix elements of x3 and x4. The matrix elements
hn9jx3jni are as follows:
hn‡ 3jx3jni ˆ hnjx3jn‡ 3i ˆ "
2mø
\ufffd \ufffd3=2 
(n‡ 1)(n‡ 2)(n‡ 3)
p
(4:50a)
hn‡ 1jx3jni ˆ hnjx3jn‡ 1i ˆ 3 (n‡ 1)"
2mø
\ufffd \ufffd3=2
(4:50b)
hnÿ 1jx3jni ˆ hnjx3jnÿ 1i ˆ 3 n"
2mø
\ufffd \ufffd3=2
(4:50c)
hnÿ 3jx3jni ˆ hnjx3jnÿ 3i ˆ "
2mø
\ufffd \ufffd3=2 
n(nÿ 1)(nÿ 2)
p
(4:50d)
hn9jx3jni ˆ 0, n9 6ˆ n\ufffd 1, n\ufffd 3 (4:50e)
The matrix elements hn9jx4jni are as follows
hn‡ 4jx4jni ˆ hnjx4jn‡ 4i ˆ "
2mø
\ufffd \ufffd2 
(n‡ 1)(n‡ 2)(n‡ 3)(n‡ 4)
p
(4:51a)
hn‡ 2jx4jni ˆ hnjx4jn‡ 2i ˆ 1
2
"
mø
\ufffd \ufffd2
(2n‡ 3)

(n‡ 1)(n‡ 2)
p
(4:51b)
hnjx4jni ˆ 3
2
"
mø
\ufffd \ufffd2
n2 ‡ n‡ 1
2
\ufffd \ufffd
(4:51c)
hnÿ 2jx4jni ˆ hnjx4jnÿ 2i ˆ 1
2
"
mø
\ufffd \ufffd2
(2nÿ 1)

n(nÿ 1)
p
(4:51d)
hnÿ 4jx4jni ˆ hnjx4jnÿ 4i ˆ "
2mø
\ufffd \ufffd2 
n(nÿ 1)(nÿ 2)(nÿ 3)
p
(4:51e)
hn9jx4jni ˆ 0, n9 6ˆ n, n\ufffd 2, n\ufffd 4 (4:51f)
124 Harmonic oscillator
4.5 Heisenberg uncertainty relation
Using the results of Section 4.4, we may easily verify for the harmonic
oscillator the Heisenberg uncertainty relation as discussed in Section 3.11.
Specifically, we wish to show for the harmonic oscillator that
\u2dcx\u2dcp > 1
2
"
where
(\u2dcx)2 ˆ h(xÿ hxi)2i
(\u2dcp)2 ˆ h( p^ÿ hpi)2i
The expectation values of x and of p^ for a harmonic oscillator in eigenstate
jni are just the matrix elements hnjxjni and hnj p^jni, respectively. These matrix
elements are given in equations (4.45c) and (4.46c). We see that both vanish,
so that (\u2dcx)2 reduces to the expectation value of x2 or hnjx2jni and (\u2dcp)2
reduces to the expectation value of p^2 or hnj p^2jni. These matrix elements are
given in equations (4.48b) and (4.49b). Therefore, we have
\u2dcx ˆ "
mø
\ufffd \ufffd1=2
(n‡ 1
2
)1=2
\u2dcp ˆ (m"ø)1=2(n‡ 1
2
)1=2
and the product \u2dcx\u2dcp is
\u2dcx\u2dcp ˆ (n‡ 1
2
)"
For the ground state (n ˆ 0), we see that the product \u2dcx\u2dcp equals the
minimum allowed value "=2. This result is consistent with the form (equation
(3.85)) of the state function for minimum uncertainty. When the ground-state
harmonic-oscillator values of kxl, k pl, and º are substituted into equation
(3.85), the ground-state eigenvector j0i in equation (4.31) is obtained. For
excited states of the harmonic oscillator, the product \u2dcx\u2dcp is greater than the
minimum allowed value.
4.6 Three-dimensional harmonic oscillator
The harmonic oscillator may be generalized to three dimensions, in which case
the particle is displaced from the origin in a general direction in cartesian
space. The force constant is not necessarily the same in each of the three
dimensions, so that the potential energy is
V ˆ 1
2
kxx
2 ‡ 1
2
kyy
2 ‡ 1
2
kzz
2 ˆ 1
2
m(ø2xx
2 ‡ ø2y y2 ‡ ø2z z2)
4.6 Three-dimensional harmonic oscillator 125
where kx, ky, kz are the respective force constants and øx, ø y, øz are the
respective classical angular frequencies of vibration.
The Schro¨dinger equation for this three-dimensional harmonic oscillator is
ÿ "
2
2m
@2\u142
@x2
‡ @
2\u142
@ y2
‡ @
2\u142
@z2
 !
‡ 1
2
m(ø2xx
2 ‡ ø2y y2 ‡ ø2z z2)\u142 ˆ E\u142
where \u142(x, y, z) is the wave function. To solve this partial differential equation
of three variables, we separate variables by making the substitution
\u142(x, y, z) ˆ X (x)Y (y)Z(z) (4:52)
where X (x) is a function only of the variable x, Y (y) only of y, and Z(z) only
of z. After division by ÿ\u142(x, y, z), the Schro¨dinger equation takes the form
"2
2mX
d2 X
dx2
ÿ 1
2
mø2xx
2
\ufffd \ufffd
‡ "
2
2mY
d2Y
dy2
ÿ 1
2
mø2y y
2
 !
‡ "
2
2mZ
d2 Z
dz2
ÿ 1
2
mø2z z
2
\ufffd \ufffd
ˆ E
The first term on the left-hand side is a function only of the variable x and
remains constant when y and z change but x does not. Similarly, the second
term is a function only of y and does not change in value when x and z change
but y does not. The third term depends only on z and keeps a constant value
when only x and y change. However, the sum of these three terms is always
equal to the constant energy E for all choices of x, y, z. Thus, each of the three
independent terms must be equal to a constant
"2
2mX
d2 X
dx2
ÿ 1
2
mø2xx
2 ˆ Ex
"2
2mY
d2Y
dy2
ÿ 1
2
mø2y y
2 ˆ Ey
"2
2mZ
d2 Z
dz2
ÿ 1
2
mø2z z
2 ˆ Ez
where the three separation constants Ex, Ey, Ez satisfy the relation
Ex ‡ Ey ‡ Ez ˆ E (4:53)
The differential equation for X (x) is exactly of the form given by (4.13) for a
one-dimensional harmonic oscillator. Thus, the eigenvalues Ex are given by
equation (4.30)
Enx ˆ (nx ‡ 12)"øx, nx ˆ 0, 1, 2, . . .
and the eigenfunctions are given by (4.41)
126 Harmonic oscillator
Xnx(x) ˆ (2nx nx!)ÿ1=2
møx
ð"
\ufffd \ufffd1=4
Hnx(î)e
ÿî2=2
î ˆ møx
"
\ufffd \ufffd1=2
x
Similarly, the eigenvalues for the differential equations for Y (y) and Z(z) are,
respectively
Eny ˆ (ny ‡ 12)"ø y, ny ˆ 0, 1, 2, . . .
Enz ˆ (nz ‡ 12)"øz, nz ˆ 0, 1, 2, . . .
and the corresponding eigenfunctions are
Yny(y) ˆ (2ny ny!)ÿ1=2
mø y
ð"
\ufffd \ufffd1=4
Hny(ç)e
ÿç2=2
ç ˆ mø y
"
\ufffd \ufffd1=2
y
Znz(z) ˆ (2nz nz!)ÿ1=2
møz
ð"
\ufffd \ufffd1=4
Hnz(æ)e
ÿæ2=2
æ ˆ møz
"
\ufffd \ufffd1=2
z
The energy levels for the three-dimensional harmonic oscillator are, then,
given by the sum (equation (4.53))
Enx,n y,nz ˆ (nx ‡ 12)"øx ‡ (n y ‡ 12)"ø y ‡ (nz ‡ 12)"øz (4:54)
The total wave functions are given by equation (4.52)
\u142nx,n y,nz(x, y, z) ˆ (2nx‡n y‡nz nx!ny!nz!)ÿ1=2
m
ð"
\ufffd \ufffd3=4
(øxø yøz)
1=4
3 Hnx(î)Hny(ç)Hnz(æ)e
ÿ(î2‡ç2‡æ2)=2 (4:55)
An isotropic oscillator is one for which the restoring force is independent of
the direction