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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a suitably large, but
otherwise arbitrary value of º, say ç, and continually apply the lowering
operator a^ to the eigenstate jçii, thereby forming a succession of eigenvectors
a^jçii, a^2jçii, a^3jçii, . . .
with respective eigenvalues çÿ 1, çÿ 2, çÿ 3, . . . We have already shown
that if jçii is an eigenfunction of N^ , then a^jçii is also an eigenfunction. By
iteration, if a^jçii is an eigenfunction of N^ , then a^2jçii is an eigenfunction, and
so forth, so that the members of the sequence are all eigenfunctions. Eventually
this procedure gives an eigenfunction a^kjçii with eigenvalue (çÿ k), k being a
positive integer, such that 0 < (çÿ k) , 1. The next step in the sequence
would yield the eigenfunction a^k‡1jçii with eigenvalue º ˆ (çÿ k ÿ 1) , 0,
which is not allowed. Thus, the sequence must terminate by the condition
a^k‡1jçii ˆ a^[a^kjçii] ˆ 0
112 Harmonic oscillator
The only circumstance in which a^ operating on an eigenvector yields the
value zero is when the eigenvector corresponds to the eigenvalue º ˆ 0, as
shown in equation (4.29). Since the eigenvalue of a^kjçii is çÿ k and this
eigenvalue equals zero, we have çÿ k ˆ 0 and ç must be an integer. The
minimum value of º ˆ çÿ k is, then, zero.
Beginning with º ˆ 0, we can apply the operator a^y successively to j0ii to
form a series of eigenvectors
a^yj0ii, a^y2j0ii, a^y3j0ii, . . .
with respective eigenvalues 0, 1, 2, . . . Thus, the eigenvalues of the operator N^
are the set of positive integers, so that º ˆ 0, 1, 2, . . . Since the value ç was
chosen arbitrarily and was shown to be an integer, this sequence generates all
the eigenfunctions of N^ . There are no eigenfunctions corresponding to non-
integral values of º. Since º is now known to be an integer n, we replace º by n
in the remainder of this discussion of the harmonic oscillator.
The energy eigenvalues as related to º in equation (4.25) are now expressed
in terms of n by
En ˆ (n‡ 12)&quot;ø, n ˆ 0, 1, 2, . . . (4:30)
so that the energy is quantized in units of &quot;ø. The lowest value of the energy or
zero-point energy is &quot;ø=2. Classically, the lowest energy for an oscillator is
zero.
Non-degeneracy of the energy levels
To determine the degeneracy of the energy levels or, equivalently, of the
eigenvalues of the number operator N^, we must first obtain the eigenvectors
j0ii for the ground state. These eigenvectors are determined by equation (4.29).
When equation (4.18a) is substituted for a^, equation (4.29) takes the form
d
dî
‡ î
\ufffd \ufffd
j0ii ˆ d
dî
‡ î
\ufffd \ufffd
ö0i(î) ˆ 0
or
dö0i
ö0i
ˆ ÿî dî
This differential equation may be integrated to give
ö0i(î) ˆ ceÿî2=2 ˆ eiÆðÿ1=4eÿî2=2
where the constant of integration c is determined by the requirement that the
functions öni(î) be normalized and eiÆ is a phase factor. We have used the
standard integral (A.5) to evaluate c. We observe that all the solutions for the
ground-state eigenfunction are proportional to one another. Thus, there exists
4.2 Quantum treatment 113
only one independent solution and the ground state is non-degenerate. If we
arbitrarily set Æ equal to zero so that ö0(î) is real, then the ground-state
eigenvector is
j0i ˆ ðÿ1=4eÿî2=2 (4:31)
We next show that if the eigenvalue n of the number operator N^ is non-
degenerate, then the eigenvalue n‡ 1 is also non-degenerate. We begin with
the assumption that there is only one eigenvector with the property that
N^ jni ˆ njni
and consider the eigenvector j(n‡ 1)ii, which satisfies
N^ j(n‡ 1)ii ˆ (n‡ 1)j(n‡ 1)ii
If we operate on j(n‡ 1)ii with the lowering operator a^, we obtain to within a
multiplicative constant c the unique eigenfunction jni,
a^j(n‡ 1)ii ˆ cjni
We next operate on this expression with the adjoint of a^ to give
a^y a^j(n‡ 1)ii ˆ N^ j(n‡ 1)ii ˆ (n‡ 1)j(n‡ 1)ii ˆ ca^yjni
from which it follows that
j(n‡ 1)ii ˆ c
n‡ 1 a^
yjni
Thus, all the eigenvectors j(n‡ 1)ii corresponding to the eigenvalue n‡ 1 are
proportional to a^yjni and are, therefore, not independent since they are
proportional to each other. We conclude then that if the eigenvalue n is non-
degenerate, then the eigenvalue n‡ 1 is non-degenerate.
Since we have shown that the ground state is non-degenerate, we see that the
next higher eigenvalue n ˆ 1 is also non-degenerate. But if the eigenvalue n ˆ 1
is non-degenerate, then the eigenvalue n ˆ 2 is non-degenerate. By iteration, all
of the eigenvalues n of N^ are non-degenerate. From equation (4.30) we observe
that all the energy levels En of the harmonic oscillator are non-degenerate.
4.3 Eigenfunctions
Lowering and raising operations
From equations (4.27) and (4.28) and the conclusions that the eigenvalues of N^
are non-degenerate and are positive integers, we see that a^jni and a^yjni are
eigenfunctions of N^ with eigenvalues nÿ 1 and n‡ 1, respectively. Accor-
dingly, we may write
a^jni ˆ cnjnÿ 1i (4:32a)
114 Harmonic oscillator
and
a^yjni ˆ c9njn‡ 1i (4:32b)
where cn and c9n are proportionality constants, dependent on the value of n, and
to be determined by the requirement that jnÿ 1i, jni, and jn‡ 1i are normal-
ized. To evaluate the numerical constants cn and c9n, we square both sides of
equations (4.32a) and (4.32b) and integrate with respect to î to obtain…1
ÿ1
ja^önj2 dî ˆ jcnj2
…1
ÿ1
ö\ufffdnÿ1önÿ1 dî (4:33a)
and …1
ÿ1
ja^yönj2 dî ˆ jc9nj2
…1
ÿ1
ö\ufffdn‡1ön‡1 dî (4:33b)
The integral on the left-hand side of equation (4.33a) may be evaluated as
follows…1
ÿ1
ja^önj2 dî ˆ
…1
ÿ1
(a^ö\ufffdn )(a^ön) dî ˆ hnja^y a^jni ˆ hnjN^ jni ˆ n
Similarly, the integral on the left-hand side of equation (4.33b) becomes…1
ÿ1
ja^yönj2 dî ˆ
…1
ÿ1
(a^yö\ufffdn )(a^yön) dî ˆ hnja^a^yjni ˆ hnjN^ ‡ 1jni ˆ n‡ 1
Since the eigenfunctions are normalized, we obtain
jcnj2 ˆ n, jc9nj2 ˆ n‡ 1
Without loss of generality, we may let cn and c9n be real and positive, so that
equations (4.32a) and (4.32b) become
a^jni ˆ np jnÿ 1i (4:34a)
a^yjni ˆ n‡ 1p jn‡ 1i (4:34b)
If the normalized eigenvector jni is known, these relations may be used to
obtain the eigenvectors jnÿ 1i and jn‡ 1i, both of which will be normalized.
Excited-state eigenfunctions
We are now ready to obtain the set of simultaneous eigenfunctions for the
commuting operators N^ and H^ . The ground-state eigenfunction j0i has already
been determined and is given by equation (4.31). The series of eigenfunctions
j1i, j2i, . . . are obtained from equations (4.34b) and (4.18b), which give
jn‡ 1i ˆ [2(n‡ 1)]ÿ1=2 îÿ d
dî
\ufffd \ufffd
jni (4:35)
Thus, the eigenvector j1i is obtained from j0i
4.3 Eigenfunctions 115
j1i ˆ 2ÿ1=2 îÿ d
dî
\ufffd \ufffd
(ðÿ1=4eÿî
2=2) ˆ 21=2ðÿ1=4îeÿî2=2
the eigenvector j2i from j1i
j2i ˆ 1
2
îÿ d
dî
\ufffd \ufffd
(21=2ðÿ1=4îeÿî
2=2) ˆ 2ÿ1=2ðÿ1=4(2î2 ÿ 1)eÿî2=2
the eigenvector j3i from j2i
j3i ˆ 6ÿ1=2 îÿ d
dî
\ufffd \ufffd
(2ÿ1=2ðÿ1=4(2î2 ÿ 1)eÿî2=2)
ˆ 3ÿ1=2ðÿ1=4(2î3 ÿ 3î)eÿî2=2
and so forth, indefinitely. Each of the eigenfunctions obtained by this procedure
is normalized.
When equation (4.18a) is combined with (4.34a), we have
jnÿ 1i ˆ (2n)ÿ1=2 î‡ d
dî
\ufffd \ufffd
jni (4:36)
Just as equation (4.35) allows one to go \u2018up the ladder\u2019 to obtain jn‡ 1i from
jni, equation (4.36) allows one to go \u2018down the ladder\u2019 to obtain jnÿ 1i from
jni. This lowering procedure maintains the normalization of each of the
eigenvectors.
Another, but completely equivalent, way of determining the series of eigen-
functions may be obtained by first noting that equation (4.34b) may be written
for the series n ˆ 0, 1, 2, . . . as follows
j1i ˆ a^yj0i
j2i ˆ 2ÿ1=2 a^yj1i ˆ 2ÿ1=2(a^y)2j0i
j3i ˆ 3ÿ1=2 a^yj2i ˆ (3!)ÿ1=2(a^y)3j0i
..
.
Obviously, the expression for jni is
jni ˆ (n!)ÿ1=2(a^y)nj0i
Substitution of equation (4.18b) for a^y and (4.31) for the ground-state eigen-
vector j0i gives
jni ˆ (2n n!)ÿ1=2ðÿ1=4