1 Slide of 15
Chem. 328, Physical Chemistry
II, Quantum Chemistry
Prof. E. Castner, Rutgers, The
State University of New Jersey
Chapter 5, part B: Quantum Harmonic
Oscillators
The Schroedinger equation for a 1dimensional harmonic oscillator is
2
2
2
2
x
n x
12
k x n x
2
(1)
En n x
Reorganizing terms provides the
following second-order differential
equation:
2
n x
x2
2
2
En
12
k x n x
2
The quantum harmonic oscillator
0 (2)
The quantum harmonic oscillator
Schroedinger eq. does not have constant
coefficients, so more sophisticated means
of solving the diff. eqn. must be used.
A power series solution is an exact means
of solving the equation.
An elegant solution is obtained using
raising and lowering operators, shown in
the Appendix to Chap. 5. Both of these
methods are typically done in a first
graduate level QM course.
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Slide 2 of 15
Energy eigenvalues for the quantum
harmonic oscillator
The energy eigenvalues for the quantum
H.O. are
1
En h n
,
2
for n 0, 1, 2,
where
1
2
k
(3)
.
2
We can also write this in terms of the
angular frequency:
1
En
n
,
2
for n 0, 1, 2,
(4)
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Slide 3 of 15
Energy levels of a quantum harmonic
oscillator
Note that the energy between quantum
numbers is fixed at E
h
.
Fig. 5-7, "Quantum Chemistry", 2nd ed.,
D. A. McQuarrie.
The other extremely significant fact is
that the ground state of the quantum
harm. osc. is NOT ZERO, but is
1
E0 1 h
. This leads to the
2
2
existence of zero point energies.
If the ground state energy were zero, we
If the ground state energy were zero, we
would know that both position and
momentum would be zero, thus
contradicting the uncertainty principle.
Thus, zero point energies result from the
Heisenberg uncertainty principle.
The existence of a non-zero value for E0
means that even at absolute zero
temperature, all molecules are vibrating!
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Slide 4 of 15
The Quantum Harmonic Oscillator
describes vibrating molecules
The quantum harmonic oscillator model
describes the vibrational spectra (FT-IR
and Raman) of diatomic molecules.
Vibrational spectroscopy uses light to
cause transitions between energy levels in
vibrating molecules. For example, a
diatomic molecule can often undergo a
transition from the E0 energy level to the
E1 level, by absorbing a photon of energy
E
h obs
obs.
Later we will show that the selection rule
for the quantum harmonic oscillator
permits transitions to occur only between
adjacent energy levels; i.e., n
1.
Absorption of light occurs when
n
1, so
E En 1 En h
.
Absorption of light occurs when
n
1, so
E En 1 En h
.
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Slide 5 of 15
Frequencies of vibrating diatomic
molecules
The observed frequencies for vibrating
molecules are given by obs
1
2
k
.
It is common to write the observed
spectroscopic energies as
2
Gn
1
hc
En
n
1
2
1
2c
.
k
n
1
2
The terms and G n have units of cm 1,
or wavenumbers. The tilde symbol "~"
above a quantity means that we should
expect wavenumber units.
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Slide 6 of 15
Force constants from vibrational
frequencies
We can now calculate the force constants
k for diatomic molecules from a
measurement of the vibrational frequency
or . Remember when using
wavenumber units that it's always more
convenient to use c 3.0 x 1010 cm/s .
Typical diatomic force constants range
from about 10 to 2300 N/m.
Example: carbon monoxide (CO)- obs=
2,169.81 cm 1.
k
c
2
2c
2
12 amu 16 amu
12 16 amu
1.661 10 27 kg amu
2 2169.81 cm 1 2.9979
1010
2
1010 s
2
1903 cm N m
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Slide 7 of 15
More on anharmonicities
Another limitation of the harmonic
oscillator model is that it predicts that
only fundamental transitions will be
observed in the vibrational spectrum for
diatomic molecules.
Specifically, this model predicts that only
transitions between energy levels En and
En 1 could be observed. This does not
reflect what experiments tell us.
Experimental spectra for the vibrations of
diatomic molecules show overtones, or
vibrational transitions for which the
change in quantum number is n = 2, 3,
4, or larger.
Real molecular force laws are
anharmonic, requiring us to extend the
content of the potential energy beyond
1
Vx
k x2 to include higher order
2
content of the potential energy beyond
1
Vx
k x2 to include higher order
2
(anharmonic) terms. I.e.,
2V
x2
1
2
Vx
1
3
3V
x3
1
2
1
6
x0
x2
x0
2
4V
x4
1
4
3
x
x0
x4
kx
3 x3
1
24
4 x4
Note that the anharmonic coefficients are
defined as j
jV
xj
x0
.
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Slide 8 of 15
Vibrational overtones
Vibrational spectroscopy can be used to
measure the anharmonicity coefficients
for the potential energy curve for a given
molecule. For a triatomic or larger
molecule, we have a potential energy
surface.
Experimentally, we observe
G n e n 1
xe e n
2
12
2
,
where n {0, 1, 2, } and xe is the
anharmonicity constant. xe 1.
This energy equation shows that as the
quantum number n increases, the energy
levels are no longer (nearly) equally
spaced, but decrease so that near the
dissociation energy, they become very
tightly packed in energy.
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Slide 9 of 15
Vibrational overtones (harmonic vs.
anharmonic HCl energy levels)
Figs. 5-8 and 5-9, respectively, from
"Quantum Chemistry", 2nd ed., D. A.
McQuarrie.
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Slide 10 of 15
Quantum harmonic oscillator
wavefunctions: Hermite polynomials
The quantum Hamiltonian for the
harmonic oscillator is:
2
H
2
H n x
n x
2
x2
1
2
k x2. The solution to
En n x is:
n
.
Nn Hn 1 2 x exp
k 12
2
and Nn
1
2
x2 where
1
2n n 1 2
14
Hn are the Hermite polynomials,
Nn are the normalization coefficients
exp 1 x2 is the (symmetric, even)
2
Gaussian function.
It is standard to use the variable to
substitute as 1 2 x .
The Hermite polynomials are solutions to
the Hermite equation below, with the
The Hermite polynomials are solutions to
the Hermite equation below, with the
Gaussian function as a weight function:
2y
x2
2x
y
x
2n y
0.
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Slide 11 of 15
Hermite polynomials for n
{0, 1, 2,
, 10}
Column Table HermiteH n, x ,
n, 0, 10
1
2x
4 x2 2
8 x3 12 x
16 x4 48 x2 12
32 x5 160 x3 120 x
64 x6 480 x4 720 x2 120
128 x7 1344 x5 3360 x3 1680 x
256 x8 3584 x6
13 440 x4 13 440 x2
1680
512 x9 9216 x7
48 384 x5 80 640 x3
30 240 x
1024 x10 23 040 x8 161 280 x6
403 200 x4 302 400 x2 30 240
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Slide 12 of 15
Harmonic oscillator wavefunctions for
v = 0, 1, 2, , 6
1 x2 4
2
4
2
1 x2
2
4
x 3 4
1 x2
2
4
4
2
1 x2
2
2 x2 1
x 3 4 2 x2 3
4
3
1 x2
2
4
6
4
x 3 4 4 x2 5 x2 15
2
1 x2
2
4 x2 3 x2 3
2
1 x2
2
4
15
4
8 3 x6 60 2 x4 90 x2 15
12
5
4
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Slide 13 of 15
Visualizing the harmonic oscillator
wavefunctions
8
6
4
2
4
2
2
4
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Slide 14 of 15
Visualizing the harmonic oscillator
probability amplitudes
8
6
4
2
4
2
2
4
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Slide 15 of 15
Quantum harmonic oscillator
wavefunctions & probabilities
Quantum Number
1.0
0.8
0.6
0.4
0.2
0.0
6
4
2
0
2
4
6
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