Download Review Vibrational energy of a quantum oscillator Diatomic Molecule

Quantum Oscillators and Diatomic Molecules
Objective: To describe a “quantum harmonic oscillator”; to calculate the vibrational energy in a quantum
oscillator; to calculate the energy of a photon absorbed or emitted by a quantum oscillator; to describe
the types of energy of a diatomic molecule and what the spectrum and absorption spectra for a diatomic
molecule would look like.
Review
We have studied a “classical harmonic oscillator”, basically a ball connected to a spring. The total energy
of the system is
E = K + U = K + 1/2ks2 + U0
(1)
However, if it is a closed system, the kinetic energy and potential are related. When the spring is most
stretched, the ball’s kinetic energy is zero. At this point, E = 1/2 ∗ k ∗ A2 + U0 where A is called the
amplitude which is the greatest amount the spring is stretched. Since the total energy doesn’t change (i.e.
closed system), then the energy is constant at every point. Any total energy is allowed for a given spring,
just choose a different amplitude of oscillation.
We used the ball-spring system as a model of a solid. We considered a solid as being composed of balls
(atoms) connected by springs (bonds). By measuring Young’s modulus for a solid, we can calculated the
interatomic spring (bond) stiffness, k.
Vibrational energy of a quantum oscillator
Suppose we have a system of two atoms connected by a bond, and we model the bond as a spring of stiffness
k. This is called a quantum harmonic oscillator. Its energy, in order to distinguish it from other types of
energy, is called vibrational energy. Unlike the classical oscillator, the quantum oscillator may not have any
value of energy. It may only have discrete values (i.e. certain values called levels). The energy of a quantum
oscillator is
r
k
(2)
E = N h̄ω0 + E0
where N = 0, 1, 2, 3... and ω0 =
m
where k is the bond stiffness and the constant h̄ is Planck’s constant divided by 2π. The lowest energy is
when N=0. The energy, E0 , is called the ground state. We will generally calculate the change in energy
between levels, however, as the system makes a transition from one energy to another. The change in energy,
or energy spacing, between successive levels is
∆E = h̄ω0
(3)
What would cause the energy of a quantum oscillator to increase? What would cause its energy to
decrease?
Diatomic Molecule
The bond between two neutral atoms in a diatomic molecule acts as a spring only for small amplitude
oscillations (also called vibrations). The potential energy of a diatomic molecule is more accurately described
by the Morse potential energy
2
UM = E0 1 − e−α(r−req ) − E0
(4)
where α is a parameter used to adjust the width of the curve (to fit experimental results), req is the
equilibrium separation distance, and E0 is the energy of the system at r = req . Note that as the atoms get
very far apart UM approaches zero, as it should in accordance with the principle of relativity.
The energy spacing (energy difference between successive levels) is nearly constant for low energy (where
the Morse potential energy is approximately the same as spring potential energy) but get closer together at
higher energies.
Not only can molecules vibrate, they can also rotate and they have an energy associated with the configuration of the electron clouds of each atom. Thus, there are 3 distinct types of energy, and each is
quantized.
We’ll call the electronic energy the energy associated with the electrons’ kinetic energies and the molecule’s
potential energy (of each electron with each atom and with each other). As the configuration of the electron
clouds changes (this is like the “orbits” of the hydrogen atom), the electronic energy changes.
At each electronic energy (configuration of the electron cloud), the molecule can vibrate at various
energies. At each vibrational state (energy), the electron can rotate with various energies. Therefore, there
are many possible transitions because there are so many different allowed energies; however, there are large
gaps between these allowed energy states. The resulting emission spectrum, instead of being lines like you
get with atomic hydrogen, contains bands with large gaps between bands that are the forbidden regions.
The energy differences between the “lines” in a band are so small that the band appears to be continuous.
Application
1. Describe the emission spectrum of a diatomic molecule.
2. Describe the absorption spectrum of a diatomic molecule.
3. Problem 6.3
4. Exercise 6.8
5. Problem 6.4
6. What is the energy of the photon emitted when a harmonic oscillator with stiffness k and mass m loses
3 quanta of energy? Use typical values for the interatomic spring stiffness k and mass m to calculate
the energy of the photon. In what region of the spectrum is the energy of this photon?