# 5: The Harmonic Oscillator and the Rigid Rotor

The harmonic oscillator is common: It appears in many everyday examples: Pendulums, springs, electronics (such as the RLC circuit), standing waves on a string, etc. It's trivial to set up demonstrations of these phenomena, and we see them constantly. The harmonic oscillator is intuitive: We can picture the forces on systems such as pendulum or a plucked string. This makes it simple to study in the classroom. In contrast, there are many "everyday" examples that are not intuitive. The harmonic oscillator is mathematically simple: Math is part of physics. In studying simple harmonic motion, students can immediately use the formulas that describe its motion. These formulas are understandable: for example, the equation for frequency shows the intuitive result that increasing spring stiffness increases frequency.

• 5.1: A Harmonic Oscillator Obeys Hooke's Law
The simple harmonic oscillator, a nonrelativistic particle in a quadratic potential, is an excellent model for a wide range of systems in nature. Indeed, it was for this system that quantum mechanics was first formulated: the blackbody radiation formula of Planck.
• 5.2: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule
Viewing the multi-body system as a single particle allows the separation of the motion: vibration and rotation, of the particle from the displacement of the center of mass. This approach greatly simplifies many calculations and problems.
• 5.3: The Harmonic Oscillator Approximates Vibrations
The quantum harmonic oscillator is the quantum analog of the classical harmonic oscillator and is one of the most important model systems in quantum mechanics. This is due in partially to the fact that an arbitrary potential curve V(x) can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it
• 5.4: The Harmonic Oscillator Energy Levels
In this section we contrast the classical and quantum mechanical treatments of the harmonic oscillator, and we describe some of the properties that can be calculated using the quantum mechanical harmonic oscillator model.
• 5.5: The Harmonic Oscillator and Infrared Spectra
Infrared (IR) spectroscopy is one of the most common and widely used spectroscopic techniques employed mainly by inorganic and organic chemists due to its usefulness in determining structures of compounds and identifying them. Chemical compounds have different chemical properties due to the presence of different functional groups.
• 5.6: The Harmonic Oscillator Wavefunctions involve Hermite Polynomials
The quantum-mechanical description of vibrational motion using the harmonic oscillator model will produce vibrational quantum numbers, vibrational wavefunctions, quantized vibrational energies, and a zero-point energy.
• 5.7: Hermite Polynomials are either Even or Odd Functions
Hermite polynomials were defined by Laplace (1810) though in scarcely recognizable form, and studied in detail by Chebyshev (1859). Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new. They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.
• 5.8: The Energy Levels of a Rigid Rotor
Rigid Rotor means when the distance between particles do not change as they rotate.
• 5.9: The Rigid Rotator is a Model for a Rotating Diatomic Molecule
To develop a description of the rotational states, we will consider the molecule to be a rigid object, i.e. the bond lengths are fixed and the molecule cannot vibrate. This model for rotation is called the rigid-rotor model. It is a good approximation (even though a molecule vibrates as it rotates, and the bonds are elastic rather than rigid) because the amplitude of the vibration is small compared to the bond length.
• 5.E: The Harmonic Oscillator and the Rigid Rotor (Exercises)
These are homework exercises to accompany Chapter 5 of McQuarrie and Simon's "Physical Chemistry: A Molecular Approach" Textmap.