5: The Harmonic Oscillator and the Rigid Rotor
- Page ID
- 11782
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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
- This page discusses the motions of diatomic molecules, including translational, vibrational, and rotational aspects. It highlights the classical harmonic oscillator's role in modeling molecular vibrations, paralleling mass-spring systems, while noting its limitations regarding dissociation energy. The frequency of oscillation is affected by force constant and mass, and the oscillator's energy distribution is confined.
- 5.2: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule
- This page explores reduced mass in diatomic heteronuclear molecules, focusing on a proton and electron for simplification in calculations of vibrational and rotational motions. It introduces the quantum harmonic oscillator model and highlights isotope effects in vibrational frequencies, specifically comparing \(\ce{H^{35}Cl}\) and \(\ce{H^{37}Cl}\), with calculated frequencies of 2886 cm-1 and 2081 cm-1, respectively.
- 5.3: The Harmonic Oscillator Approximates Molecular Vibrations
- This page discusses the quantum harmonic oscillator as a model for molecular vibrations, highlighting its analytical solvability and approximation capabilities but noting limitations like equal energy spacing and failure to predict bond dissociation. It describes how anharmonic perturbations improve the model and introduces the Morse potential, which accounts for bond-breaking effects and offers a more accurate representation of energy levels near dissociation.
- 5.4: The Harmonic Oscillator Energy Levels
- This page discusses the differences between classical and quantum harmonic oscillators. Classical oscillators define precise position and momentum, while quantum oscillators have quantized energy levels and operate on probabilities, exhibiting zero-point energy.
- 5.5: The Harmonic Oscillator and Infrared Spectra
- This page explains infrared (IR) spectroscopy as a vital tool for identifying molecular structures through absorption patterns. It details the quantum harmonic oscillator model relevant to diatomic molecules, such as hydrogen halides, and the calculation of bond stretching in HCl. The page also covers selection rules for IR transitions that require changes in dipole moment and discusses factors affecting molar absorptivity and transition probabilities.
- 5.6: The Harmonic Oscillator Wavefunctions involve Hermite Polynomials
- This page discusses the quantum mechanical model of a diatomic molecule modeled as a harmonic oscillator, detailing the Hamiltonian operator, time-independent Schrödinger equation, and the significance of Hermite polynomials in wavefunction solutions. It emphasizes normalization, energy eigenvalues, and tunneling, with wavefunctions extending beyond classical limits.
- 5.7: Hermite Polynomials are either Even or Odd Functions
- This page explores Hermite polynomials, focusing on their orthogonality, symmetry, and applications in quantum mechanics, especially as solutions for harmonic oscillators. It explains their historical context, definitions, and recurrence relations, alongside the significance of even and odd functions. The text illustrates the orthogonality of specific Hermite polynomials using integral methods over symmetric intervals, emphasizing how the product of even and odd functions affects their integrals.
- 5.8: The Energy Levels of a Rigid Rotor
- This page covers the rigid rotor in classical and quantum mechanics, emphasizing the fixed distances in the rotor approximation and the separation of variables in solving the 3D Schrödinger Equation. It discusses angular variables and derives solutions as Associated Legendre Functions, highlighting energy levels' quantization and degeneracy linked to quantum numbers. The relationship between increasing \(J\) and energy spacing is explored, challenging classical rotation concepts.
- 5.9: The Rigid Rotator is a Model for a Rotating Diatomic Molecule
- This page outlines learning objectives on rotational states in diatomic molecules using the rigid-rotor model and microwave spectroscopy, explaining the role of permanent electric dipole moments and selection rules for transitions. It covers rotational spectra, equations for energy levels, and practical applications, including calculating bond lengths and analyzing molecular properties like reduced mass and centrifugal stretching.
- 5.E: The Harmonic Oscillator and the Rigid Rotor (Exercises)
- This page presents calculations of reduced mass and harmonic oscillator properties for diatomic molecules, including methods for determining vibrational frequency and zero-point energy. It encompasses quantum mechanics concepts like angular momentum for hydrogen-like atoms, provides insights on using angular momentum operators, and details Rydberg constant calculations for various isotopes.
Thumbnail: The rigid rotor model for a diatomic molecule. (CC BY-SA 3.0 Unported; Mysterioso via Wikipedia)