4: QM for Rotational and Vibrational Motion
- Page ID
- 521299
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: 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.
- 4.2: 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.
- 4.3: 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.
- 4.4: 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.
- 4.5: 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.
- 4.6: 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.
- 4.7: The Wavefunctions of a Rigid Rotator are Called Spherical Harmonics
- This page discusses the solutions to the hydrogen atom Schrödinger equation, detailing wavefunctions dependent on quantum numbers \(n\), \(l\), and \(m_l\), which define electron properties and probability densities. It covers angular momentum orientations, atomic orbitals, and the corresponding wavefunctions, emphasizing their visualization and implications in molecular interactions.
- 4.8: 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.