1: The Basic Tools of Quantum Mechanics
- Page ID
- 60508
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Quantum Mechanics Describes Matter in Terms of Wavefunctions and Energy Levels and physical Measurements are Described in Terms of Operators Acting on Wavefunctions
- 1.1: Operators
- This page discusses the relationship between operators and measurable physical quantities in quantum mechanics, highlighting that eigenvalues represent observable values. It explains the process of deriving operators from classical expressions and the complexities involved in curvilinear systems.
- 1.2: Wavefunctions
- This page covers the concept of wavefunctions in quantum mechanics, represented as \(\Psi\), which describe a system's state based on spatial coordinates and time. It explains that probability densities for observing specific coordinates at a given time are calculated using |\(\Psi(q_j, t)|^2\). For complex systems like the \(H_2O\) molecule, the wavefunction incorporates multiple coordinates.
- 1.3: The Schrödinger Equation
- This page covers the Time-Dependent and Time-Independent Schrödinger Equations in quantum mechanics, focusing on deriving wavefunctions through operator formalism and using the Hamiltonian operator to represent total energy. It highlights eigenfunctions and eigenvalues for quantifying quantum systems.
- 1.4: Free-Particle Motion in Two Dimensions
- This page discusses the quantization of energy levels for an electron in a two-dimensional system, emphasizing how confinement leads to discrete energy states. It covers the application of the Schrödinger equation, separation into independent x and y dimensions, and the derivation of energy equations in the context of both quantum and classical mechanics.
- 1.5: Particles in Boxes
- This page discusses the particle-in-a-box model in chemistry, which illustrates electron states and nuclear motion in various systems. It explains how this model applies to different dimensions and is relevant to \(\pi\)-electron behavior in polyenes, linking box length to carbon-carbon bonds. While it accurately approximates excitation energies, adjustments are required for aligning energy levels with ionization energies.
- 1.6: One Electron Moving About a Nucleus
- This page covers the Schrödinger equation for a particle in central potential, focusing on spherical coordinates and the resulting angular momentum quantization. It derives Legendre polynomials and spherical harmonics through differential equations, emphasizing quantization conditions based on boundary constraints.
- 1.7: Harmonic Vibrational Motion
- This page explores the radial motion of diatomic molecules in their lowest rotational state through the Schrödinger equation. It introduces the harmonic oscillator model, noting its limitations in handling anharmonicity and bond dissociation. The Morse potential is proposed as a more accurate alternative for modeling vibrational states, illustrating how its energy levels vary as they approach the dissociation threshold, thereby correcting the deficiencies of the harmonic oscillator.
- 1.8: Rotational Motion for a Rigid Diatomic Molecule
- This page explains the rigid rotor model for diatomic molecules, detailing their rotational dynamics through a Schrödinger equation. It highlights the connection between rotation angles and energy levels, showing how energy eigenvalues are influenced by the rotational quantum number J. The rotational constant B is linked to bond length and reduced mass. Additionally, it discusses how vibrational motion and centrifugal distortions impact ro-vibrational energy levels.
- 1.9: The Physical Relevance of Wavefunctions, Operators and Eigenvalues
- This page provides an overview of key concepts in quantum mechanics, including wavefunctions, operators, and eigenvalues. It details measurement principles, emphasizing that only Hermitian operators yield observable quantities. The behavior of eigenvalues and eigenfunctions, along with the significance of angular momentum quantization and the role of commuting and non-commuting operators in measurements, is explored.