7: So You Can't Solve the Schrodinger Equation Exactly...What Next? (Approximation Methods)
- Page ID
- 521145
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 7.1: The Helium Atom Cannot Be Solved Exactly
- This page addresses the complexities of solving Schrödinger equations for multi-electron atoms like helium, which lacks an analytic solution unlike hydrogen. It discusses the non-separable nature of multi-electron Hamiltonians, the orbital approximation for wavefunctions, and the effects of electron-electron interactions. Key components of the helium Hamiltonian are explored, along with the significance of neutron variations on electron energies.
- 7.2: The Variational Method Approximation
- This page explains the variational method in quantum mechanics, which refines helium atom approximations by incorporating electron interactions through an effective nuclear charge. It optimizes wavefunction parameters to minimize energy, yielding better ground-state energy estimates than basic models. The method highlights the complexities of multi-electron atoms, making accurate predictions possible.
- 7.3: Trial Functions Can Be Linear Combinations of Functions That Also Contain Variational Parameters
- This page explores variational methods in quantum mechanics, particularly the construction of wavefunctions using linear combinations of basis functions. It differentiates between normal and nonlinear variational methods, emphasizing the computational advantages of the former. The significance of adjustable zeta parameters in modeling electron interactions, especially in multi-electron atoms, is highlighted.
- 7.4: Perturbation Theory Expresses the Solutions in Terms of Solved Problems
- This page examines perturbation theory in quantum mechanics, emphasizing its role in approximating energy and wavefunction changes from small Hamiltonian alterations. It focuses on first-order perturbation, detailing energy shifts and wavefunction changes through specific examples, including harmonic oscillators and different potentials.
- 7.5: Perturbation Theory and the Variational Method for Helium
- This page explores methods for solving the helium atom's electron structure using perturbation theory and variational methods. It highlights the refinement of energy estimates by incorporating electron-electron interactions, notably showing that using complex trial wavefunctions leads to results within 0.08% of experimental values. The Chandrasakar wavefunction notably reduces electron-electron repulsion and achieves an accuracy of 0.07%.
- 7.6: Atomic and Molecular Calculations are Expressed in Atomic Units
- This page discusses the benefits of using atomic units (au) in atomic physics, emphasizing their role in simplifying calculations compared to SI units. It details how atomic units standardize mass, charge, and Planck's constant, allowing for a clearer focus on important physical factors. The article compares the Hamiltonian of a helium atom in both unit systems, illustrating the clarity atomic units provide.
- 7.7: Hartree-Fock Equations are Solved by the Self-Consistent Field Method
- This page explores the Hartree approximation's role in determining wavefunctions and energies for multi-electron atoms by treating electrons as independent entities interacting through an average potential. It outlines the Self-Consistent Field (SCF) method and its iterative approach to solving the Schrödinger equation. The Hartree-Fock method enhances accuracy with antisymmetrized wavefunctions and addresses the challenges of electron-electron repulsion.
- 7.8: An Electron has an Intrinsic Spin Angular Momentum
- This page explores electron spin, an intrinsic angular momentum linked to magnetic properties and quantum mechanics. It highlights two spin states (α and β) with different energies influenced by magnetic fields, demonstrated by the Zeeman effect and the Stern-Gerlach experiment. The g-factor, approximately 2.0023, relates to the spin gyromagnetic ratio, indicating that electron spin is more complex than simple orbital motion.
- 7.9: Wavefunctions must be Antisymmetric to Interchange of any Two Electrons
- This page explores quantum mechanics principles for multi-electron atoms, highlighting indistinguishable particles and the Pauli Exclusion Principle. It explains the unchanged probability density when electrons swap positions and introduces the Schrödinger equation for helium. Additionally, it discusses exchange symmetry in particles, distinguishing between bosons and fermions through the exchange operator, \(\hat{P}_{12}\).