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7: Approximation Methods

Last updated

Apr 16, 2022
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- 6.E: The Hydrogen Atom (Exercises)
- 7.1: The Variational Method Approximation

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The Schrödinger equation for realistic systems quickly becomes unwieldy, and analytical solutions are only available for very simple systems - the ones we have described as fundamental systems in this module. Numerical approaches can cope with more complex problems, but are still (and will remain for a good while) limited by the available computer power. Approximations are necessary to cope with real systems. Within limits, we can use a pick and mix approach, i.e. use linear combinations of solutions of the fundamental systems to build up something akin to the real system. There are two mathematical techniques, perturbation and variation theory, which can provide a good approximation along with an estimate of its accuracy. These two approximation techniques are described in this chapter.

7.1: 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.
7.2: Linear Variational Method and the Secular Determinant
This page discusses the linear variational method for approximating molecular wavefunctions using trial wavefunctions as linear combinations of basis functions. It explains variational energy minimization through secular equations involving Hermitian operators, particularly the Hamiltonian. The text highlights the importance of determinant properties in finding energy levels and mentions the use of Gaussian orbitals for testing the method on Hydrogen, despite the exactness of Slater orbitals.
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.E: Approximation Methods (Exercises)
This page covers various applications of the variational method and perturbation theory in quantum mechanics. It explores trial wavefunctions for harmonic and anharmonic oscillators, including calculations of ground state energies and properties using Hamiltonian operators and integrals. The text details the evaluation of determinants and the conceptual framework for particle systems.

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