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Cambridge University Press eBooks

Approximation Methods

Uri Peskin

Published May 11, 2023• 0 citations
Publisher

Abstract

Exact solutions to the Schrödinger equation for realistic nanoscale systems are beyond reach, hence, different strategies for approximating the solutions are necessary. Perturbation theory relies on a “zero-order Hamiltonian” to express the desired eigenvalues and eigenvectors of “the full Hamiltonian.” We derive working equations for the Rayleigh–Schrödinger perturbation theory, and the validity of the approach is analyzed for a generic two-level system. Applications are given to atoms perturbed by point charges or static fields, and for electrons in quantum wells. An alternative strategy is the variation method, which replaces exact solutions by their projection on a reduced space of “trial functions,” varied to minimize the associated error. Particularly important is the method of linear variation, which can potentially converge to the exact Hamiltonian eigenstates. The mean-field approximation, commonly used for many-particle systems, is derived by optimizing a trial function in the form of product of single-particle functions.

DOI
10.1017/9781108877787.013

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The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.

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The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation can be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges. The splitting of an energy level due to the interaction of the nuclear magnetic dipole and the magnetic field Be produced by the electron orbit current is called the fine structure. The term hyperfine structure is used to describe extremely small splitting of spectral lines.

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Many physical systems cannot be solved exactly. In this case, we look for approximate methods to gain insight into their properties. Perturbation theory is one such method. When the Hamiltonian of the system doesn't change over time, one approximate method is the Time Independent Perturbation Theory, which is the subject of Chapter 10 . The infinite square well, the harmonic oscillator, and the hydrogen atom are among the few physical systems for which the Schrodinger equation can be solved exactly. For this reason, they are described in detail in most of the introductory quantum mechanics textbooks. However, most physical systems are not exactly solvable, i.e., they are described by the Schrodinger equation, whose solutions cannot be expressed in terms of commonly used functions. In order to study such systems, we try to decompose the corresponding Hamiltonian into a solvable Hamiltonian plus a small perturbation. For such cases, we say that the Hamiltonian is “close enough” to an exactly solvable Hamiltonian to warrant using perturbative methods to approximate its eigenenergies and eigenfunctions.

CRC Press eBooks•Jul 28, 2025•0 citations
Exact solutions to the Schrödinger equation for realistic nanoscale systems are beyond reach, hence, different strategies for approximating the solutions are necessary. Perturbation theory relies on a “zero-order Hamiltonian” to express the desired eigenvalues and eigenvectors of “the full Hamiltonian.” We derive working equations for the Rayleigh–Schrödinger perturbation theory, and the validity of the approach is analyzed for a generic two-level system. Applications are given to atoms perturbed by point charges or static fields, and for electrons in quantum wells. An alternative strategy is the variation method, which replaces exact solutions by their projection on a reduced space of “trial functions,” varied to minimize the associated error. Particularly important is the method of linear variation, which can potentially converge to the exact Hamiltonian eigenstates. The mean-field approximation, commonly used for many-particle systems, is derived by optimizing a trial function in the form of product of single-particle functions.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
A self-contained pedagogical introduction to the functional Schrödinger picture method of many-body theory is given at a level suitable for graduate students and also for many-body physicists who have not been exposed to the functional Schrödinger picture method previously. Mathematical tools necessary for the functional Schrödinger picture calculation are introduced. The method is first applied to various condensed matter problems including the electron gas, the Hubbard model, the BCS superconductivity, and dilute bose gas within the variational approximation. It is shown that the variational approximation with the Gaussian trial functional invariably leads into the Hartree-Fock results for both zero and finite temperature cases. In order to go beyond the Gaussian results, concepts of variational and optimized perturbation theories are introduced using simple quantum mechanical languages. Then, the variational perturbation theory in the functional Schrödinger picture is applied to the λϕ4λϕ^4λϕ4 model to yield the effective potential to the third order. The optimized perturbation theory is also applied to the λϕ4λϕ^4λϕ4 model to yield a general expression up to the second order.
Approximation of eigen values and eigenvectors Many realistic problems that arise in theoretical physics lead to equations which cannot be given exact general solutions. Therefore, many attempts have been made to produce satisfactory methods to provide approximate solutions. The most common ones search for solutions in terms of power series and in terms of asymptotic expansions. The main idea behind approximation methods is to try to find approximate solutions of a given system by means of exact solutions of an approximate system (a ‘nearby’ system). We start with the eigenvalue equation associated with the Hamiltonian H and consider an approximating one, say H 0 , whose equations of motion can be explicitly solved. We introduce an interpolating family of Hamiltonians H ε = H 0 + ε H 1 such that, for a given value of ε, say ε, H ε ≡ H . Both eigenvalues and eigenvectors for H ε may be expanded as a series in ε and solved order by order. The simplest example to illustrate what we have in mind is provided by a second-order algebraic equation, e.g. x 2 − 3.03 x + 2.02 = 0. This equation may be written by means of an interpolating family x 2 − 3(1 + ε) x + 2(1 + ε) = 0, which yields the equation we want to solve when ε = 10 −2 . One can consider the approximate equation to be given by ε = 0, i.e. x 2 − 3 x + 2 = 0.
Exactly solvable quantum systems are rare—the harmonic oscillator, the hydrogen atom, the square well—and the physics of everything else is the physics of approximation. This article presents a narrative review of the primary literature that built quantum mechanics' approximation methods, from Rayleigh's Theory of Sound, whose treatment of slightly perturbed vibrating systems gave perturbation theory its name and scheme, and Ritz's 1909 variational method, which converted boundary-value problems into energy minimizations, through Schrödinger's 1926 wave mechanics, which equipped the new quantum theory with perturbation theory from the start, Born and Oppenheimer's 1927 adiabatic separation of electronic and nuclear motion, the 1926 papers of Wentzel, Kramers, and Brillouin that independently derived the semiclassical expansion bearing their initials, Hartree's 1928 self-consistent field and Hylleraas's 1929 variational computation of helium, which made the many-electron atom calculable, Kato's 1949 mathematical foundation of the perturbation method's convergence, Landau and Lifshitz's codifying textbook, and Bender and Wu's 1969 demonstration that the anharmonic oscillator's perturbation series diverges—the asymptotic character that defines the method's limits. The synthesis is organized around three themes: the perturbation program, in which small parameters turned the solvable cases into engines for the unsolvable; the variational program, in which upper bounds and trial functions turned minimization into the many-body atom's working instrument; and the semiclassical bridge, in which the WKB expansion connected wave mechanics to action-angle quantization and exposed tunneling. It is concluded that the three programs are complementary solutions to one problem—the gap between what can be solved and what must be calculated—and that their asymptotic character, made precise by Bender and Wu, is not a defect but the field's realistic self-understanding.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation can be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges. The splitting of an energy level due to the interaction of the nuclear magnetic dipole and the magnetic field Be produced by the electron orbit current is called the fine structure. The term hyperfine structure is used to describe extremely small splitting of spectral lines.
Many physical systems cannot be solved exactly. In this case, we look for approximate methods to gain insight into their properties. Perturbation theory is one such method. When the Hamiltonian of the system doesn't change over time, one approximate method is the Time Independent Perturbation Theory, which is the subject of Chapter 10 . The infinite square well, the harmonic oscillator, and the hydrogen atom are among the few physical systems for which the Schrodinger equation can be solved exactly. For this reason, they are described in detail in most of the introductory quantum mechanics textbooks. However, most physical systems are not exactly solvable, i.e., they are described by the Schrodinger equation, whose solutions cannot be expressed in terms of commonly used functions. In order to study such systems, we try to decompose the corresponding Hamiltonian into a solvable Hamiltonian plus a small perturbation. For such cases, we say that the Hamiltonian is “close enough” to an exactly solvable Hamiltonian to warrant using perturbative methods to approximate its eigenenergies and eigenfunctions.