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CRC Press eBooks

Perturbation Theories without Wavefunction

Francisco Marcelo Fernández

Published Sep 19, 2000• 0 citations
Publisher

Abstract

If the Hamiltonian operator depends on a parameter λ, which may be a particle charge or mass, a field strength, or simply a dummy parameter artificially introduced to apply perturbation theory, then the eigenvalues and eigenfunctions of Hˆ will also depend on that parameter. If we differentiate < |(Hˆ − E)| >= 0 with respect to λ 〈 ∂ ∂λ ∣∣∣(Hˆ − E)∣∣∣ 〉 + 〈 ∣∣∣∣∣ ∂(Hˆ − E) ∂λ ∣∣∣∣∣ 〉 + 〈 ∣∣∣(Hˆ − E)∣∣∣ ∂ ∂λ 〉 = 0 (3.5) and take into account equation (3.2), then we obtain the Hellmann-Feynman theorem [25] ∂E ∂λ < | >= 〈 ∣∣∣∣∣ ∂Hˆ ∂λ ∣∣∣∣∣ 〉 , (3.6) or ∂E ∂λ = 〈 ∂Hˆ ∂λ 〉 (3.7) if is normalized to unity. In what follows we show that the general results just derived facilitate the application of pertur- bation theory.

DOI
10.1201/9781420039641-6

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If the Hamiltonian operator depends on a parameter λ, which may be a particle charge or mass, a field strength, or simply a dummy parameter artificially introduced to apply perturbation theory, then the eigenvalues and eigenfunctions of Hˆ will also depend on that parameter. If we differentiate < |(Hˆ − E)| >= 0 with respect to λ 〈 ∂ ∂λ ∣∣∣(Hˆ − E)∣∣∣ 〉 + 〈 ∣∣∣∣∣ ∂(Hˆ − E) ∂λ ∣∣∣∣∣ 〉 + 〈 ∣∣∣(Hˆ − E)∣∣∣ ∂ ∂λ 〉 = 0 (3.5) and take into account equation (3.2), then we obtain the Hellmann-Feynman theorem [25] ∂E ∂λ < | >= 〈 ∣∣∣∣∣ ∂Hˆ ∂λ ∣∣∣∣∣ 〉 , (3.6) or ∂E ∂λ = 〈 ∂Hˆ ∂λ 〉 (3.7) if is normalized to unity. In what follows we show that the general results just derived facilitate the application of pertur- bation theory.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.
The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]
H R W R y y( ) ( )= (11.15) This system is the general case of a stationary state. This equation is also called the eigenvalue equation in which W is the eigenvalue of the system (constant) and y is the eigenfunction (wavefunction). If a possible value of observable A is defined as · Ò = =ÚA R t A R t dR R t A R tY Y Y Y* *( , ) ( , ) ( , ) ( , ) (11.16)substituting Eq. 11.16 in Eq. 11.15, we get A R t A R e iWt Y System ( , ) ( )= · Ò -y (11.17) Next, if the Hamiltonian operator is separated into unperturbed and perturbation parts as H H H = +0 ¢ (11.18) we will finally have a solution to the energy W and the wavefunction y as W W m H m k H m = + + - + π ¢ ¢ 2 (11.19) m k k H m W W = + - + π  ¢ (11.20) 11.3 Semi-Classical Quantum Mechanical TreatmentHere we will determine the dielectric response of matter under the light frequency by solving a time-dependent perturbation theory (Fig. 11.2).
Abstract The generalized Hellmann–Feynman and curvature theorems are derived for the generalized Sturm–Liouville‐type eigenvalue equation which is widely encountered in many branches of physics. In classical mechanics, the Sturm–Liouville eigenvalue equation primarily arises from applying the theory of small oscillations to vibration problems while, in quantum mechanics, the most important example is the time‐independent Schrödinger equation. The generalized theorems, respectively, provide simple useful expressions for the first and second λ derivatives of the eigenvalue where λ is an arbitrary real parameter appearing in the eigenvalue equation. These results, which apply to both classical and quantal systems, include as special cases the well‐known quantum mechanical Hellmann–Feynman and curvature theorems for the time‐independent Schrödinger equation, thus unifying the formalism. This connection between classical and quantum mechanics is of interest per se, and also because classical eigenvalue equations find application in the treatment of some quantum chemical problems. Two derivations of both generalizations are presented, the first proof applying to exact and Rayleigh–Ritz (linear variational) solutions of the eigenvalue problem, and the second, more general one, to arbitrary optimal variational as well as exact solutions. Necessary and sufficient conditions for the validity of the results are discussed. To demonstrate the insight they afford, several applications of the theorems in conjunction with perturbation theory are described.
The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.