The authors show how the perturbative-variational approximation of eigenvalues and eigenvectors, that has provided very good results for one-dimensional problems, can be generalized to Hamiltonians with degenerate states. This generalization will be illustrated by applying it to a pair of coupled harmonic oscillators. The obtained results for the ground and first excited states have a very good precision when they are compared with the results of direct diagonalization of the Hamiltonian with 190 states in the same auxiliary basis used in the perturbative-variational approximation.
The authors show how the perturbative-variational approximation of eigenvalues and eigenvectors, that has provided very good results for one-dimensional problems, can be generalized to Hamiltonians with degenerate states. This generalization will be illustrated by applying it to a pair of coupled harmonic oscillators. The obtained results for the ground and first excited states have a very good precision when they are compared with the results of direct diagonalization of the Hamiltonian with 190 states in the same auxiliary basis used in the perturbative-variational approximation.
Perturbation theory for non-degenerate and degenerate systems, variational methods, and WKB approximation are covered in this chapter. First, second, and kth order corrections to energy and wave function ket of non-degenerate states, energy eigenvalues, and wave functions kets of anharmonic oscillator, first-order perturbation theory for degenerate states and linear stark effect are written under perturbation theory. Second part of the chapter includes the variational methods for computation of the energy for ground state of helium atom, Rayleigh-Ritz variational method and its application to hydrogen molecule ion, calculation of energy of excited states, and its application to 1D harmonic oscillator. The topics which are covered under WKB approximation are the classical and non-classical regions of particle motion, alternative derivation of WKB formulae, connecting formulae, and quantum condition for bound state. The kth order corrections to energy and wave function kets for non-degenerate perturbation theory, treatment of WKB approximation from two different approaches and the detailed derivation of connecting formulae, application of variational method to compute energy for excited states are topics which are not generally found in existing text books on quantum mechanics. A good number of solved examples illustrating the importance of approximation methods in contemporary fields are added at the end of chapter.
Although the weakly coupled double pendulum seems to be a standard illustration in classical mechanics, it is seldom mentioned in quantum mechanics. It shares this fate with the damped harmonic oscillator, but while there is a good reason for avoiding the last one in quantum mechanics, the first can be treated with the same procedure as in classical mechanics. The exact solution is then compared with the solutions obtained by time-independent and time-dependent perturbation methods. It turns out that there are some additional steps to be taken, compared to the usual textbook treatment of the time-independent perturbation theory, due to the fact that all the levels of the unperturbed problem are degenerate. It is shown that the diagonalization of the secular matrix in the time-independent problem is equivalent to a problem of rotation of angular momentum operators in function space. The time-dependent problem needs, again, additional steps due to the degeneracy, and it yields the well-known beat-behavior.
Starting from the study of one-dimensional potentials in quantum mechanics having a small distance behavior described by a harmonic oscillator, we extend this way of analysis to models where such a behavior is not generally expected. In order to obtain significant results we approach the problem by a renormalization group method that can give a fixed point Hamiltonian that has the shape of a harmonic oscillator. In this way, good approximations are obtained for the ground state both for the eigenfunction and the eigenvalue for problems like the quartic oscillator, the one-dimensional Coulomb potential having a not normalizable ground state solution and for the one-dimensional Kramers-Henneberger potential. We keep a coupling constant in the potential and take it running with a generic cut-off that goes to infinity. The solution of the Callan-Symanzik equation for the coupling constant generates the harmonic oscillator Hamiltonian describing the behavior of the model at very small distances (ultraviolet behavior). This approach, although algorithmic in its very nature, does not appear to have a simple extension to obtain excited state behavior. Rather, it appears as a straightforward non-perturbative method.
The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.
A new approach to the eigenvalue problem in quantum mechanics is proposed. This approach is based on three propositions: 1) a perturbation theory which does not require knowledge of the entire eigenvalue spectrum of the unperturbed problem and which uses a nonlinearization procedure (leaving some latitude in the choice of a zeroth order approximation); 2) a relationship between the perturbation theory and a variational principle, namely that any variational calculation is none other than the first two terms of some nontrivial perturbation theory which, when developed further, can reveal the accuracy of the variational calculations and can refine them by an iterative procedure; 3) Dyson's argument, which serves as a criterion for the reasonableness of the choice of a zeroth order approximation (the unperturbed problem). The realization of this perturbation theory in a k-dimensional space is equivalent to the solution of a k-dimensional electrostatic problem with a variable dielectric permittivity. In the one-dimensional case and in cases which reduce to the one-dimensional case, all the corrections are written in quadratures. It is shown that the construction of an ordinary perturbation theory (in which the zeroth order approximation is an exactly solvable problem) within the framework of this perturbation theory is a purely algebraic procedure, which reduces to the solution of some simple recurrence relations. An approximation analogous to the leading logarithmic approximation of quantum field theory is constructed. Some standard problems of quantum mechanics—the anharmonic oscillator, the Zeeman effect, and the Stark effect—are treated as examples. It is shown that this new approach makes it possible to develop systematically a theory for strong coupling and large perturbations.