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Sum rules and the domain after the last node of an eigenstate

C V Sukumar

Published Oct 24, 2006• 0 citations
OpenAlex

Abstract

It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schroedinger equation has two linearly independent solutions one of which is normalisable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalisable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules give conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on Supersymmetric Quantum Mechanics are discussed.

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Sum rules and the domain after the last node of an eigenstate

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It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schrödinger equation has two linearly independent solutions one of which is normalizable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalizable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules gives conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on supersymmetric quantum mechanics are discussed.

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It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schroedinger equation has two linearly independent solutions one of which is normalisable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalisable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules give conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on Supersymmetric Quantum Mechanics are discussed.
It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schrödinger equation has two linearly independent solutions one of which is normalizable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalizable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules gives conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on supersymmetric quantum mechanics are discussed.
There are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ϕ n ( x ) and eigenvalues α n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ϕ n ( x ) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.
Starting from the orthonormal eigenfunctions which are the solutions of the Schrödinger equation for a given potential it is shown that sets of functions orthogonal to the eigenstates inside a region of varying radius R may be constructed. These sets of functions may in turn be used to construct a new set of functions which satisfy bound-state boundary conditions and are themselves solutions of the Schrödinger equation in a new potential. The bound-state spectrum of the new potential is related to the spectrum of the original potential in a definite manner. The relationship of this construction of a new potential to the inverse scattering theory approach based on the Gelfand–Levitan equation and the potentials constructed using supersymmetric quantum mechanics (SUSYQM) is explored. The connection of this approach to other approaches based on the implementation of orthogonality and their relation to Pauli exclusion principle is examined.
We discuss the solutions of the Schroedinger equation for piecewise potentials, given by the harmonic oscillator potential for |x|>a and by an arbitrary function for |x|<a, using elementary methods. The study of this problem sheds light on usual errors made in discussions of the asymptotic behavior of the eigenfunctions of the quantum harmonic oscillator and can also be used for the analysis of the eigenfunctions of the hydrogen atom. We present explicit results for the energy levels of a potential of this class, used to model the confinement of electrons in nanostructures.
A general formulation of the eigenvalue condition is pointed out for a one-dimensional stationary part of a separable Schrodinger equation with finite bounds, i.e. where the wavefunction vanishes at two arbitrary values of the independent variable. The procedure is general and applies equally to other eigenvalue problems of second-order, homogeneous differential equations. Where appropriate a continuous density of eigenvalues can be defined and evaluated without first solving the eigenvalue problem. Illustrative examples are provided, including gravitational and harmonic oscillator potentials.