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Journal of Shandong University

Study on conditions for the existence of bound states in central field with the variational method

Chen Gang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In quantum physics,the variational method can be used to study the conditions for the existence of bound states in a central field.We take the following six kinds of potentials,V1(r)=-V0a δ(r-a),V2(r)=-V0e-r/a,V3(r)=-V0ae-r/a/r,V4(r)=-V0e-r2/a2,V5(r)=-e2/r,and V6(r)=-A/r2 as examples to demonstrate the usage of this method.The derived conditions for V1 to V4 are μγa2/h-20.68,0.84,1,1.95 respectively;for V5,the bound states can exist for e with arbitrary value,while for V6,there is no bound states for any value of A.We also get the precise conditions by solving the Schroedinger equation for these potentials.A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient,but not necessary conditions.

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In quantum physics,the variational method can be used to study the conditions for the existence of bound states in a central field.We take the following six kinds of potentials,V1(r)=-V0a δ(r-a),V2(r)=-V0e-r/a,V3(r)=-V0ae-r/a/r,V4(r)=-V0e-r2/a2,V5(r)=-e2/r,and V6(r)=-A/r2 as examples to demonstrate the usage of this method.The derived conditions for V1 to V4 are μγa2/h-20.68,0.84,1,1.95 respectively;for V5,the bound states can exist for e with arbitrary value,while for V6,there is no bound states for any value of A.We also get the precise conditions by solving the Schroedinger equation for these potentials.A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient,but not necessary conditions.
A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]
We propose a new method to obtain approximate solutions for the Schrödinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows to find in many cases, analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which greatly improves the corresponding formulae that can be found in the literature.
Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.