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This paper is addressed to the problem of Galilei invariant basis construction for identical fermions systems. The recently introduced method for spurious state elimination from expansions in harmonic oscillator basis [1] is adopted and applied to bound states of two particles system with Coulomb potential description. Traditional expansions in this case demonstrate the extremely well-known slow convergence, and hence this is the best problem with known exact solutions for the test of the method. Obtained results demonstrate the significant simplification of the problem and fast convergence of expansions. We show that the application of this general method is very efficient in a test case of the energy spectrum calculation problem of two particles with different masses interacting with Coulomb potential.

The recent development of parallel-computing facilities has promoted a rapid evolution of the large-scale quantum systems’ calculations. Therefore, there is a chance to check the precision of well-known methods for these systems’ description, as a rule, violating principle of Galilei invariance.

Matter is invariant with respect to translations in space; therefore, the operators of observables and wave- functions of the self-bound system must be translationally invariant. However, this invariance significantly complicates the problem. As a result, the best-known methods for the quantum systems’ description, such as the Shell Model or the Hartree-Fock Self-Consistent Field Method [^{−4}, applying the wave-functions dependent on redundant (i.e. center-of-mass of entire system) variable, one cannot be sure about the declared high precision of calculations. Obviously, for nuclei and hadrons the translational invariance appears as an even more significant problem. Common is the statement claiming that the error in this case will be of order 1/A, where A is baryon charge of corresponding nucleus. Obviously, now such precision of atomic nuclei description is unacceptable.

Some methods, developed for few-body systems, are designed to produce translationally invariant wave functions. These are methods based on the Faddeev [

Therefore, the basis of wave functions written in one-particle degrees off reedom is the best solution for quantum systems, composed of identical fermions, description. The well-known and traditional solution of this problem is Shell Model (SM), based on the fact that wave function is written as a product of one particle functions, antisymmetrized applying fractional parentage coefficients [

The only choice for the solution of this problem is center of mass state fixation. The center-of-mass Hamiltonian and intrinsic Hamiltonian operators commute, because they are functions of different variables, hence center-of-mass Hamiltonian commutes also with total Hamiltonian, expressed in one-particle variables and both have coinciding sets of eigen functions. The wave function of bound state has to be square integrable, hence square integrable has to also be the center-of-mass wave function. The easiest way for such function constructionis modification of Hamiltonian adding and subtracting some potential, dependent on center-of-mass radius vector. The best choice for this operator is harmonic oscillator potential because this operator, proportional the square of center-of-mass radius vector can be easily transformed to form of operator, dependent on one-particle radius- vectors, as defined in [

Therefore, for Galilei invariant quantum system description one needs the following operations. The first one is the construction of harmonic oscillator shell model functions basis, antisymmetrized in respect of identical fermions, composing system under investigation or it’s subsystem degrees of freedom permutations and equipped by necessary quantum numbers like total number of oscillator quanta, total momentum, parity and so on. Later on, linear combinations of these functions, corresponding the ground state of mentioned above three dimensional center of mass harmonic oscillator Hamiltonian are necessary. As is shown in [

The standard procedure, applying for quantum system wave function expansion the basis of many particle harmonic oscillator functions, cannot ensure Galilei invariant results of calculations because the basis without mentioned modification contains significant amount of basic functions with excited center of mass, called as spurious states. Well known in such approaches are huge matrices of Hamiltonian and bad convergence of expansions, first of all -due to Gaussian asymptotic of harmonic oscillator functions.

The reduction of basis dimensions after operation, as it follows from [

The purpose of this paper is to describe and illustrate the application of this method for handling the energy spectrum calculation problem of two particles with different masses interacting with Coulomb potential. Since the treatment of Coulomb potential in the harmonic oscillator basis is known as a hard computational problem, the obtained in this case peculiarities of the method will be valuable for further investigations of many-particle systems.

The Hamiltonian of two particles system, with masses

where

Here

The problem under consideration is investigation of spectrum and eigenfunctions of this Hamiltonian expansion in basis of three-dimensional harmonic oscillator functions. The bound angular moment a two-particle har- monic oscillator functions can be expressed in terms of single-particle functions utilizing the angular momentum Clebsch-Gordan coefficients

Here

where

The expectation values of kinetic energy operators of Hamiltonian (2.3) after application of the Wigner-Eckart theorem, may be expressed by reduced matrix element

represented as double-bar matrix element. The reduced matrix element of operator acting only on coordinates of first particle may be expressed in terms of 6j coefficient and the reduced matrix element between the single-particle functions depending only on coordinates of the first particle [

Since the considered Hamiltonian is a scalar in angular-momentum space, the tensorial rank of the kinetic energy operator is zero. After application of Equation (2.7), the expression of the reduced matrix element of the first kinetic energy term takes the form

Due to zero value of one of the parameters of the 6j coefficient, it may be significantly simplified [

Therefore, the reduced matrix element of the first kinetic energy term may be written as

After these manipulations the matrix element assumes the form

Here we present its abbreviated form as well. The second kinetic energy term of Hamiltonian (2.3) may be evaluated using the formula that expresses the reduced matrix element of operator acting only on coordinates of second particle of the two-particle coupled state in terms of 6j coefficient and the reduced matrix element between the single-particle functions depending only on coordinates of second particle [

Then for the matrix element of the second kinetic energy term, between the coupled two-particle harmonic oscillator functions, we find that

The expectation value of the two-particle Coulomb operator

at

In these new coordinates the Coulomb potential takes a single-particle form due to relation

where the reduced mass is introduced

The same transformation to Jacobi coordinates should be accomplished for the wave functions. By means of Talmi-Moshinsky brackets the two particle harmonic oscillator functions may be expanded in terms of vector coupled products of the single Jacobi variable functions [

Here ε_{0} and

The matrix element in Equation (2.19) may be simplified similarly as the kinetic energy terms

Here the integral may be obtained most easily by direct numerical integration. Finally, the matrix element of the first Coulomb term between the coupled two-particle harmonic oscillator functions takes the form

Here, the summation parameters are under restrictions:

The obtained abbreviated form for the Hamiltonian matrix element becomes

Here the dimensionless harmonic oscillator length parameter:

Let us use the formation of the matrix (2.23), its diagonalization, and calculation of the eigenvalues for the Hamiltonian (2.3) as a benchmark. In this case the calculated eigenvalues can be compared directly with well- known spectrum:

The dependence of the first five eigenvalues on the dimensionless harmonic oscillator length parameter

sis functions with larger values of

The dependence of the ground-state eigenvalue of the Hamiltonian on the size of the basis and the parameter

For calculation of eigenvalues and eigenfunctions of the two particles with different masses bound by Coulomb potential, we apply the efficient method proposed in [

Here the orthogonal transformation (2.14) to the Jacobi coordinates is applied. The obtained result illustrates the most distinct advantage of the method, i.e. the significant reduction and simplification of modified basis. It should be stressed that unlike the two-particle basis case, the modified basis is strictly decomposed in products of the center-of-mass and relative coordinates terms. So, the spurious states in the modified basis are explicitly separated. The Hamiltonian in the introduced Jacobi coordinates takes the form

dim | g.s. eigenvalue | ||
---|---|---|---|

5 | 2.1286 | 10 | −0.65199 |

10 | 1.9320 | 56 | −0.79018 |

15 | 1.8856 | 120 | −0.83607 |

20 | 1.8546 | 286 | −0.87777 |

25 | 1.8462 | 455 | −0.89599 |

30 | 1.8414 | 816 | −0.91541 |

Following the method [

Now, it is obvious that due to the applied method the significant simplification of the initial problem is obtained. The problem is reduced to the problem of motion of one particle in the potential with fixed coordinates. The matrix elements of the second kinetic energy term may be easily evaluated

Here we introduce more compact notation for the matrix element of the kinetic energy term depending on the Jacobi coordinate with the index equal to one. The radial harmonic-oscillator wave functions of the form (2.5) may be used for numerical evaluation of the Coulomb matrix elements

Here the more compact notation of Coulomb matrix elements is introduced as well. It should be noted that Coulomb matrix elements may be evaluated analytically in terms of the hypergeometricfunction, which is computationally more efficient especially in the case of large values of quantum numbers

Here widely accepted notations [

and

The resulting abbreviated form for the Hamiltonian matrix element in the units of the first eigenvalue (2.22) becomes

Let us illustrate the accuracy of calculations by computing the eigenvalues of Hamiltonian (2.30) in the basis of functions (2.25). In

Eigenvalues | Exact | ||
---|---|---|---|

0.32661 | −0.99999595 −0.23247534 | −1 −0.25 | −0.00000405 −0.01752466 |

0.75626 | −0.99995744 −0.24999395 −0.10872288 −0.02208709 | −1 −0.25 −0.11111111 −0.0625 | −0.00004256 −0.00000605 −0.00238823 −0.04041291 |

1.2946 | −0.99979056 −0.24997379 −0.11110222 −0.06161077 −0.02601705 | −1 −0.25 −0.11111111 −0.0625 −0.04 | −0.00020944 −0.00002621 −0.00000889 −0.00088923 −0.01398295 |

1.9563 | −0.99929603 −0.24991175 −0.11108495 −0.06248778 −0.03954675 −0.02141062 | −1 −0.25 −0.11111111 −0.0625 −0.04 −0.02777778 | −0.00070397 −0.00008825 −0.00002616 −0.00001222 −0.00045325 −0.00636716 |

2.7020 | −0.99820520 −0.24977440 −0.11104419 −0.06247176 −0.03998363 −0.02744921 −0.01661333 −0.00263523 | −1 −0.25 −0.11111111 −0.0625 −0.04 −0.02777778 −0.02040816 −0.015625 | −0.00179480 −0.00022560 −0.00006692 −0.00002824 −0.00001637 −0.00032857 −0.00379483 −0.01298977 |

The purpose of the present work is to demonstrate the efficiency of the spurious state elimination method [

The attempt to solve the eigenvalue problem for deuteron in the single particle basis variationally for up to 20 harmonic oscillator quanta using the standard Lawson method [

AlgirdasDeveikis,Gintautas P.Kamuntavičius， (2015) New Possibilities of Harmonic Oscillator Basis Application for Quantum System Description. Two Particles with Coulomb Interaction。 Journal of Modern Physics，06，403-413. doi: 10.4236/jmp.2015.64044