Comment on: Uncommonly accurate energies for the general quartic oscillator, Int. J. Quantum Chem., e26554 (2020), by P.Okun and K.Burke
aa r X i v : . [ qu a n t - ph ] F e b Comment on:
Uncommonly accurate energies for the generalquartic oscillator , Int. J. Quantum Chem., e26554 (2020), byP. Okun and K. Burke
Alexander V. Turbiner ∗ and Juan Carlos del Valle † Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 70-543, 04510 M´exico, D.F., Mexico (Dated: February 25, 2021)It is shown that for the one-dimensional quantum quartic anharmonic oscillatorthe numerical results obtained by Okun-Burke in Ref. 2 are easily reproduced and canbe significantly improved in Lagrange Mesh Method (based on non-uniform lattice). ∗ [email protected] † [email protected] There is a common opinion among people working on one-dimensional quantum dynam-ics that the spectra of eigenvalues of one-dimensional Schr¨odinger equation can be foundnumerically with any desirable accuracy (unlike eigenfunctions), see e.g. [1] and referencestherein . The question is how much time is ready to be invested and what is the mosteconomic way to do so . Variational method is the most common candidate. In Ref.[2]the simplest trial function in the form of linear superposition of the harmonic oscillatoreigenfunctions, we denote it as Ψ HO , was used for quartic anharmonic oscillator − d Ψ dx + V ( x ) Ψ = E Ψ , V ( x ) = − λ x + 14 x , to find energies at nine values of parameter λ , ranging from -1 to 16, for 20 lowest eigenvalues.It was obtained 41 figures in each energy . For this type of trial functions the variationalenergy (saying differently, the expectation value of the Hamiltonian) can be calculated an-alytically, which simplifies the process of minimization if it is needed. However, it is wellknown that this type of trial functions has an intrinsic deficiency: it does not reproduce thecorrect asymptotic behavior of the exact eigenfunctions at large distances, which is ∼ e − √ | x | x . It leads to increasingly slow convergence with the increase of the number of correct figuresin eigenvalues . Furthermore, it gives wrong expectations values: for any α > < exp (cid:8) ax α (cid:9) > | Ψ OB = ∞ , for a >
0, although for α < V trial = 1Ψ HO (cid:18) d dx Ψ HO (cid:19) , behaves at large distances like ∼ x , while the perturbation potential ( V − V trial ) grows In particular, it is shown that in the so-called Riccati-Pade method one can easily reach several hundredfigures in energies for polynomial potentials. Hence, the use in the title of [2] the wording
Uncommonlyaccurate energies . . . is misleading. It is necessary to mention that high-accuracy calculations (say more than 10 figures) suffer from thepresence of different corrections (relativistic, non-pointness of masses, thus, mass distribution effects etc)which has to be taken into account to get physically relevant results. CPU time needed for calculations was not indicated. Following numerous critical remarks by W. Kutzelnigg we think that due to this reason (as well asimpossibility to reproduce the cusp parameters correctly) the Gaussian orbitals were not used to getbenchmark results for the helium atom, see [3], contrary to the Coulomb orbitals. like ∼ x , hence, dominant with respect to V trial and the perturbation theory in deviation( V − V trial ) is divergent. It implies that the first correction to variational energy does not provide the correct estimate of the accuracy in variational energy (for discussion see [4]).The goal of this Comment is to demonstrate that in Lagrange Mesh Method [5] (andreferences therein), based on non-uniform discretization following the zeroes of Hermitepolynomials, the results obtained in [2] can be easily reproduced - it is a matter of seconds inCPU time - and easily overtaken reaching the accuracy &
200 figures in a matter of minutesin CPU time in standard laptop with Mathematica code. Present authors have definitedoubts that similar accuracies can be reached in the method used in [2] in comparabletimes.To simplify presentation let us introduce the following notation X Y Z : X = Digit , Y = Mesh Points , Z = Decimal Place of X , all marked by red color. Digit X indicates the maximal digit in energy which is reproducedwith number of mesh points Y . Maximal accuracy of the energy is calculated with 2000mesh points and checked with 2020 mesh points. Maximal digit, reached in calculations byOkun & Burke (O&B), see [2], is marked by bold, see below.Concrete calculations we made for λ = − λ = 1 ,
16 (double well potential case) for the ground state ( λ = 1 ,
16) andfor 19th excited state ( λ = 1) energies. Computations were carried out in Mathematica-12using iMac with 2.7 GHz Intel Core i5 with 8GB RAM. Quartic Anharmonic Oscillator: λ = − As for the ground state energy: E = 0 .
620 927 02
32 987 120
698 200
O&B
17 25
188 768 883 979 391
351 303 479 456 083 601 618 760 073 476 624 891 085768 308 099 065 938 402 5
80 084 530 397 024 737 474 347 663 406 954 493 075 566 093052 396 859 302 472 486 392 601 975 136 357 293 108 871 529 439
117 092 275 Accuracy of 300 digits in definition of mesh points used. In Mathematica,
WorkingPrecision → It is worth noting that the use of the 25 mesh points allows to get 9 figures (8 decimaldigits (d.d.)) in ∼ . ∼ ∼
35 min , see Table I. It was checked that similar accuracies with similar CPU times arereached for the first 80 eigenstates for both energies and node positions with similar CPUtimes. Let us emphasize that the rate of convergence is about 10-11 correct digits withincrement of the number of mesh points in 100.
Double Well Potential: λ = 1 As for the ground state energy: E = 0 .
147 235 1
40 090 03
69 124
897 756 466 017 325
O&B
55 318 874 539 254 9
92 800 263120 981
357 377 138 079 999 9
82 297 179 296 021 890 349 762 419 246 096 725 129 055929 407 582 589 84
It is worth noting that the use of the 25 mesh points allows to get 8 figures (7 d.d.) in ∼ . ∼
16 sec , see Table II. For 2000 mesh points 239 d.d. are reached, it takes ∼
41 min , seeTable II.While as for the 19th excited state energy: E = 42 .
87 460 39
360 748 4
60 339 151 3
40 412 5 O&B
74 939 156 835
342 873 143 4758
34 442 346 776 630 858 786 014 482 728 909 852 009 515 813 795 919 753 312 2
44 257330 182 061 161 689 338 128 957 261 369 362 484 027 548 806 789 503 865 374 787 715466 718 578 447 536 669 8
10 863 978 041
It is worth noting that in this case the use of 50 mesh points allows to get 3 figures (1d.d.) in ∼ . sec , see Table II. The result by [2] of 40 d.d. requires about 330 mesh points: ittakes ∼
31 sec , see Table II. For 2000 mesh points 219 d.d. are reached, it takes ∼
41 min ,see Table II.
Double Well Potential: λ = 16 It can be called the extreme double well potential case. As for the ground state energy: E = − .
09 723
934 704
051 951
487 83 O&B
44 866 26
34 499630 898 026 753 728 525 948 951 003 309 559 623 352 261 458 5
67 996 340 964 347 302074 068 801 017 081 360 119 109 362 199 469 453 000 146 444 413 730 116 152 941 7989
36 942 637
In this case, by using 25 mesh points it allows to get 4 figures (2 d.d.) in ∼ . ∼
15 sec , seeTable III. For 2000 mesh points 198 d.d. are reached, it takes ∼
34 min , see Table III.
Running Time
Table I: CPU time (R. Time) needed to compute the ground state energy at λ = − versus the number of mesh points (Mesh P.). Mesh P. R. Time Mesh P. R. Time25 0.23 s 250 7.20 s50 0.38 s 300 11.25 s75 0.64 s 400 23.05 s100 1.06 s 500 41.77 s150 2.13 s 1000 4.85 min200 4.15 s 2000 35.85 min
Table II: CPU time (R. Time) needed to compute the energy of any of the first 20 low-lyingstates at λ = 1 in Lagrange-Mesh Method versus the number of mesh points (Mesh P.). Mesh P. R. Time Mesh P. R. Time25 0.22 s 250 21.42 s50 4.12 s 300 28.16 s75 7.50 s 400 47.53 s100 9.55 s 500 1.20 min150 12.32 s 1000 6.10 min200 16.09 s 2000 41.06 min
Table III: CPU time (R. Time) needed to compute the ground state energy at λ = 16 inLagrange-Mesh Method versus the number of mesh points (Mesh P.). Mesh P. R. Time Mesh P. R. Time25 0.27 s 250 8.59 s50 0.48 s 300 12.76 s75 1.65 s 400 24.73 s100 1.06 s 500 42.86 s150 3.14 s 1000 4.76 min200 5.13 s 2000 34.06 min
As the conclusion we have to mention that there are no real obstacles to increase thenumber of mesh points in the Lagrange Mesh Method further, hence, to increase accuracyin eigenvalues of the quartic oscillator. For − ≤ λ ≤
16, the CPU time needed to calculatede ground state energy with given accuracy is basically independent on λ . This method canbe easily applied to any polynomial potential, which has the discrete spectra, for calculationof the eigenvalues. It was checked for the general radial anharmonic oscillator [6] leadingto the benchmark results for cubic, quartic and sextic radial oscillators. Furthermore, theLagrange Mesh Method, which uses non-uniform lattice, allows to get easily the highly-accurate results comparable (or better) the existing benchmark results for the low-lyingstates of hydrogen atom in a constant uniform magnetic field of arbitrary strength [7]. ACKNOWLEDGMENTS
This work is partially supported by CONACyT grant A1-S-17364 and DGAPA grantIN113819 (Mexico). [1] F.M. Fern´andez, J. Garcia,
Highly accurate calculation of the real and complex eigenvalues of one-dimensional anharmonicoscillators , Acta Polytechnica : 391–398, 2017[2] P. Okun, K. Burke,
Uncommonly accurate energies for the general quartic oscillator , Int. J. Quantum Chem. , e26554 (2020)ArXiv: 2007.04762[3] Ch. Schwartz,
Experiment and theory in computations of the he atom ground state , Int Journ Mod Phys E 15 , 877-888 (2006);H. Nakashima, H. Nakatsuji,
Solving the Schr¨odinger equation for helium atom and its isoelectronic ions with the free iterativecomplement interaction (ICI) method,J. Chem. Phys. , 224104 (2007);D.T. Aznabaev, A.K. Bekbaev, and V.I. Korobov,
Nonrelativistic energy levels of helium atoms , Phys. Rev.
A 98 (2018) 012510[4] A. V. Turbiner,
Soviet Phys. – ZhETF , 1719-1745 (1980). JETP , 868-876 (1980) (English Translation); Soviet Phys. - Usp. Fiz. Nauk. , 35-78 (1984),
Sov. Phys. Uspekhi , 668-694 (1984) (English Translation)[5] D. Baye, Phys. Rep. , 1-108 (2015) [6] J.C. del Valle, A. V. Turbiner,
Int.Journ.Mod.Phys.
A34 (2019) 1950143 (43pp); ibid
A35 (2020) 1950143 (45pp)[7] J.C. del Valle, A. V. Turbiner, M.A.Escobar-Ruiz,
Two-body neutral Coulomb system in a magnetic field at rest: from Hydrogen atom to positro-nium ,ArXiv: 2012.00044, pp.49 (December 2020 - February 2021)