PHYSICAL REVIEW E 91, 027301 (2015)
Comment on “Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit” M. A. Carrillo-Bemal* Facultad de Fi'sica, Universidad Veracruzana, Xalapa de Enriquez CP 91000 Veracruz, Mexico
H. N. Nunez-Yepezt Departamento de Fi'sica, Universidad Autonoma Metropolitana, Unidad Iztapalapa, Apartado Postal 55-534. Iztapalapa CP 09340 D. F., Mexico
A. L. Salas-BriuT Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, CP 04510 Mexico D. F. Mexico
Didier A. Solis5 Facultad de Matemdticas, Universidad Autonoma de Yucatan, Periferico Norte, Tablaje C 13615, Merida, Yucatan, Mexico (Received 18 July 2014; published 18 February 2015) In the referred paper, the authors use a numerical method for solving ordinary differential equations and a softened Coulomb potential —1/\Jx^ + fi- to study the one-dimensional Coulomb problem by approaching the parameter fl to zero. We note that even though their numerical findings in the soft potential scenario are correct, their conclusions do not extend to the one-dimensional Coulomb problem (0 = 0). Their claims regarding the possible existence of an even ground state with energy - o o with a Dirac-5 eigenfunction and of well-defined parity eigenfunctions in the one-dimensional hydrogen atom are questioned. DOI: 10.1103/PhysRevE.91.027301
PACS number(s): 02.60.Lj, 03.65.Ge, 02.70.Dh
In a recent work, Gebremedhin and Weatherford [1] (GW) addressed controversial issues related to the eigenstates of the one-dimensional (1D) Coulomb problem that have arisen from the well-known work of Loudon [2], The GW discussion illustrates their proposed numerical algorithm for solving ordi nary differential equations. Though their numerical proposal is excellent, their related discussion of the ID Coulomb problem is flawed. Some of the issues they bring about have already been extensively discussed and ultimately solved. Both the supposed negative infinite energy ground state and the parity eigenstates for the problem have been shown to be nonexistent on mathematical grounds using techniques that are very important for both ID systems and their applications [3-12], The ID Coulomb Hamiltonian / / ,DC [Eq. (14) in [1] with
0 = 0], „ 1 2 Z n\DC = -zP" — t t z
lx |
Z is the atomic number,
(1)
is not self-adjoint [13], but GW do not seem to pay enough attention to this crucial fact and its consequences [10]. Let us recall that conservation of probability is a direct consequence of the self-adjointness of the Hamiltonian according to Stone’s theorem [14]. Besides, H lDC has a singularity at the origin
' [email protected] [email protected] ^Author to whom all correspondence should be addressed: [email protected]; On leave from Departamento de Cien cias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Azcapotzalco D. F. Mexico, Apartado Postal 21-267 Coyoacan CP 04000 D. F., Mexico. §[email protected] 1539-3755/2015/91 (2)/027301 (2)
(x = 0), but the sequence of Hamiltonians of the regularized Coulomb potential Hp [Eq. (14) in [1]], Z
(2 )
with the softening parameter p ± 0, are all self-adjoint and nonsingular. As ft -* 0, the sequence of operators Hp con verges to the non-self-adjoint operator H w c . In such a limiting process not much can be said about the relation between the spectra of the operators Hp and that of Hl0C, since spurious eigenvalues may appear in the spectrum of the operator limit, a phenomenon known as spectral pollution [15,16]. To describe the physics of the problem at the hard Coulomb limit, a specific self-adjoint extension H of H tDC has to be chosen. To guarantee the self-adjointness of (1), the Friedrichs extension (FE) H D of the operator may be considered as argued in [4], The FE of the Hamiltonian operator does not change the functional form of H [DC, but it has to be defined on the set of functions / such that /(0 ) = 0, which is the Dirichlet condition making the operator self-adjoint, and assuming that the triplet / , / ' , and f " are all square integrable and absolutely continuous, except at x = 0 [17,18]. Based on the form of the probability density (Figs. 3, 4, and 5 of [I]), GW suggest that, as p —» 0, Vq°(x = 0) = 0, which is the condition making the origin impermeable [5,6,8-10]. The boundary condition i/f(0) = 0, at the hard Coulomb limit, implies the vanishing of the quantum flux,
• J=l^
,*3^
,3
r
8 7 -* 1 7
=
0.
(3)
*=0
at the origin [8,9], So there cannot be particle flow from the right to the left of the origin or vice versa, in agreement with the impenetrable wall that, as GW mention, appears to develop as
027301-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 027301 (2015)
COMMENTS
fi 0. Unfortunately, GW do not take into account all the con sequences of such behavior.
=
On the other hand, a breakdown of the symmetry arises due to a superselection rule that divides the x axis in two subsets (-o o ,0 ] and [0,oo) with known eigenfunctions
2n_5/123456789102x L ^_,(2x/n) exp (—x /n ),
x > 0,
0,
x^0,
2n_5/2xL ;,)_ ,(-2 x /« )e x p (x/n),
x < 0,
0,
x ^ 0,
(4)
and =
where L'n_ l (± 2 x / n ) are the associated Laguerre polynomials, and n > 1 since f ^ =0(x) is not square integrable, i.e. the E0 eigenvalue with —oo energy is not possible in the hard Coulomb limit [6]. When fi ± 0, the E0 energy eigenvalue is both finite and physically relevant—as suggested by GW—but in the case when fi = 0 such a state is removed from the Hmc spectrum. This phenomenon can be interpreted as a form of spectral pollution. Note that since the support [19] of the above functions is not the whole x axis, they cannot have a definite parity. So, neither r/c±(jc) = —i/f±( - x ) nor t/r±(x) = t//±( - x ) occur in spite of [H,V] = 0, where V is the parity operator. It has been previously proved that odd or even eigenfunctions cannot be defined as a combination of (4) and (5) [9]. All the energy eigenvalues are given by E„ = - ^ ,
n = 1 ,2 ,3 ,...
(6)
(5)
their conclusions for the hard Coulomb limit, when fi = 0, are incorrect for the most part. The aforementioned GW conclusion about the definite parity eigenfunctions of the ID hydrogen atom is mistaken, though it is correct for the 1D soft Coulomb problem. Properties of the ID soft Coulomb problem, such as energy eigenvalues with definite parity and a ground-state energy that increases as the softening parameter of the soft problem diminishes, were previously found in the analysis made by Loudon [2], We should recall that the lowest energy state of a system is defined as its ground state. Then, since Eq — —oo is not present in the hard Coulomb limit, the correct ground state of this system must be E\ = - 1 /2 . GW obtained numerically an energy value of E\ % —0.5, but they consider it as an excited state of the ID hydrogen atom. In summary, the novelty and value of the work commented upon must be limited to the numerical methods and findings related to the soft Coulomb problem, since a more detailed mathematical analysis must be carried out in order to draw valid conclusions for the hard 1D hydrogen atom.
and the corresponding eigenfunctions have support only to the left or only to the right side of the singularity, depending on the initial position of the particle [4,6,7,20]. The algorithm developed by GW can solve numerically in a very effective way the ID soft Coulomb problem, but
M.A. C.-B. is grateful for support from the scholarship granted to him by the Academia Mexicana de Ciencias through the XXIV Verano de la Investigation Cientffica 2014 program.
[1] D. H. Gebremedhin and C. A. Weatherford, Phys. Rev. E 89, 053319(2014). [2] R. Loudon, Am. J. Phys. 27, 649 (1959). [3] M. Andrews, Am. J. Phys. 34, 1194 (1966). [4] F. Gesztezy, J. Phys. A 13, 867 (1980). [5] R. G. Newton, J. Phys. A 27, 4717 (1994). [6] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 83, 064101 (2011). [7] D. Xianxi, J. Dai, and J. Dai, Phys. Rev. A 55, 2617 (1997). [8] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Soli's, J. Phys. A 46, 208003 (2013). [9] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 89, 049908(E) (2014). [10] W. Fischer, H. Leschke, and P. Muller, J. Math. Phys. 36, 2313 (1995). [11] T. Giamarchi, Quantum Physics in One-Dimension (Oxford University Press, Oxford, UK, 2014). [12] W. Zhao, L. Wang, J. Bai, J. S. Francisco, and X. Cheng Zeng, J. Am. Chem. Soc. (to be published).
[13] An operator, O, is self-adjoint if O = 0 ] and the domains of definition of both O and Of are the same [18]. [14] Stone’s theorem is as follows: “Let U be a strongly continuous one-parameter group of unitary operators; then, there exists a unique self-adjoint operator H such that U, — e"HJ e R. Conversely, if H is a self-adjoint operator on a Hilbert space, then U, = ehH ,t e R, is a strongly continuous one-parameter group of unitary operators [ 18].” Notice the relationship between the self-adjointness of a quantum Hamiltonian and unitary time evolution of the system described by it. [15] M. Lewin andE. Sere, Proc. London Math. Soc. 100,864 (2010). [16] E. Davies and M. Plum, IMA J. Numer. Anal. 24, 417 (2004). [17] M. A. Robdera, A Concise Approach to Mathematical Analysis (Springer-Verlag, Berlin, 2004). [18] M. Reed and B. Simon, Methods o f Modem Mathematical Physics: Fourier Analysis, Self-Adjointness, Vol. 2 (Academic, New York, 1975). [19] The support of a function g is the subset, S , of its domain of definition, where g(x), x 6
Comment on “Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit” M. A. Carrillo-Bemal* Facultad de Fi'sica, Universidad Veracruzana, Xalapa de Enriquez CP 91000 Veracruz, Mexico
H. N. Nunez-Yepezt Departamento de Fi'sica, Universidad Autonoma Metropolitana, Unidad Iztapalapa, Apartado Postal 55-534. Iztapalapa CP 09340 D. F., Mexico
A. L. Salas-BriuT Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, CP 04510 Mexico D. F. Mexico
Didier A. Solis5 Facultad de Matemdticas, Universidad Autonoma de Yucatan, Periferico Norte, Tablaje C 13615, Merida, Yucatan, Mexico (Received 18 July 2014; published 18 February 2015) In the referred paper, the authors use a numerical method for solving ordinary differential equations and a softened Coulomb potential —1/\Jx^ + fi- to study the one-dimensional Coulomb problem by approaching the parameter fl to zero. We note that even though their numerical findings in the soft potential scenario are correct, their conclusions do not extend to the one-dimensional Coulomb problem (0 = 0). Their claims regarding the possible existence of an even ground state with energy - o o with a Dirac-5 eigenfunction and of well-defined parity eigenfunctions in the one-dimensional hydrogen atom are questioned. DOI: 10.1103/PhysRevE.91.027301
PACS number(s): 02.60.Lj, 03.65.Ge, 02.70.Dh
In a recent work, Gebremedhin and Weatherford [1] (GW) addressed controversial issues related to the eigenstates of the one-dimensional (1D) Coulomb problem that have arisen from the well-known work of Loudon [2], The GW discussion illustrates their proposed numerical algorithm for solving ordi nary differential equations. Though their numerical proposal is excellent, their related discussion of the ID Coulomb problem is flawed. Some of the issues they bring about have already been extensively discussed and ultimately solved. Both the supposed negative infinite energy ground state and the parity eigenstates for the problem have been shown to be nonexistent on mathematical grounds using techniques that are very important for both ID systems and their applications [3-12], The ID Coulomb Hamiltonian / / ,DC [Eq. (14) in [1] with
0 = 0], „ 1 2 Z n\DC = -zP" — t t z
lx |
Z is the atomic number,
(1)
is not self-adjoint [13], but GW do not seem to pay enough attention to this crucial fact and its consequences [10]. Let us recall that conservation of probability is a direct consequence of the self-adjointness of the Hamiltonian according to Stone’s theorem [14]. Besides, H lDC has a singularity at the origin
' [email protected] [email protected] ^Author to whom all correspondence should be addressed: [email protected]; On leave from Departamento de Cien cias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Azcapotzalco D. F. Mexico, Apartado Postal 21-267 Coyoacan CP 04000 D. F., Mexico. §[email protected] 1539-3755/2015/91 (2)/027301 (2)
(x = 0), but the sequence of Hamiltonians of the regularized Coulomb potential Hp [Eq. (14) in [1]], Z
(2 )
with the softening parameter p ± 0, are all self-adjoint and nonsingular. As ft -* 0, the sequence of operators Hp con verges to the non-self-adjoint operator H w c . In such a limiting process not much can be said about the relation between the spectra of the operators Hp and that of Hl0C, since spurious eigenvalues may appear in the spectrum of the operator limit, a phenomenon known as spectral pollution [15,16]. To describe the physics of the problem at the hard Coulomb limit, a specific self-adjoint extension H of H tDC has to be chosen. To guarantee the self-adjointness of (1), the Friedrichs extension (FE) H D of the operator may be considered as argued in [4], The FE of the Hamiltonian operator does not change the functional form of H [DC, but it has to be defined on the set of functions / such that /(0 ) = 0, which is the Dirichlet condition making the operator self-adjoint, and assuming that the triplet / , / ' , and f " are all square integrable and absolutely continuous, except at x = 0 [17,18]. Based on the form of the probability density (Figs. 3, 4, and 5 of [I]), GW suggest that, as p —» 0, Vq°(x = 0) = 0, which is the condition making the origin impermeable [5,6,8-10]. The boundary condition i/f(0) = 0, at the hard Coulomb limit, implies the vanishing of the quantum flux,
• J=l^
,*3^
,3
r
8 7 -* 1 7
=
0.
(3)
*=0
at the origin [8,9], So there cannot be particle flow from the right to the left of the origin or vice versa, in agreement with the impenetrable wall that, as GW mention, appears to develop as
027301-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 027301 (2015)
COMMENTS
fi 0. Unfortunately, GW do not take into account all the con sequences of such behavior.
=
On the other hand, a breakdown of the symmetry arises due to a superselection rule that divides the x axis in two subsets (-o o ,0 ] and [0,oo) with known eigenfunctions
2n_5/123456789102x L ^_,(2x/n) exp (—x /n ),
x > 0,
0,
x^0,
2n_5/2xL ;,)_ ,(-2 x /« )e x p (x/n),
x < 0,
0,
x ^ 0,
(4)
and =
where L'n_ l (± 2 x / n ) are the associated Laguerre polynomials, and n > 1 since f ^ =0(x) is not square integrable, i.e. the E0 eigenvalue with —oo energy is not possible in the hard Coulomb limit [6]. When fi ± 0, the E0 energy eigenvalue is both finite and physically relevant—as suggested by GW—but in the case when fi = 0 such a state is removed from the Hmc spectrum. This phenomenon can be interpreted as a form of spectral pollution. Note that since the support [19] of the above functions is not the whole x axis, they cannot have a definite parity. So, neither r/c±(jc) = —i/f±( - x ) nor t/r±(x) = t//±( - x ) occur in spite of [H,V] = 0, where V is the parity operator. It has been previously proved that odd or even eigenfunctions cannot be defined as a combination of (4) and (5) [9]. All the energy eigenvalues are given by E„ = - ^ ,
n = 1 ,2 ,3 ,...
(6)
(5)
their conclusions for the hard Coulomb limit, when fi = 0, are incorrect for the most part. The aforementioned GW conclusion about the definite parity eigenfunctions of the ID hydrogen atom is mistaken, though it is correct for the 1D soft Coulomb problem. Properties of the ID soft Coulomb problem, such as energy eigenvalues with definite parity and a ground-state energy that increases as the softening parameter of the soft problem diminishes, were previously found in the analysis made by Loudon [2], We should recall that the lowest energy state of a system is defined as its ground state. Then, since Eq — —oo is not present in the hard Coulomb limit, the correct ground state of this system must be E\ = - 1 /2 . GW obtained numerically an energy value of E\ % —0.5, but they consider it as an excited state of the ID hydrogen atom. In summary, the novelty and value of the work commented upon must be limited to the numerical methods and findings related to the soft Coulomb problem, since a more detailed mathematical analysis must be carried out in order to draw valid conclusions for the hard 1D hydrogen atom.
and the corresponding eigenfunctions have support only to the left or only to the right side of the singularity, depending on the initial position of the particle [4,6,7,20]. The algorithm developed by GW can solve numerically in a very effective way the ID soft Coulomb problem, but
M.A. C.-B. is grateful for support from the scholarship granted to him by the Academia Mexicana de Ciencias through the XXIV Verano de la Investigation Cientffica 2014 program.
[1] D. H. Gebremedhin and C. A. Weatherford, Phys. Rev. E 89, 053319(2014). [2] R. Loudon, Am. J. Phys. 27, 649 (1959). [3] M. Andrews, Am. J. Phys. 34, 1194 (1966). [4] F. Gesztezy, J. Phys. A 13, 867 (1980). [5] R. G. Newton, J. Phys. A 27, 4717 (1994). [6] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 83, 064101 (2011). [7] D. Xianxi, J. Dai, and J. Dai, Phys. Rev. A 55, 2617 (1997). [8] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Soli's, J. Phys. A 46, 208003 (2013). [9] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 89, 049908(E) (2014). [10] W. Fischer, H. Leschke, and P. Muller, J. Math. Phys. 36, 2313 (1995). [11] T. Giamarchi, Quantum Physics in One-Dimension (Oxford University Press, Oxford, UK, 2014). [12] W. Zhao, L. Wang, J. Bai, J. S. Francisco, and X. Cheng Zeng, J. Am. Chem. Soc. (to be published).
[13] An operator, O, is self-adjoint if O = 0 ] and the domains of definition of both O and Of are the same [18]. [14] Stone’s theorem is as follows: “Let U be a strongly continuous one-parameter group of unitary operators; then, there exists a unique self-adjoint operator H such that U, — e"HJ e R. Conversely, if H is a self-adjoint operator on a Hilbert space, then U, = ehH ,t e R, is a strongly continuous one-parameter group of unitary operators [ 18].” Notice the relationship between the self-adjointness of a quantum Hamiltonian and unitary time evolution of the system described by it. [15] M. Lewin andE. Sere, Proc. London Math. Soc. 100,864 (2010). [16] E. Davies and M. Plum, IMA J. Numer. Anal. 24, 417 (2004). [17] M. A. Robdera, A Concise Approach to Mathematical Analysis (Springer-Verlag, Berlin, 2004). [18] M. Reed and B. Simon, Methods o f Modem Mathematical Physics: Fourier Analysis, Self-Adjointness, Vol. 2 (Academic, New York, 1975). [19] The support of a function g is the subset, S , of its domain of definition, where g(x), x 6
