PHYSICAL REVIEW E 91,027302 (2015)
Reply to “Comment on ‘Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit’ ” Daniel H. Gebremedhin* and Charles A. Weatherford1 Physics Department, Florida A&M University, Tallahassee, Florida 32307, USA (Received 16 December 2014; published 18 February 2015) This is a response to the comment we received on our recent paper “Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit.” In that paper, we introduced a computational algorithm that is appropriate for solving stiff initial value problems, and which we applied to the one-dimensional time-independent Schrodinger equation with a soft Coulomb potential. We solved for the eigenpairs using a shooting method and hence turned it into an initial value problem. In particular, we examined the behavior o f the eigenpairs as the softening param eter approached zero (hard Coulomb limit). The commenters question the existence o f the ground state of the hard Coulomb potential, which we inferred by extrapolation o f the softening parameter to zero. A key distinction between the com m enters’ approach and ours is that they consider only the half-line while we considered the entire x axis. Based on mathematical considerations, the commenters consider only a vanishing solution function at the origin, and they question our conclusion that the ground state o f the hard Coulomb potential exists. The ground state we inferred resembles a 0, no matter how small fi is. Since the Hamiltonian is symmetric, and since we are considering the entire x axis, we emphasize that the eigenfunctions are symmetric or antisymmetric with respect to interchanging x and - x for all values of /?, including ^ = 0; we call this limit the hard-Coulomb (HC) limit. It is in this sense that the authors use the terms “even parity” and “odd parity.” Finally, we note that the sign of the eigenfunction defined for the whole axis is
*daniel 1.gebremedhin @ famu .edu fcharles. [email protected] 1539-3755/2015/91(2)7027302(3)
arbitrary. The authors expect that everyone can agree on what we have just stated. We can all also agree that the problem of the 1D H atom has a long history with much disagreement about valid eigenpairs. Clearly the fundamental difficulty is that the problem is ID and that x = 0 is an irregular singular point of the ID Schrodinger equation. The authors believe that much of the disagreement stems from considering slightly different definitions of the 1D H atom. The primary objective of our paper was to present the algorithm for solving stiff differential equations by applying it to a challenging problem. The HC limit of the SC problem was the problem we chose—that is, we had to treat very small values of fi in order to get a convincing and clear /3 = 0 limit. In the process of doing so, we suggested that this HC limit can serve as a unique and interesting definition of the ID H atom. We emphasize that we did not impose the half-line partition of the problem. We imposed the boundary conditions shown below for the soft-Coulomb problem. These are basic boundary conditions applicable to any smooth functions of even and odd symmetry,
< ( x = 0) = i ,
4£(* =
o) = o,
(i)
^ ( x = 0) = 0,
4r|(jt = 0) = 1.
(2)
Changing the values from 0 1 on any one of these entries while still demanding the respective symmetry of the functions leads to either trivial solutions that lie along the vertical or horizontal axis, or to discontinuity of the function at the origin. Hence, the above boundary conditions are used to calculate nontrivial, well-behaved eigenpairs for the soft Coulomb potential in both symmetries. The authors suggest that we all agree on the odd-parity eigenpairs. They are the same whether or not we impose a half line solution. They are unchanged whether or not an impen etrable wall exists at x = 0. The disagreement pertains to the even-parity eigenpairs. It was the question of what happened
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©2015 American Physical Society
PHYSICAL REVIEW E 91, 027302 (2015)
COMMENTS
to the even-parity SC eigenpairs as fi —> 0 that motivated this application of the algorithm. The authors long suspected that in some sense, the n > 1 even-parity eigenfunctions “disappeared” at fi — 0, but it was only with the construction and application of the algorithm that we could determine how this disappearance occurred. We also suspected that in some sense, the n = 0 solution was real and seemed to grow without bound as fi -> 0. The authors’ previous attempts to solve for the eigenpairs involved diagonalization of the Hamiltonian matrix in a basis set. However, a basis set adequate to get to very small fi values could not be found, and a clearly defined fi — 0 limit could not be determined. It was the invention of the spectral-element shooting algorithm with first derivative conti nuity that allowed for very small fi values and a clear inference of the fi — 0 limit. Thus, there is no possibility of “spectrum pollution” or “intruder” states since we did not diagonalize and since the SC eigenpairs are well-defined. We continuously followed each SC eigenpair as it evolved to smaller fi and there was no possibility of an anomalous intrusion of eigenstates. So let us revisit what happened to the even functions as we decreased the magnitude of fi. At the origin, the n — 0 and the n > 1 solutions began heading in opposite directions. Namely, ^ ° =0(x = 0) -* +oo and = 0) -» 0. Now at x = 0 the derivatives of the eigenfunctions are zero, but as we move away from x = 0 in positive or negative directions, the derivatives start to drop sharply (for all n solutions), which is an indication of a discontinuity with opposite slopes such that 4/“(x = 0+) = —4>°(x = 0“ ). It appears (and this is clearly an inference) that the zero slope in fact disappears at fi = 0. We suggest that this is a signature of a formation of an impenetrable wall at the origin. Note that we did not impose this result. It appeared as a consequence of the boundary conditions and the accurate solution as fi —> 0. Let us examine the HC limit further using the full-line plots instead of the half-line plots shown in the paper in question [1]. Figures 4, 1, 3, and 2 are the full-line versions of Figs. 2, 3, 4, and 5 in [1], respectively. Consider Fig. 1. The dashed line is then = 1 odd-parity state for fi = 1CT8. Examine the behavior of the n = 1 even-parity state as fi gets smaller. For negative x, the even-parity state approaches the odd-parity state and
FIG. 1. (Color online) Comparison of states T,1 and shown values of fi.
for the
-0.08 -0.06 -0.04 -0.02
0.00
0.02
0.04
0.06
0.08
x FIG. 2. (Color online) Tj* near the origin. The area under the curve gets smaller with fi.
we infer that they become exactly the same for fi = 0. For positive x, they appear to become exactly the same except that they have a different sign. If an impenetrable wall is developing at fi = 0, as it appears to do, then since the wave function is determined only up to its sign, we infer that the even-parity and the odd-parity n = 1 states become exactly the same at 0 = 0. This is the subtle behavior that the authors could not discern until the algorithm was implemented. Figure 2 is a blow-up of the Fig. 1 behavior around x — 0. Figure 3 is the n — 2 correspondent of Fig. 1, and it exhibits the same type of HC limit behavior as in Fig. 1. Thus, the authors’ results in [1] seem to agree with the commenters’ conclusions about the ID H atom excited states, including the impenetrable wall at x = 0, although the commenters would not call the 1D H atom n = 1 even-parity state an excited state since we disagree about the ID H atom n = 0 state. Also, the commenters would not use the term “parity” for the 1D H atom states because of the impenetrable
FIG. 3. (Color online) Comparison of states T-J and shown values of fi.
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COMMENTS
PHYSICAL REVIEW E 91, 027302 (2015)
wall. The authors have shown how the 1D H atom states evolve from the SC states and how the impenetrable wall appears to develop at p = 0. Concerning the n = 0 even-parity state, for p ^ 0, the authors' results are clear and highly accurate. In [1], the n — 0 even-parity eigenpair is followed as p approaches zero. Again, the authors never reach p = 0 and thus they cannot definitively claim that the eigenvalue is - 0 0 and that the eigenfunction is a Dirac 8 function. Nevertheless, the eigenvalue seems to be growing in magnitude without bounds, and the eigenfunction seems to approach a shape that resembles a function whose P = 0 limit could be a Dirac 8 function (Fig. 4). The ground state, assuming it behaves like a Dirac 8 function with eigenvalue —00, is uniquely important because it is the only state that would signify a particle fixed at the origin. It is simply intuitive for a particle to be trapped at the center of an attractive |x |-1 potential. Previous mathematical studies that the commentators cite impose the Dirichlet boundary condition ^ ( x = 0) = 0 in order to make the (extension) Hamiltonian thereby considered to be self-adjoint [2—4]. They do not directly allow the wave function to behave like a Dirac delta function (x = 0) = 0 is also sufficient for self-adjointness.
[1] D. H. Gebremedhin and C. A. Weatherford, Phys. Rev. E 89, 053319(2014). [2] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 83. 064101 (2011).
FIG. 4. (Color online) The even ground state for different values of p.
We believe that our numerical results are useful and com pelling enough to motivate such rigorous mathematical investigation.
[3] W. Fischer, H. Leschke, and P. MUer, J. Math. Phys. 36, 2313 (1995). [4] F. Gesztesy, J. Phys. A 13, 867 (1980).
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Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
Reply to “Comment on ‘Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit’ ” Daniel H. Gebremedhin* and Charles A. Weatherford1 Physics Department, Florida A&M University, Tallahassee, Florida 32307, USA (Received 16 December 2014; published 18 February 2015) This is a response to the comment we received on our recent paper “Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit.” In that paper, we introduced a computational algorithm that is appropriate for solving stiff initial value problems, and which we applied to the one-dimensional time-independent Schrodinger equation with a soft Coulomb potential. We solved for the eigenpairs using a shooting method and hence turned it into an initial value problem. In particular, we examined the behavior o f the eigenpairs as the softening param eter approached zero (hard Coulomb limit). The commenters question the existence o f the ground state of the hard Coulomb potential, which we inferred by extrapolation o f the softening parameter to zero. A key distinction between the com m enters’ approach and ours is that they consider only the half-line while we considered the entire x axis. Based on mathematical considerations, the commenters consider only a vanishing solution function at the origin, and they question our conclusion that the ground state o f the hard Coulomb potential exists. The ground state we inferred resembles a 0, no matter how small fi is. Since the Hamiltonian is symmetric, and since we are considering the entire x axis, we emphasize that the eigenfunctions are symmetric or antisymmetric with respect to interchanging x and - x for all values of /?, including ^ = 0; we call this limit the hard-Coulomb (HC) limit. It is in this sense that the authors use the terms “even parity” and “odd parity.” Finally, we note that the sign of the eigenfunction defined for the whole axis is
*daniel 1.gebremedhin @ famu .edu fcharles. [email protected] 1539-3755/2015/91(2)7027302(3)
arbitrary. The authors expect that everyone can agree on what we have just stated. We can all also agree that the problem of the 1D H atom has a long history with much disagreement about valid eigenpairs. Clearly the fundamental difficulty is that the problem is ID and that x = 0 is an irregular singular point of the ID Schrodinger equation. The authors believe that much of the disagreement stems from considering slightly different definitions of the 1D H atom. The primary objective of our paper was to present the algorithm for solving stiff differential equations by applying it to a challenging problem. The HC limit of the SC problem was the problem we chose—that is, we had to treat very small values of fi in order to get a convincing and clear /3 = 0 limit. In the process of doing so, we suggested that this HC limit can serve as a unique and interesting definition of the ID H atom. We emphasize that we did not impose the half-line partition of the problem. We imposed the boundary conditions shown below for the soft-Coulomb problem. These are basic boundary conditions applicable to any smooth functions of even and odd symmetry,
< ( x = 0) = i ,
4£(* =
o) = o,
(i)
^ ( x = 0) = 0,
4r|(jt = 0) = 1.
(2)
Changing the values from 0 1 on any one of these entries while still demanding the respective symmetry of the functions leads to either trivial solutions that lie along the vertical or horizontal axis, or to discontinuity of the function at the origin. Hence, the above boundary conditions are used to calculate nontrivial, well-behaved eigenpairs for the soft Coulomb potential in both symmetries. The authors suggest that we all agree on the odd-parity eigenpairs. They are the same whether or not we impose a half line solution. They are unchanged whether or not an impen etrable wall exists at x = 0. The disagreement pertains to the even-parity eigenpairs. It was the question of what happened
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©2015 American Physical Society
PHYSICAL REVIEW E 91, 027302 (2015)
COMMENTS
to the even-parity SC eigenpairs as fi —> 0 that motivated this application of the algorithm. The authors long suspected that in some sense, the n > 1 even-parity eigenfunctions “disappeared” at fi — 0, but it was only with the construction and application of the algorithm that we could determine how this disappearance occurred. We also suspected that in some sense, the n = 0 solution was real and seemed to grow without bound as fi -> 0. The authors’ previous attempts to solve for the eigenpairs involved diagonalization of the Hamiltonian matrix in a basis set. However, a basis set adequate to get to very small fi values could not be found, and a clearly defined fi — 0 limit could not be determined. It was the invention of the spectral-element shooting algorithm with first derivative conti nuity that allowed for very small fi values and a clear inference of the fi — 0 limit. Thus, there is no possibility of “spectrum pollution” or “intruder” states since we did not diagonalize and since the SC eigenpairs are well-defined. We continuously followed each SC eigenpair as it evolved to smaller fi and there was no possibility of an anomalous intrusion of eigenstates. So let us revisit what happened to the even functions as we decreased the magnitude of fi. At the origin, the n — 0 and the n > 1 solutions began heading in opposite directions. Namely, ^ ° =0(x = 0) -* +oo and = 0) -» 0. Now at x = 0 the derivatives of the eigenfunctions are zero, but as we move away from x = 0 in positive or negative directions, the derivatives start to drop sharply (for all n solutions), which is an indication of a discontinuity with opposite slopes such that 4/“(x = 0+) = —4>°(x = 0“ ). It appears (and this is clearly an inference) that the zero slope in fact disappears at fi = 0. We suggest that this is a signature of a formation of an impenetrable wall at the origin. Note that we did not impose this result. It appeared as a consequence of the boundary conditions and the accurate solution as fi —> 0. Let us examine the HC limit further using the full-line plots instead of the half-line plots shown in the paper in question [1]. Figures 4, 1, 3, and 2 are the full-line versions of Figs. 2, 3, 4, and 5 in [1], respectively. Consider Fig. 1. The dashed line is then = 1 odd-parity state for fi = 1CT8. Examine the behavior of the n = 1 even-parity state as fi gets smaller. For negative x, the even-parity state approaches the odd-parity state and
FIG. 1. (Color online) Comparison of states T,1 and shown values of fi.
for the
-0.08 -0.06 -0.04 -0.02
0.00
0.02
0.04
0.06
0.08
x FIG. 2. (Color online) Tj* near the origin. The area under the curve gets smaller with fi.
we infer that they become exactly the same for fi = 0. For positive x, they appear to become exactly the same except that they have a different sign. If an impenetrable wall is developing at fi = 0, as it appears to do, then since the wave function is determined only up to its sign, we infer that the even-parity and the odd-parity n = 1 states become exactly the same at 0 = 0. This is the subtle behavior that the authors could not discern until the algorithm was implemented. Figure 2 is a blow-up of the Fig. 1 behavior around x — 0. Figure 3 is the n — 2 correspondent of Fig. 1, and it exhibits the same type of HC limit behavior as in Fig. 1. Thus, the authors’ results in [1] seem to agree with the commenters’ conclusions about the ID H atom excited states, including the impenetrable wall at x = 0, although the commenters would not call the 1D H atom n = 1 even-parity state an excited state since we disagree about the ID H atom n = 0 state. Also, the commenters would not use the term “parity” for the 1D H atom states because of the impenetrable
FIG. 3. (Color online) Comparison of states T-J and shown values of fi.
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COMMENTS
PHYSICAL REVIEW E 91, 027302 (2015)
wall. The authors have shown how the 1D H atom states evolve from the SC states and how the impenetrable wall appears to develop at p = 0. Concerning the n = 0 even-parity state, for p ^ 0, the authors' results are clear and highly accurate. In [1], the n — 0 even-parity eigenpair is followed as p approaches zero. Again, the authors never reach p = 0 and thus they cannot definitively claim that the eigenvalue is - 0 0 and that the eigenfunction is a Dirac 8 function. Nevertheless, the eigenvalue seems to be growing in magnitude without bounds, and the eigenfunction seems to approach a shape that resembles a function whose P = 0 limit could be a Dirac 8 function (Fig. 4). The ground state, assuming it behaves like a Dirac 8 function with eigenvalue —00, is uniquely important because it is the only state that would signify a particle fixed at the origin. It is simply intuitive for a particle to be trapped at the center of an attractive |x |-1 potential. Previous mathematical studies that the commentators cite impose the Dirichlet boundary condition ^ ( x = 0) = 0 in order to make the (extension) Hamiltonian thereby considered to be self-adjoint [2—4]. They do not directly allow the wave function to behave like a Dirac delta function (x = 0) = 0 is also sufficient for self-adjointness.
[1] D. H. Gebremedhin and C. A. Weatherford, Phys. Rev. E 89, 053319(2014). [2] H. N. Nunez-Yepez, A. L. Salas-Brito, and D. A. Solis, Phys. Rev. A 83. 064101 (2011).
FIG. 4. (Color online) The even ground state for different values of p.
We believe that our numerical results are useful and com pelling enough to motivate such rigorous mathematical investigation.
[3] W. Fischer, H. Leschke, and P. MUer, J. Math. Phys. 36, 2313 (1995). [4] F. Gesztesy, J. Phys. A 13, 867 (1980).
027302-3
Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
