COMMENT pubs.acs.org/JPCA
Comment on “New Generalization of Supersymmetric Quantum Mechanics to Arbitrary Dimensionality or Number of Distinguishable Particles” Vladimir A. Mandelshtam Chemistry Department, University of California at Irvine, Irvine, California 92697, United States For an m-dimensional system they first define a “vector-charge operator” Q B1 with components
arlier publications of Kouri, Bittner, and co-workers 1,2 explored possible numerical implications of the supersymmetric quantum mechanics (SUSY QM).4 In the previous developments of SUSY QM, starting with a Hamiltonian
E
H1 ¼ -
2
d þ V1 ðxÞ dx2
ð1Þ
Q1i := ∂i - ∂i lnðΨ0 Þ so that
ð1Þ
one constructs a family of isospectral Hamiltonians Hk ¼ -
d þ Vk ðxÞ dx2
ðk ¼ 1; 2; :::Þ
d d ðkÞ lnðΨ0 Þ dx dx
H1 ¼ QB1 3 QB1
ð2Þ
¼ 0
ð3Þ
ð4Þ
is then always satisfied if Qk is not singular. The “second sector Hamiltonian” is constructed using the ground state (Ψ(1) 0 ) of H1: H2 ¼ Q1 Q1†
ð5Þ
These equations are correct. However, from the numerical perspective, their utility is still to be seen as besides an enormous increase in the complexity of the eigenvalue problem, _ ! ð2Þ ! ð2Þ H 2 Ψ ð2Þ ð14Þ n ¼ En Ψ n
The “third sector Hamiltonian” is constructed using the ground state (Ψ(2) 0 ) of H2: H3 ¼ Q 2 Q †2 ð6Þ
(as a simple numerical example _ presented in the paper shows) the ground state (ΨB(2) 0 ) of H 2 may have nodes. To construct the third-sector Hamiltonian, the authors introduce a vector-charge operator Q B2 with components
and so on. In the family of the eigenvalue problems ðkÞ ðkÞ Hk ΨðkÞ n ¼ E n Ψn
ðk ¼ 1; 2; :::Þ
ð7Þ
the eigenfunctions (as well as the eigenvalues) of different sector Hamiltonians are related according to ðk þ 1Þ
Qk ΨðkÞ n µ Ψn - 1
ðk þ 1Þ
Qk† Ψn - 1 µ ΨðkÞ n
ð2Þ
Q2i ¼ ∂i - ∂i lnðΨ0i Þ so that for a nonsingular Q B2 (see below) ! ð2Þ QB2 3 Ψ 0 ¼ 0
ðn ¼ 1; 2; :::Þ ð8Þ
One remarkable feature of SUSY QM is that the ground states of all sector Hamiltonians (Ψ(k) 0 ) are nodeless, so, in principle, one could perform sequential calculations for only the nodeless ground states of Hk (k = 1, 2, 3, ...) to obtain the corresponding excited states of H1. (The reader interested in learning more about SUSY QM is referred to the reviews.5-8) Unfortunately, this original formulation of SUSY QM is only one-dimensional. In a recent paper,3 which from now on we will refer to as “the paper”, Kouri et al presented a generalization of 1D SUSY QM to the case of multiple dimensions. Following the above ideas these authors also try to construct a family of isospectral Hamiltonians. r 2011 American Chemical Society
ð11Þ
Next, the authors define the sector 2 Hamiltonian: _ † H 2 := QB1 X QB1 ð12Þ _ which is though a tensor. Moreover, H 2 does not have the standard form of a Schr€odinger operator (i.e., a-Laplacianplus-a-potential); its eigenfunctions Ψ B(2) n also have m compo(2) nents, (Ψni ) (i = 1, ..., m). The authors establish the following relationships between sector 1 and sector 2 eigenfunctions: † ! ð2Þ ð1Þ ! ð2Þ QB1 Ψð1Þ QB1 3 Ψ n - 1 µ Ψn ðn ¼ 1; 2; :::Þ ð13Þ n µ Ψ n-1
where Ψ(k) 0 is the ground state of Hk. The relationship ðkÞ
ð10Þ
is satisfied automatically. It then follows that
using the so-called “charge operators”
Qk Ψ 0
¼ 0 †
2
Qk :=
ð1Þ
Q B1 Ψ0
ð9Þ
ð15Þ
ð16Þ
is satisfied automatically. Then they write (without derivation) _ † ð2Þ ð17Þ H 2 := QB2 X QB2 þ E0 †
H3 := QB2 3 QB2
ð18Þ
Received: November 8, 2010 Revised: December 13, 2010 Published: January 18, 2011 948
dx.doi.org/10.1021/jp110681x | J. Phys. Chem. A 2011, 115, 948–949
The Journal of Physical Chemistry A
COMMENT
_ Although H 2 is a tensor, H3 appears to be a normal Schr€odinger operator of the same dimensionality as H1. If correct, these equations (also eqs 27 and 30 in the paper) together with eq 14 above imply the new set of relationships between sector 2 and sector 3 eigenfunctions: ! ð3Þ QB2 3 Ψ ð2Þ n µ Ψn - 1
† ð3Þ ! QB2 Ψn - 1 µ Ψ ð2Þ ðn ¼ 1; 2; :::Þ n
(4) Witten, E. Nucl. Phys. B 1981, 188, 513. (5) Haymaker, R. W.; Rau, A. R. P. Am. J. Phys 1986, 54, 10. (6) Lahiri, A.; Numar Roy, P.; Bagchi, B. Int. J. Mod. Phys. A 1990, 5, 4579. (7) Cooper, F.; Khare, A.; Sukhatme, U. Phys.Rep. 1995, 251, 267. (8) Junker, G. Supersymmetric Methods in Quantum and Statistical Physics; Springer-Verlag: Berlin, 1996.
ð19Þ
Moreover, if correct, the procedure could be continued to construct the higher-sector Hamiltonians. Unfortunately, the key equation, eq 17 above (and eq 27 in the paper), appears to be incorrect. Besides lacking any derivation, eq 17 is not supported in the paper by any numerical example (all calculations _ in the paper are concerned with analyzing the properties of H 2 as defined by eq 12). To justify our claim, we considered several tests based on either completely analytical models that could be handled with Mathematica or numerical models, implemented using Fortran. While the first-sector relationships (13) were always confirmed, the second-sector relationships (19) failed in all cases. In particular, we considered a nondegenerate (ω 6¼1, ω ∼ 1) 2D harmonic oscillator H1 ¼ - Δ þ ðx cos θ - y sin θÞ2 þ ω2 ðx sin θþy cos θÞ2 ð20Þ where the rotated frame (θ 6¼ 0) was used to avoid having to deal with the zero component of ΨB(2) 0 . For this model all the computations were done analytically using Mathematica. We found ! ð2Þ ð1Þ QB2 3 Ψ 1 ¼ QB2 3 QB1 Ψ2 ¼ 0 ! ð2Þ ð1Þ QB2 3 Ψ 2 ¼ QB2 3 QB1 Ψ3 is a nodeless function, and ! ð2Þ ð1Þ QB2 3 Ψ 3 ¼ QB2 3 QB1 Ψ4 ¼ 0 This is clearly contradictory with the expectation that these functions should be proportional to the three lowest eigenfunctions of H3. In addition, we present another (indirect) evidence for why eqs 12 and 17 cannot define the same object. Namely, for a operator nodeless ground state (Ψ(1) _ 0 ) of H1, the vector-charge Q B1 (eq 9) is always well-behaved, and so is H 2 defined by eq 12. However, as supported by both a numerical example in the paper and our tests, _ for a coupled 2D problem, the ground state B2 singular, which (ΨB(2) 0 ) of H 2 may have nodes. This will make Q will, in turn, make the operators in eq 17 singular (unless the singularities are magically canceled).
’ ACKNOWLEDGMENT NSF support, grant CHE-0108823, is acknowledged. I am grateful to Craig Martens for helpful suggestions and moral support. ’ REFERENCES (1) Bittner, E. R.; Maddox, J. B.; Kouri, D. J. J. Phys. Chem. A 2009, 113, 15276–15280. (2) Kouri, D. J.; Markovich, T.; Maxwell, N.; Bittner, E. R. J. Phys. Chem. A 2009, 113, 15257–15264. (3) Kouri, D. J.; Maji, K.; Markovich, T.; Bittner, E. R. J. Phys. Chem. A 2010, 114, 8202–8216. 949
dx.doi.org/10.1021/jp110681x |J. Phys. Chem. A 2011, 115, 948–949
