New Generalization of Supersymmetric Quantum ... - ACS Publications

8202

J. Phys. Chem. A 2010, 114, 8202–8216

New Generalization of Supersymmetric Quantum Mechanics to Arbitrary Dimensionality or Number of Distinguishable Particles Donald J. Kouri,*,†,‡,§ Kaushik Maji,† Thomas Markovich,‡,§ and Eric R. Bittner*,† Departments of Chemistry, Mathematics, and Physics, UniVersity of Houston, Houston, Texas 77204 ReceiVed: April 13, 2010; ReVised Manuscript ReceiVed: June 29, 2010

We present here a new approach to generalize supersymmetric quantum mechanics to treat multiparticle and multidimensional systems. We do this by introducing a Vector superpotential in an orthogonal hyperspace. In the case of N distinguishable particles in three dimensions this results in a vector superpotential with 3N orthogonal components. The original scalar Schro¨dinger operator can be factored using a 3N-component gradient operator b1 and Q b†1. Using these operators, we can write the original (scalar) and introducing vector “charge” operators: Q † b (1) (1) b Hamiltonian as H1 ) Q1 · Q1 + E0 , where E0 is the ground-state energy. The second sector Hamiltonian is b †1 + E(1) 52 ) Q b 1Q b (2) a tensor given by H 0 and is isospectral with H1. The vector ground state of sector 2, ψ0 , can † b be used with the charge operator Q1 to obtain the excited-state wave function of the first sector. In addition, 5 2 can also be factored in terms of a sector 2 vector superpotential with components W2j ) we show that H (2) b (2) b b† -(∂ ln ψ(2) )/∂x 0j j. Here ψ0j is the jth component of ψ0 . Then one obtains charge operators Q 2 and Q 2 so that 52 ) Q b †2Q b 2 + E(2) the second sector Hamiltonian can be written as H . This allows us to define a third 0 b2 · Q b †2 + E(2) sector Hamiltonian which is a scalar, H3 ) Q . This prescription continues with the sector 0 Hamiltonians alternating between scalar and tensor forms, both of which can be treated by the variational method to obtain approximate solutions to both scalar and tensor sectors. We demonstrate the approach with examples of a pair of separable 1D harmonic oscillators and the example of a nonseparable 2D anharmonic oscillator (or equivalently a pair of coupled 1D oscillators). We consider both degenerate and nondegenerate cases. We also present a generalization to arbitrary curvilinear coordinate systems in the Appendix. Introduction

Q1 )

Supersymmetric quantum mechanics (SUSY-QM) provides a fascinating approach for solving one-dimensional problems.1 It is appealing because it reduces the problem to solving a firstorder (rather than second-order) differential equation for the ground-state wave function. This is achieved by factoring the Schro¨dinger Hamiltonian operator in terms of so-called “charge” operators, Q and Q†. The simplest and best known example is the one-dimensional harmonic oscillator where Q and Q† are the well-known lowering and raising operators. There are a number of 1D problems that have been solved using this approach, and it is interesting to point out that Schro¨dinger used a similar approach in his original solution of the hydrogen atom problem. Recently we have pointed out that one may approach SUSY in a different light as a way to generate numerical solutions starting from a sufficiently accurate ground-state wave function, 2,3 ψ(1) 0 , that can be used to generate an approximate superpotential

W1 ) -

and charge operators †

Department of Chemistry. Department of Mathematics. § Department of Physics. ‡

d ln ψ(1) 0 dx

d + W1 dx

and

Q†1 ) -

d + W1 dx

(2)

Here ψ0(1) is the ground-state wave function for a system described by the Hamiltonian

H1 ) -

d2 + V1(x) dx2

(3)

and it is straightforward to show that

V1(x) ) W12 -

dW1 dx

(4)

We then rewrite H1 using the charge operators

H1 ) Q†1Q1 + E(1) 0

(5)

One can also show that defining H2

(1) H2 ) Q1Q†1 + E(1) 0

(6)

leads to a second Hamiltonian system with potential V2 ) W12 + dW1/dx that is isospectral with H1 except that its lowest energy state is isoenergetic with the first excited state of H1. This hierarchy of isospectral Hamiltonian operators is continued by 10.1021/jp103309p  2010 American Chemical Society Published on Web 07/27/2010

Generalization of Supersymmetric Quantum Mechanics

J. Phys. Chem. A, Vol. 114, No. 32, 2010 8203

defining a second set of charge operators and an associated superpotential, viz.

W2 ) -

d ln ψ(2) 0 dx

) 1, ..., 3n. Note that, e.g., x1, y1, z1 f u1, u2, u3, etc. We take the Hamiltonian for this system to be given by

H1 ) -∇2 + V1

(7)

(10)

where

d Q2 ) + W2 dx

and

Q†2

d ) - + W2 dx

(8) b) ∇

and so on until all of the states of H1 have been exhausted. In other words, for every eigenstate of H1 with energy eigenvalue En(1) there is a corresponding Hamiltonian Hn in the hierarchy (1) whose ground-state E0(n) has the same energy eigenvalue (En-1 ) E(n) ) for n > 1. Correspondingly, there is a connection between 0 the wave functions of neighboring sectors, e.g. † † (3) ψ(1) 2 ∝ Q1Q2ψ0

(9)

† (3) (1) † (2) That is, ψ(2) 1 ∝ Q2ψ0 and then ψ2 ∝ Q1ψ1 . The primary point of this is provided by the realization that one can generate excitation energies and excited-state wave functions for H1 by only solving for the nodeless ground-state wave functions of the higher sector Hamiltonians. We recently took advantage of this for one-dimensional systems, calculating accurate excitation energies and wave functions using both the quantum Monte Carlo technique and the Rayleigh-Ritz variational principle while avoiding all of the complications introduced by the presence of nodes in the excited-state wave functions of H1. From a numerical point of view, determining the eigenspectrum of a one-dimensional system is a trivial matter, and while the SUSY approach provides an elegant route, it is of no practical utility unless it can be extended well beyond one dimension. As far as we know, there has not been a generalization of SUSY beyond 1D that has led to numerical applications.4-13 In this paper we provide a new solution that allows the generalization of SUSY to deal with general multidimensional systems that may include more than one (distinguishable) particle. This paper is organized as follows. We first present the generalized approach to SUSY that allows us to deal with fully 3D systems involving any number of distinguishable particles. We then derive the Rayleigh-Ritz variational principle for the higher sector equations. Following this, we apply our rigorous approach to an analytically solvable problem and show that our formulation enables us to obtain all of the desired excitation energies along with the excited-state wave functions. We then consider nontrivial, 2D degenerate and nondegenerate examples that can represent either a 2D nonseparable anharmonic oscillator or two coupled 1D oscillators. We conclude by discussing the results. In the Appendix, we generalize the formulation to arbitrary curvilinear coordinates.

Generalization of SUSY-QM to N-Dimensional Systems Previous attempts to generalize SUSY-QM to treat more than one spatial dimension and more than one particle generally have involved introducing additional “spinlike” degrees of freedom.4-13 In our approach, we make use of a vectorial approach that simultaneously treats more than one dimension and any number of distinguishable particles. We consider, therefore, a system of n particles in three-dimensional space. We denote the coordinates of particle i by (xi, yi, zi). We then define an orthogonal hyperspace of dimension 3n denoted by {uj} with j

∑ bεj ∂u∂ j

(11)

j

εk ) δjk. For simplicity we take the masses of the particles and b εj · b to be equal and use units such that p2/2m ) 1. For the development here we assume a flat coordinate space and provide an extension to more general curvilinear coordinates in the Appendix. As per usual in quantum mechanics, the ground-state wave function is a solution of the Schro¨dinger equation (1) (1) H1ψ(1) 0 ) E0 ψ0

(12)

where we index the Hamiltonian and its eigenstates with the sector index “(1)” with a view toward defining new sectors. We also emphasize that the lowest energy state, ψ(1) 0 , is nodeless. b 1, as We now introduce a Vector superpotential, W

b1 ) -∇ bln ψ(1) W 0

(13)

that is to say

b1 ) W

3n

∑ j)1

3n

bεjW1,j ) -

∑ bεj ∂u∂ j ln ψ(1)0

(14)

j)1

b1 It is straightforward to see that one can write H1 in terms of W as

f f f b (H1 - E(1) 0 ) ) (-∇ + W1) · (+∇ + W1) ) (-∂i + W1, i)(∂i + W1, i)

(15)

where, according to the Einstein convention, we sum over b 1)ψ0(1) ≡ b b+W repeated indices. Since (∇ 0, it is clear that (H1 (1) (1) E0 )ψ0 ) 0 as required. The SUSY charge operators are also vectors with components

Q1i ) ∂i + W1i

(16)

† Q1i ) -∂i + W1i

(17)

We now define the sector 2 Hamiltonian such that, above the ground-state (E0(1)), it is isospectral with H1. For the first excited state in sector 1 we can write † (1) (1) (1) Q1i Q1iψ(1) 1 ) (E1 - E0 )ψ1

(18)

We then form the tensor product by operating on the left b 1 so that with Q

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J. Phys. Chem. A, Vol. 114, No. 32, 2010 (1) (1) b (1) b 1Q b†1) · Q b1ψ(1) (Q 1 ) (E1 - E0 )Q1ψ1

Kouri et al.

(19)

That is to say † (1) (1) (1) (Q1iQ1j )Q1jψ(1) 1 ) (E1 - E0 )Q1iψ1

(20)

b 1ψ1(1) is an eigenstate of the tensor It then follows that Q b b 5 Hamiltonian H2 ) (Q1Q1†) with energy E0(2) ) E1(1) - E0(1). Since we are free to set the energy origin, taking E0(1) ) 0 gives E0(2) b (1) ) E(1) 1 . It is also clear that Q1ψ0 cannot generate a lower energy b b 1ψ1(1) is indeed 5 0, so that Q eigenstate of H2 since Q1ψ0(1) ) b 5 proportional to the ground state of H2. We next form the scalar b 0(2) ) E0(2)ψ b 0(2) with Q b 1† 52 · ψ product of H

b†1 · (Q b1Q b†1) · ψ Q b(2) 0

)

b† b(2) E(2) 0 Q1 · ψ 0

where n labels a given eigenvalue/eigenvector and we sum over repeated indices. At this point, to distinguish various sub- and superscripted indices decorating the various terms appearing in (2) this paper, we adopt a standard notation such that ψ(n)j will denote the jth component of the nth eigenstate of the sector 2 b 2, we Hamiltonian. Upon forming the scalar product with Q obtain (2) (2) b (2) b2 · Q b†2)(Q b2 · ψ (Q b(n) ) ) (E(2) b(n) ) n - E0 )(Q2 · ψ

(29)

Therefore, we have a third sector Hamiltonian which is a scalar as given by

b2 · Q b†2 H3 ) Q

(30)

(3) (2) En-1 ) E(2) n - E0

(31)

(2) (3) b2 · ψ Q b(n) ∝ ψ(n-1)

(32)

(21) with energy eigenvalues

and deduce that (2) b† b†1 · ψ H1(Q b(2) b(2) 0 ) ) E0 (Q1 · ψ 0 )

(22)

(1) b n(2) and ψn+1 , This analysis holds in general for any eigenstate ψ 5 so that H2 is isospectral with H1 above the ground state of H1. Indeed, the precise relation between the two sets of states is given by

(1) ψn+1 )

1 (1) - E(1) √En+1 0

b†1 · ψ Q b(2) n

(23)

Again, writing this in component form (1) ψn+1 )

1

√

(1) En+1

-

E(1) 0

† (2) (Q1i ψni )

(24)

5 2 is It remains to be shown whether the ground state of H (1) nodeless as is the case with ψ0 . We shall return to this issue later. The next step in our generalization of SUSY-QM is to show how the next sector (sector 3) is obtained from sector 2. To do b 2, with components so, we define a second vector superpotential, W (2) W2j ) -∂j ln ψ0j

(25)

We stress that, in this equation, the index “j” is not summed over and, in fact, occurs on both sides of the equation. It is then apparent that we can define a new set of charge operators

Q2j ) ∂j + W2j

(26)

(2) ≡ 0. Therefore, we can define such that Q2jψ0j

and eigenstates

Note that E0(3) ) E1(2) - E0(2) (cf. eq 18). It is important to note that, like H1, H3 is a scalar. In fact, using eqs 26 and 30, it is b2 · W b2 + ∇ b 2 ) -∇2 + b·W easy to establish that H3 ) -∇2 + W V3. Here, the potential is a real function solely of position, and thus, H3 is the sum of the usual kinetic energy operator and potential energy operator. It follows that H3 is a standard Schro¨dinger operator, with all attendant properties. The 5 4 being a tensor, H5 a scalar, etc. procedure continues with H Clearly, the generalization, as it stands, does not include identical particle symmetry and spin effects. That extension is under consideration. SUSY Intertwining Relations We now consider in more detail the algebra associated with the b1 and Q b1† and the sector Hamiltonians H1 and charge operators, Q 52. An important consequence of the algebra is the existence H of “intertwining” relations. These are of fundamental importance because they underlie the isospectral property and in addition they can be used to establish the unique correspondence between the eigenstates of sectors 1 and 2. Indeed, they are responsible b n(2)} of for establishing the completeness of the eigenvectors {ψ 5 H2. It is of interest to note that intertwining relations are essential to the fact that, in ordinary quantum scattering, the continua of the full Hamiltonian, H, and the unperturbed Hamiltonian, H0 (where H ) H0 + V, with V the perturbation responsible for scattering), coincide. In that case, the intertwining relation is

Ω+H0 ) HΩ+

T b†2Q b2 + E(2) H2 ) Q 0

(27)

(2) (2) (H2)ijψ(n)j ) E(2) n ψ(n)i

(28)

and it follows that

(33)

where Ω+ is the Møller wave operator. It is useful to derive the SUSY intertwining relations explicitly. We have (1) (1) H1ψ(1) n ) En ψn

(34)



where H1 is a standard Schro¨dinger operator (comprised of a Laplacian for the kinetic energy and a Hermitian potential, V1). One result of this fact is that the ground state of H1 is nodeless. In addition, H1 is Hermitian, and a well-known postulate of quantum mechanics asserts that its eigenstates are complete. Essentially from a physical standpoint (as opposed to pure mathematics) H1 is required to be Hermitian because (a) it represents an observable, implying only real eigenvalues, and (b) quantum mechanics further asserts that these eigenvalues are the only values that can be obtained when the energy for the physical system represented by H1 is measured. This implies that any physically realizable state, ψ, of the system must be a superposition (in general) of these and only these eigenvectors. This then implies that the set {ψ(1) n } is complete on the physically allowed space of state vectors. To derive the intertwining relation, we again recall that the charge operator (which exactly factors H1) is such that

b†1 · Q b1 + E(1) H1 ) Q 0

(35)

b1ψ(1) b Q 0 ≡ 0

(36)

where

The general sector 1 Schro¨dinger equation is (1) (1) (1) b†1 · Q b1 + E(1) (Q 0 )ψn ) En ψn

(37)

where we now assume that n > 0; i.e., ψn(1) is an excited state of b 1 to eq 37 to find H1. We apply Q (1) (1)b (1) b1(Q b†1 · Q b1 + E(1) Q 0 )ψn ) En Q1ψn

(38)

5 2 as We define H

T b1Q b†1 + E(1) H2 ) Q 0

(39)

T ·Q b1H1 ) H b1 Q 2

(40)

so that eq 38 yields

This is the intertwining relation. Let us explore intertwining consequences further. Suppose we consider an eigenstate, ψn(1), of H1. It follows from eq 40 that there is also a unique eigenstate b 1ψn(1), with the same energy. Next, assume that H 52 5 2, Q of H possesses an eigenvalue, Eλ(2), that differs from all of the En(1) values. Then we have

T (2) (2) H2 · ψ b(2) bλ λ ) Eλ ψ

(41)

5 2 and H1 are manifestly Hermitian. Taking the adjoint Now H of eq 40 yields

T b†1 ) Q b†1 · H H1Q 2

(42)

which is again an intertwining relation. We then take the scalar b 1† product of eq 41 with Q

T ·ψ (2) (2)b† b†1 · H Q b(2) 2 bλ ) Eλ Q1 · ψ λ

(43)

but by the adjoint intertwining relation, we have (2)b† b†1 · ψ H1Q b(2) b(2) λ ) Eλ Q1 · ψ λ

(44)

Thus, we find that H1 also has the scalar eigenvector

b† b(2) ψ(1) λ ∝ Q1 · ψ λ

(45)

and its eigenvalue is equal to Eλ(2). The final step of the proof is b λ(2) ≡ 0. In that case, by eqs b 1† · ψ to consider the possibility that Q 39-44 we see that Eλ(2) ≡ E0(1). This result again violates our initial assertion that H1 did not have an eigenvalue equal to E(2) λ . In fact, such a possibility was considered for 1D systems in ref 3. In that case, if the domain of the position was finite (e.g., from 0 to 2π), the ground state of sector 2 did satisfy Q1†ψ0(2) ) 0 and E0(2) ≡ E0(1). However, for an infinite domain such as -∞ < x < +∞, the function satisfying Q†1ψ(2) 0 ) 0 is not normalizable and therefore does not occur. Then the spectrum of H2 starts with E0(2) ) E1(1). We speculate that the same holds true in the multidimensional case, although the proof depends on requiring † (2) b 0(2) ) 0 be satisfied separately for each term: Q1j b 1† · ψ ψ0j that Q 5 2 starts at ) 0. Computationally, we find that the spectrum of H b n(2), there E0(2) ) E1(1). We conclude that, for eigenvectors ψ (1) corresponds a unique eigenvector ψn′ , where n′ ≡ n + 1. That b 0(2) must have the same energy as the first excited state ψ1(1). is, ψ It cannot be lower than E1(1) because it is the lowest eigenvalue 5 2, and it cannot equal E0(1). In fact, the intertwining relation of H 5 2 is a Schro¨dinger operator and, is sufficient to establish that H b . Note as such, its eigenvectors must be complete on the space ψ b n(2) span the space generated by H1. that we are not saying that ψ They are completely separate vector spaces arising from two distinct Hermitian Hamiltonians. All of the above can be made mathematically rigorous, but our purpose here is to supply a physically reasonable argument for the properties of the tensor sector. Finally, at no point in this discussion have we imposed 5 2) are strictly discrete. a condition that the spectra of H1 (and H The intertwining relations hold for systems with mixed discrete and continuous spectra and even for systems with a purely continuous spectrum. Derivation of the Rayleigh-Ritz Variational Principle for the Tensor Sectors An important issue is whether the standard approximation methods of quantum mechanics can be applied to the tensorsector Schro¨dinger equations. In particular, can we use the Rayleigh-Ritz method to carry out calculations of the energies and vector eigenstates? This question hinges first on whether 5 2p, p ) 1, 2, ... are Hermitian. The the tensor Hamiltonians H general structure is given by

T † b2p-1Q b2p-1 H2p ) Q + E(2p-1) 0

p ) 1, 2, ...

(46)

As noted above such a Hamiltonian is manifestly Hermitian, viz.

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J. Phys. Chem. A, Vol. 114, No. 32, 2010

T )† ) (Q † † b2p-1Q b2p-1 b2p-1 b2p-1)† (H )† ) (Q )†(Q 2p

Kouri et al.

(47)

so that

b2p)† ) (H b2p) (H

(48)

Examples Analytically Soluble Example in Two Dimensions. It is very instructive to begin by considering a simple 2D separable harmonic oscillator model problem. This is because we can learn something of how our SUSY formalism works with an exactly soluble problem. We therefore consider a system described by the Hamiltonian

(2p) b (n) , form a complete set for It follows that the eigenstates ψ physically allowed states and can be chosen to be orthonormal:

(2p) ) ) δnn ∫ dτ(ψb(n)(2p))* · (ψb(n′)

′

(49)

b(2p) The energy expectation value with any trial state, ψ (trial), is given by

H)-

∂2 ∂2 + u12 + u12 ∂u12 ∂u22

where again we set p2/2m1 ) p2/2m2 ) 1. The solution of the Schro¨dinger equation is well-known to be the product of 1D harmonic oscillator states (1) ψ(n ) Nn1,n2Hn1(u1) Hn2(u2) e-(u1 +u2 )/2 1,n2) 2

Etrial )

∫

T ·ψ (2p) (2p) dτ(ψ b(trial) )* · H 2p b(trial) (2p) (2p) )* · ψ btrial ∫ dτ(ψbtrial

(56)

2

(57)

(50) where Nn1,n2 is the normalization constant and Hn denotes a Hermite polynomial. The ground state is

(2p) b (n) The trial state can be expanded in the basis ψ , so that

(1) ψ(0,0) ) N0,0e-(u1 +u2 )/2

(58)

(1) (1) Hψ(0,0) ) 2ψ(0,0)

(59)

2

2

∞

(2p) ψ btrial )

∑ Cnψb(n)(2p)

(51)

and

n)0

and then we have

Thus, the zero-point energy in this case is 2. We next generate b 1, as the vector superpotential, W

∞

Etrial )

∑

|Cn | 2E(2p) n

n)0

(52)

∞

∑ |Cn|

2

(1) b1 ) -∇ b ln ψ(0,0) W ) u1εˆ 1 + u2εˆ 2

(60)

b ) εˆ 1 ∂ + εˆ 2 ∂ ∇ ∂u1 ∂u2

(61)

n)0

where (2p) b trial be normalized to 1, so that eq 52 We can require that ψ becomes

∞

Etrial )

∑ |Cn|2E(2p) n

(53)

n)0

∞

∑ |Cn|2 ) 1

(54)

n)0

b+b b+W b1) ) -∇2 + b b1 - ∇ b·W b1 (-∇ W1) · (∇ W1 · W (62) b1 - ∇ b 1 ) u12 + u22 - 2 ) V - 2, so b1 · W b·W We see that W that

and it immediately follows that

Etrial g E(2p) 0

We consider

(55)

This is, of course, the fundamental result underlying the Rayleigh-Ritz variational principle in quantum mechanics. We do not know at present if one can also establish a variant of the Hylleraas-Undheim theorem,14 so that, in a variational calculation, one can obtain approximate eigenvalues that provide an upper bound on the corresponding higher eigenvalues. The details of the variational calculation follow the usual quantum mechanical theory. Thus, one seeks to minimize Etrial by (2p) b (trial) . variation of the parameters embedded in ψ

b†1 · Q b1 + 2 H1 ) Q

(63)

It is easily verified that (1) (1) H1ψ(0,0) ) 2ψ(0,0)

(64)

as required. The first excited states of H1 are doubly degenerate (1) (1) (1) ) E(0,1) ) 3 and are denoted by ψ(1,0) with energy E(1,0) and (1) (1) ψ(0,1). The next excited state, ψ(1,1), is degenerate with ψ(1) (0,2) and (1) (1) (1) (1) ψ(2,0) with energy E(1,1) ) E(2,0) ) E(0,2) ) 4.



We construct the rigorous sector 2 Hamiltonian as

T b 1Q b†1 + 21 5 H2 ) Q

5 1 ) εˆ 1εˆ 1 + εˆ 2εˆ 2

(65)

52 which is a second-rank tensor in this case. The Hamiltonian H is then given by

[ [

]

[

∂2 T + u12 + 3 + H2 ) εˆ 1εˆ 1 2 ∂u1

εˆ 1εˆ 2 εˆ 2εˆ 1 -

2

] ]

in the nonseparable examples that we consider next. Indeed, we recall that, in relativistic quantum mechanics, one obtains a tensor Hamiltonian and the solutions are characterized by large and small components. In the present case, the small component is exactly zero. In the degenerate pair of solutions, which component is zero changes. We stress, however, that any linear combination of the two degenerate solutions is also a solution of the same energy. With a view toward the next section, where we consider a 2D, nonseparable anharmonic oscillator (or equivalently a pair of 1D coupled oscillators), we form the equivalent degenerate solutions

∂ ∂ ∂ + u1u2 - u1 + u2 + ∂u1 ∂u2 ∂u2 ∂u1 2

∂ ∂ ∂ + u1u2 - u2 + u1 + ∂u1 ∂u2 ∂u1 ∂u2

[

εˆ 2εˆ 2 -

]

2

2 2 (2) b(0)1 φ ) Ne-(u1 +u2 )/2[εˆ 1 + εˆ 2]

(74)

2 2 (2) b(0)2 φ ) Ne-(u1 +u2 )/2[εˆ 1 - εˆ 2]

(75)

and

∂ + u22 + 3 ∂u22

(66)

The eigenvalue equation is

T (2) (2) (2) H2 · ψ b(n) ) E(n) ψ b(n)

(67)

(2) (2) (2) ψ b(n) ) εˆ 1ψ(n)1 + εˆ 2ψ(n)2

(68)

with eigenstates

In this case, both components of the two degenerate solutions are nonzero, of the same magnitude, and of definite sign. In dealing with the 2D separable HO, the most convenient form is given by eqs 72 and 73. Nonseparable Two-Dimensional Anharmonic Oscillator Model. Degenerate Case. We next consider a model nonseparable two-dimensional anharmonic oscillator system for sector 1 for which the ground-state energy is zero and the groundstate wave function is exactly given by (1) ψ(0) (u1, u2) ) N exp(-u12u22 - u12 - u22)

It is not difficult to show that there are two degenerate groundstate solutions given by (2) ψ b(0)1 ) εˆ 1e-(u1 +u2 )/2 2

2

(76)

We can generate the superpotential corresponding to this ground state as

(69)

(1) b1 ) -∇ b ln ψ(0) W (u1, u2)

(77)

and -(u12+u22)/2

(2) ψ b(0)2 ) εˆ 2e

(70)

This is extremely interesting and in contrast to the usual situation in quantum mechanics. For most systems (excluding spin effects) the ground state is unique, i.e., nondegenerate. We shall see that the degenerate states, eqs 69 and 70, are b 1† to produce exactly what is required for the charge operator Q (1) (1) the doubly degenerate states ψ(1,0) and ψ(0,1). Thus, recall that

(

) (

)

b†1 ) εˆ 1 - ∂ + u1 + εˆ 2 - ∂ + u2 Q ∂u1 ∂u2

(71)

Then 2 2 (2) (1) b†1 · ψ Q b0(1) ) 2u1e-(u1 +u2 )/2 ∝ ψ(1,0)

(72)

2 2 (2) (1) b†1 · ψ Q b0(2) ) 2u2e-(u1 +u2 )/2 ∝ ψ(0,1)

(73)

Our results, eqs 69 and 70, possess a remarkable property. Only one component is nonzero! We shall see that this is indicative of an extremely interesting property that we observe

having the components W11 ) 2u1u22 + 2u1 and W12 ) 2u12u2 + 2u2. Now, using these components, we can generate the model potential for sector 1. Thus, we get the Hamiltonian for sector 1 of the following form:

H1 ) -∇2 + V1(u1, u2) )-

∂2 ∂2 + (2u1u22 + 2u1)2 + 2 2 ∂u1 ∂u2

(2u12u2 + 2u2)2 - 2(u12 + 1) - 2(u22 + 1)

(78) In this case, the exact ground-state energy is E(1) 0 ) 0. The sector b 1. The b and W 2 tensor Hamiltonian can be generated with ∇ calculation for sector 1 and sector 2 eigenvalues and eigenfunctions is done variationally by diagonalizing each sector Hamiltonian in an approximate truncated basis. We choose to employ a basis of the direct product of the eigenstates of a harmonic oscillator in each dimension, each with frequency ω ) 22. The trial wave function for sector 1 is (1) ψ(trial) (u1, u2) )

(1) φm(R, u1) φn(R, u2) ∑ Cm,n m,n

(79)

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where R ) (mω/p)1/2. Similarly for sector 2 the trial wave functions for each component are (2) ψ(trial)1 (u1, u2) ) (2) (u1, u2) ) ψ(trial)2

∑ C1(2) φm(R, u1) φn(R, u2) m,n ∑ C2(2) φm(R, u1) φn(R, u2) m,n

m,n

(80)

m,n

∆N ) Nref - n

Using these trial wave functions and treating Cm,n as a variational parameter, we arrive at the Hermitian eigenvalue equation for both sectors. For sector 1 the form is

H1C(1) ) EC(1)

(81)

and that for sector 2 is

(

)( ) ( )

(2) H(2) C(2) 11 H12 1

(2) H(2) C(2) 21 H22 2

)E

C(2) 1

(82)

C(2) 2

Each term of the Hamiltonian matrix can be calculated analytically in the harmonic oscillator basis. Nondegenerate Case. For completeness of our presentation, we also consider a nondegenerate, 2D anharmonic oscillator model. To generate such a Hamiltonian, we modify the ground state in eq 76 to the form (1) ψ(0) (u1, u2) ) N exp(-2u12u22 - u12 - √2u22)

TABLE 2: Comparison between Wave Function L∞ Errors for the Doubly Degenerate Sector 1 Excited State (1, 0) Generated by Standard Variational Calculation of Sector 1 and Variational SUSY Calculation for the Sector 2 Ground State, Followed by Application of the SUSY Charge Operator for Different Numbers of Basis Functions (n ) Nu1, Nu2) (Nref ) (Nu1 ) 60, Nu2 ) 60))

(83)

5 2 are readily generated. Then the exact Hamiltonians H1 and H 5 2 using the However, we also shall generate an approximate H variationally determined, approximate ground-state wave function. The formal structure of the equations is the same as that (1) b 1 and H 5 2, is used to develop W above. In the case where ψ0,approx we shall see that its accuracy is an extremely important consideration.2 Results and Discussion Degenerate Case. We have calculated energies and wave functions of the Hamiltonian in eq 78 for sectors 1 and 2 using the variational approach we just described. In all calculations, b 1 and H 5 2 exactly. In Table we use the exact ψ0(1) to generate W 1 we compare sector 1 and sector 2 energies for different harmonic oscillator basis set sizes. The notations Nu1 and Nu2 give the numbers of basis functions for the variables u1 and u2, respectively. The first row gives the approximate results for E(1) 0 . The next is the doubly degenerate first-excited-state energy, E(1) 1 , followed by the sector 2 ground-state energy, E0(2). The third row contains E2(1) and E1(2) for different basis sets. It is easily seen that the doubly degenerate ground state of sector 2 is also isospectral with the doubly degenerate first excited state of sector 1. This correspondence is clearly in accordance with the general SUSY prediction about the eigenstates for the two supersym-

40 ) 60 - 20 30 ) 60 - 30 20 ) 60 - 40 10 ) 60 - 50

error L∞ L∞ L∞ L∞

(1) b † b (2) ψ(1, 0) ) Q1 · ψ(0) -4

1.1 × 10 2.2 × 10-5 5.3 × 10-6 1.6 × 10-6

(1) ψ(1, 0)

4.9 × 10-4 8.5 × 10-5 1.9 × 10-5 3.9 × 10-6

metric partner potentials. For the higher excited states this precise correspondence between the two sectors breaks down when we use a small number of basis functions (i.e., there appear some “spurious” solutions), but it is gradually restored by increasing the basis set size. We attribute this apparent breakdown of the SUSY correspondence for higher states to the error that arises in the calculation due to the truncation of an infinite basis to a finite one. Essentially, some “extra” eigenvalues (higher in value than the ground-state energy) appear in the SUSY-QM sector 2 spectrum, but they disappear as the basis set size is increased. This may raise a question regarding the precise nature of the Hylleraas-Undheim theorem for the SUSY sector 2 tensor Hamiltonian. The accuracy of the variational results is known for the ground state of sector 1, since we know the exact energy is E0(1) ) 0. Thus, the (10, 10) basis gives an error of 9 × 10-3, while the (60, 60) basis gives an error of 4.9 × 10-9. In the case of the first excited state of sector 1, the error for the (10, 10) basis (computed relative to the (60, 60) basis result) is 0.0634. By contrast, the error in the (10, 10) basis result for the sector 2 ground state (again, relative to the (60, 60) basis result) is 2.2 × 10-4. Consequently, the use of the sector 2 Hamiltonian for a ground-state calculation enables us to obtain much improved accuracy for the first excitation energy of sector 1. Basically, we estimate an increase in accuracy (defined as the ratio of the accuracy of the sector 1 result to that of the sector 2 result) to be a factor of 280. Our exploratory calculation thus clearly reveals that, for the calculation of excited-state energies, the SUSY variational method requires a smaller number of basis functions to achieve the same order of accuracy. Of course, this level of accuracy resulted in part because we have used the exact 5 2. H As this model problem has no analytical solution for the excited states, we have taken the results of the (60, 60) basis set calculation as the reference result for both sectors to check the convergence in wave functions. In Tables 2 and 3 we compare the L∞ and L2 errors of the first excited states of sector 1 that we have obtained by the SUSY variational calculation and the simple variational calculation. The L∞ error is defined as the absolute maximum difference between the solution computed with an infinite basis set (ψ(1)(∞)), which we approximate with the (60, 60) basis, and a smaller finite (n, n) basis set (ψ(1)(n))

TABLE 1: Comparison of Energy Eigenvalues (au) of Sectors 1 and 2 for Different Numbers of Basis Functions (Nu1, Nu2) sector 1, Nu1, Nu2 ) 10

sector 2, Nu1, Nu2 ) 10





9.0 × 10-3 4.6 8.3

4.5849 8.005

4.0 × 10-7 4.58473 8.00007

4.5847275 8.0000005

5.0 × 10-9 4.5847275 8.000001

4.58472742 8.000000005

Generalization of Supersymmetric Quantum Mechanics TABLE 3: Comparison between Wave Function L2 Errors for the Doubly Degenerate Sector 1 Excited State (1, 0) Generated by Standard Variational Calculation of Sector 1 and Variational SUSY Calculation for the Sector 2 Ground State, Followed by Application of the SUSY Charge Operator for Different Numbers of Basis Functions (n ) Nu1, Nu2) (Nref ) (Nu1 ) 60, Nu2 ) 60)) ∆N ) Nref - n 40 ) 60 - 20 30 ) 60 - 30 20 ) 60 - 40 10 ) 60 - 50

(1) b † b (2) ψ(1, 0) ) Q1 · ψ(0)

error

(1) ψ(1, 0)

-6

1.8 × 10-5 5.7 × 10-7 3.0 × 10-8 1.5 × 10-9

3.0 × 10 1.1 × 10-7 6.2 × 10-9 3.5 × 10-10

L2 L2 L2 L2

TABLE 4: Comparison between Wave Function L2 and L∞ Errors for the First Sector Exact Ground-State Wave (1) Function, ψ(0, 0)(∞), and Variationally Calculated (1) Ground-State Wave Function, ψ(0, 0)(n), for Different Numbers of Basis States (n ) Nu1, Nu2) Nu1, Nu2 20, 20 30, 30 40, 40 50, 50 60, 60

L∞

L2 -5

9.3 × 10 1.5 × 10-5 3.4 × 10-6 9.1 × 10-7 2.8 × 10-7

1.5 × 10-6 3.9 × 10-8 2.0 × 10-9 1.4 × 10-10 1.3 × 10-11

L∞ ) max{|ψ(1)(∞) - ψ(1)(n)|}

The L2 error is defined by


L2 )

∫-∞∞ du1 ∫-∞∞ du2 |ψ(1)(∞) - ψ(1)(n)|2

In the first column of Table 2 we show the difference in the number of basis states used (in each degree of freedom) and the maximum, Nu1 ) Nu2 ) 60, used for the reference result. Since L2 and L∞ are computed relative to the Nu1, Nu2 ) 60, 60 basis, they measure the degree of convergence of the calculations. It is clear from Tables 2 and 3 that the state obtained b† b(2) from the SUSY relation ψ(1) (1, 0) ) Q1 · ψ(0) converges more rapidly than the result obtained directly from the variational solution for sector 1. We note that the same level of convergence is obtained for both of the degenerate wave functions. Since the analytical solution for the ground-state wave function of sector 1 is known, we also have calculated the L2 and L∞ error for this wave function, comparing the analytical and variational wave functions of sector 1 for different numbers of basis states to determine a basis set size which gives a satisfactory convergence. The results are given in Table 4. It is again clear that the variational results for the sector 1 ground-sate wave function are very well converged. In Figure 1a,b we show the two components of one of the degenerate sector 2 ground-state wave functions, and in Figure 1c,d we show the two components for the other degenerate sector 2 ground-state wave function. It may seem (2) (2) and ψ(0′)2 for the pair problematic that the components ψ(0)1 of sector 2 ground-state wave functions have nodes. We shall see below that these nodes can be eliminated in a very simple

Figure 1. (a, b) Two components of one of the degenerate sector 2 ground states. (c, d) Two components of the other sector 2 ground state. Contour shading is such that red indicates a positive amplitude and blue indicates a negative amplitude. The prime on the quantum number 0 denotes the second of the two degenerate ground states.

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Figure 2. (a, b) Doubly degenerate first excited states of sector 1. The two states are generated by the SUSY-variational method. (c, d) Corresponding states of the sector 1 Hamiltonian, which are generated variationally.

manner. However, we stress that, for each of the degenerate sector 2 ground-state wave functions, there is a large and small component. Unlike the 2D separable harmonic oscillator case, not only is the small component nonzero but it has nodes. It is roughly 10 times smaller in magnitude than the large component. The two degenerate states are 90° out of phase as far as their signs. In Figure 2a,b we show the first (1) (1) excited states ψ(0, 1) and ψ (1, 0) of sector 1 that we have obtained after applying the SUSY charge operator to the sector 2 ground states, and Figure 2c,d presents the same states that were found variationally from the sector 1 Hamiltonian. The similarity of part a to part c and part b to part d of Figure 2 clearly reflects the correctness of our method. To eliminate the nodes in the components of (2) (2) b (0′) b (0) and ψ , we note that, since they are degenerate, any ψ linear combination of them is also a valid wave function. Accordingly, in analogy to the separable 2D harmonic (2) (2) and b φ(0′) oscillator considered previously, we can define b φ(0) (2) (2) b b by combining the components of ψ(0) and ψ(0′) according to (2) (2) (2) φ(0)1 ) ψ(0)1 + ψ(0 )1

(84)

(2) (2) (2) φ(0)2 ) ψ(0)2 + ψ(0 )2

(85)

(2) (2) (2) φ(0 )1 ) ψ(0)1 - ψ(0)1

(86)

(2) (2) (2) φ(0 )2 ) ψ(0)2 - ψ(0)2

(87)

(2) (2) In Figure 3 we show the components of b φ(0) and b φ(0′) . These combinations are nodeless and have definite symmetry. We (2) (2) and b φ(0′) are analogous to stress that the forms of the above b φ(0) the results in eqs 74 and 75, obtained for the degenerate separable 2D harmonic oscillator. In principle, one can consider the next higher sector partner Hamiltonian (which is a scalar operator) starting from the sector 2 ground states to get the higher excited-state energies and wave functions. The fact that H3 is a scalar is of extreme importance because its ground-state wave function will definitely be 52 nodeless. Thus, even if one solves for the ground state of H by the variational method, the quantum Monte Carlo method can be used to obtain the sector 3 ground-state wave function (3) (2) (2) (1) (1) and energy, ψ(3) 0 and E0 ) E1 - E0 ) E2 - E0 . In addition, b 1†, and Q b 2† enable us to obtain ψ b 1(2) and ψ2(1). In knowing ψ0(3), Q the degenerate case, there are two ground states, so one may possibly obtain two partner potentials for the next higher sector. We are currently exploring these features. Nondegenerate Case. In the nondegenerate example, we have performed two distinct calculations. First, we used the exact 5 2. The results are given in Tables b 1 to construct the exact H W 5-7. In Table 5, we see basically the same behavior as was obtained in the nonseparable degenerate 2D example. The errors in the ground-state energy are of similar size for both the degenerate and nondegenerate cases, with the same variation with basis set size. This behavior extends also to the first- and second-excited-state energies. We conclude that the presence



Figure 3. Components of linear combinations of the two degenerate ground states of sector 2.

TABLE 5: Comparison of Energy Eigenvalues (au) of Sectors 1 and 2 for Different Numbers of Basis Functions (Nu1, Nu2)a

a







6.4 × 10-3 4.80 6.70

4.752 6.65

3.4 × 10-7 4.75181 6.64636

4.75180771 6.64634938

3.6 × 10-9 4.75180778 6.6463495

4.75180770 6.64634937

Exact sector 1 and sector 2 Hamiltonians are used.

TABLE 6: Comparison between Wave Function L∞ Errors for the Sector 1 Excited State Generated by Standard Variational Calculation of Sector 1 and Variational SUSY Calculation for the Sector 2 Ground State, Followed by Application of the SUSY Charge Operator for Different Numbers of Basis Functions (n ) Nu1, Nu2) (Nref ) (Nu1 ) 60, Nu2 ) 60))a ∆N ) Nref - n 40 ) 60 - 20 30 ) 60 - 30 20 ) 60 - 40 10 ) 60 - 50 a

error L∞ L∞ L∞ L∞

(1) (2) b †1 · ψ b (0) ψ(1) )Q -5

9.5 × 10 1.9 × 10-5 4.5 × 10-6 9.9 × 10-7

TABLE 7: Comparison between Wave Function L2 Errors for the Sector 1 Excited State Generated by Standard Variational Calculation of Sector 1 and Variational SUSY Calculation for the Sector 2 Ground State, Followed by Application of the SUSY Charge Operator for Different Numbers of Basis Functions (n ) Nu1, Nu2) (Nref ) (Nu1 ) 60, Nu2 ) 60))a ∆N ) Nref - n

(1) ψ(1) -4

4.5 × 10 7.5 × 10-5 1.7 × 10-5 4.3 × 10-6


or absence of degeneracy does not affect the performance of b 1 is used. our SUSY approach when the exact W In the case of the L∞ and L2 errors obtained when using the b 1, we again see the same basic behavior with regard to exact W the convergence of the wave functions. However, the situation is more interesting when we use the (1) , to variationally obtained approximate ground state, ψ0,approx approx 5 b and thereby H2,approx. These results are shown generate W1 b 1 results (the columns in Table 8 and are compared to the exact W

40 ) 60 - 20 30 ) 60 - 30 20 ) 60 - 40 10 ) 60 - 50 a

error L2 L2 L2 L2

(1) (2) b †1 · ψ b (0) ψ(1) )Q -6

1.5 × 10 5.8 × 10-8 3.2 × 10-9 1.8 × 10-10

(1) ψ(1)

8.9 × 10-6 2.7 × 10-7 1.5 × 10-8 1.1 × 10-9


(2) labeled E(1) 1 and E0 ). Results are shown for three different basis set sizes, (10,10), (20,20), and (30,30). Now because we are b 1approx, it is important to using an approximate ψ0(1) to generate W note how the accuracy depends not only on the basis set size (which affects the accuracy of ψ(1) 0 ) but also on how the accuracy b approx b approx is affected by errors in ψ(1) of W 1 0 . We have found that W1 (1) is most sensitive to errors in regions where ψ0 is small in (1) b 1approx ) -∇ b ln ψ0,approx , magnitude. This is reasonable since W (1) (1) and we expect (∂ψ0 /∂uj)/ψ0 to be most sensitive to errors in

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b 1), the Ground-State TABLE 8: Comparison among the First-Excited-State Energy of Sector 1 (Calculated Using Analytical W b 1), and a Different Sector 2 Ground-State Energy That We Obtained Using Energy of Sector 2 (Calculated Using Analytical W (1) b 1approx with Different Degrees of Approximation for ψ0,approx W for Different Basis Set Sizes (n ) Nu1, Nu2) n

E(1) 1

E(2) 0

(1) ψ(0),cutoff ) 10-10

(1) ψ(0),cutoff ) 10-5

(1) ψ(0),cutoff ) 10-3

(1) ψ(0),cutoff ) 10-2

10, 10 20, 20 30, 30

4.80 4.7532 4.752

4.752 4.75181 4.751808

4.794 4.75317 4.75187

4.791 4.75313 4.75187

4.78 4.7526 4.75186

4.756 4.75181 4.751808

regions where ψ0(1) is smallest in magnitude. In view of this, we have introduced ψcutoff levels at which we cease calculating -10 b 1. These correspond to cutoff values of ψ(1) , 10-5, W 0,cutoff ) 10 -3 -2 10 , and 10 . Those results are in columns 4-7 in Table 8. It is clear that the SUSY result is always better than the sector 1 variational result, although this is only marginally the case (1) e 10-5). The with the very small cutoff values (i.e., ψ0,cutoff -2 best results are obtained with the 10 cutoff value. While obviously this is a single computational example, it is encouraging. However, additional careful studies are under way. Finally, in Figure 4a,b, we give the two components of the (2) (2) and ψ(0)2 . In Figure 4c,d, (nondegenerate) ground state, ψ(0)1 (2) and we display the components of the first excited state, ψ(1)1 (2) . We note that they are qualitatively similar to the results ψ(1)2 obtained for the degenerate case (part a and b and parts c and d of Figure 1)! This suggests to us that the nondegenerate and degenerate cases are very similar as far as the wave functions are concerned. Again, in all cases the large component is nodeless and the small component has nodes. In Figure 5a,b (1) we show the first (ψ(1) 1 ) and second (ψ2 ) excited states of sector 1 that we have obtained after applying the SUSY charge operator

b 1(2)), b 0(2)) and first excited state (ψ to the sector 2 ground state (ψ and Figure 5c,d presents the same states that were found variationally from the sector 1 Hamiltonian. These results also reflect the similarity between the degenerate and nondegenerate cases. b 0(2) For comparison, we display in Figure 6a,b the sum of ψ (2) (2) (2) b b b and ψ1 and in Figure 6c,d the difference of ψ0 and ψ1 . The results are qualitatively the same as those in the 2D separable and 2D nonseparable degenerate cases. That is, both linear combinations are nodeless and of definite sign. We are currently exploring the consequences of this behavior for the feasibility of applying the quantum Monte Carlo method to the sector 2 tensor Hamiltonian. Conclusions and Further Research In this paper we have presented a new approach to generalizing SUSY-QM to deal with more than one dimension and more than one (distinguishable) particle. In general, previous attempts to do this have typically introduced Pauli spin matrices, and as far as we are aware, none of these has been proved useful

Figure 4. (a, b) Two components of the sector 2 ground state. (c, d) Two components for the sector 2 first excited state. Contour shading is such that red indicates a positive amplitude and blue indicates a negative amplitude.



Figure 5. (a, b) First and second excited states of sector 1. The two states are generated by the SUSY variational method. (c, d) Corresponding b 1 is used. states of the sector 1 Hamiltonian, which are generated variationally. The exact W

for the general case.4-13 One principle difficulty is that while the coordinates of different particles are independent variables, they are not defined relative to orthogonal axes. That is, there are only three independent, physical axes along which all particle positions are measured. Our approach introduces a higher dimensional vector space in which there is an orthonormal basis vector associated with each independent particle coordinate. This is analogous to the relativistic situation where each particle has its own coordinate system (and, of course, in the relativistic case, its own “proper time”). Here, however, the device is a mathematical convenience (as far as we are currently aware), and it is, of course, nonrelativistic. That is, we assume Gallilean transformations. The most striking consequence similar to relativistic quantum mechanics is that our second sector Hamiltonian becomes a tensor in the expanded space. This does not increase the number of independent variables (i.e., the wave function for the second sector is a vector in the new hyperspace). It is shown that this tensor character is then absent from the third sector Hamiltonian (which is once again a scalar operator). One in general obtains an alternating series of scalar and tensor Hamiltonians. The occurrence of a tensor sector Hamiltonian is, of course, an added computational cost to the approach. This is mitigated, to some degree, by the fact that we neVer must calculate an accurate wave function and energy except for ground states. It is this feature that makes the SUSY-QM approach attractive, since ground-state energies and wave functions are the least computationally demanding of all and typically are obtained with the highest accuracy. A second mitigating factor is that the sector 3 Hamiltonian H3 is again a scalar operator. Thus, the computational effort of obtaining the second-excited-state energy and wave function again involves

solving an equation comparable to that generated by the original H1. A complication, however, arises due to the observed fact that, for the ground-state sector 2 wave function in both the degenerate and nondegenerate cases, one of the two components possesses nodes while the other component is nodeless. This is mitigated (in terms of accuracy of the sector 2 ground-state calculation) by the fact that the component containing the node is (in the present computational examples) an order of magnitude smaller than the nodeless component. This appears to enable the variational evaluation of the ground state of sector 2 to yield better accuracy for the first-excited-state energy and wave function than a comparable basis set calculation applied to H1. An extremely important question is, however, raised by the b (2) fact that the small component of ψ 0 has nodes. This is whether the presence of nodes will prevent us from applying a simple b 0(2). We variational quantum Monte Carlo method to obtain ψ are currently exploring this question. However, in the present context, it does not appear to create difficulties for the variational approach. In this paper we present the theory in full generality for treating systems of distinguishable particles. The initial applications given are to 2D systems. We first treat the simple analytically soluble 2D separable harmonic oscillator. We demonstrate that we correctly obtain excited states of sector 1 from doubly degenerate ground states of the sector 2 tensor Hamiltonian. We then consider the application to a nontrivial, 2D anharmonic nonseparable oscillator (equivalent to coupled 1D oscillators). The results again demonstrate the validity of our generalization. The 2D nonseparable anharmonic oscillator results are very interesting.

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Figure 6. Components of linear combinations of the nondegenerate ground and first excited states.

Degenerate Case. For sector 2, in the case of a doubly degenerate first excited state of H1, we obtain a doubly degenerate ground state, and the energies obtained by the Rayleigh-Ritz method are consistently lower for all the excited states of the sector 1 Hamiltonian, for the same basis set size. In addition, the SUSY-QM sector 2 result for the first-excitedstate energy is always several orders of magnitude more accurate than the Rayleigh-Ritz result for sector 1 for any given basis set size. Assessing the accuracy of the excited-state wave functions is more difficult. We chose to do this in terms of convergence of the wave functions relative to the largest basis set results. However, we are able to assess the accuracy of our variational results quantitatively in the case of the sector 1 ground state, since it is exactly known. We report our results in terms of L2 and L∞ measures, as is typical for assessing convergence and accuracy of functions in a Hilbert space. We find that the L2 and L∞ accuracies of the SUSY-QM results are consistently better than those of the Rayleigh-Ritz results for excited states of sector 1. As a further proof of this, we also consider the accuracy for the ground state of sector 1 (where we have the exact wave function) with the variational result. In fact, we find that the convergence of the sector 2 ground state is consistently better than the convergence of the variationally obtained ground-state wave function for sector 1. Nondegenerate Case. The 2D nonseperable, nondegenerate case is interesting in that it appears that there is a relationship b1(2) similar to that which was seen for the b0(2) and ψ between ψ b(2) b(2) degenerate states ψ (0) and ψ(0′). That is, one component is nodeless and large, and the second component has nodes and is smaller in magnitude. As in the degenerate case, sums and differences yield states with both components being nodeless. In this case,

however, the nondegenerate character of the states precludes (2) states for simply using two different nodeless, orthogonal b φtrial the quantum Monte Carlo method. Thus, the implementation of the quantum Monte Carlo method in the nondegenerate case appears to require further consideration. We are currently exploring this aspect of our multidimensional SUSY approach. It is important to stress that our basic strategy is to use only the ground-state results of the higher SUSY sectors. We believe that this will allow us to obtain the best results for both excitedstate energies and wave functions of the sector 1 Hamiltonian, while requiring the least computational effort. Our upcoming computational studies will be to apply the present approach to more interesting, nonseparable higher dimensional systems such as rare-gas atomic clusters where the structure and thermodynamics seem to require a fully quantum many-body treatment.15-23 For systems composed of a single type of atom or molecule, we expect to encounter degeneracies. Thus, we expect the situation to mirror the present 2D nonseparable degenerate case. In dealing with such systems, we anticipate that, as the number of particles is increased, we will find that a Monte Carlo based approach may be preferred. In a subsequent paper, we will consider whether the Monte Carlo method is applicable to 5 2. determine the ground-state energy and eigenvector of H Finally, we stress that this is the first formulation of a general SUSY approach for multidimensional and/or multiparticle systems. There remain additional formal and computational questions, which we are continuing to explore. For example, we will explore more deeply whether the Hylleraas-Undheim theorem applies to the tensor sector. Our main conclusion is that there is sufficient promise that such studies are justified.



Acknowledgment. D.J.K., K.M., and T.M. are supported in part under R. A. Welch Foundation Grant E-0608. E.R.B. is supported in part under R. A. Welch Foundation Grant E-1337 and NSF Grant CHE-0712981. Helpful conversations with B. G. Bodmann are gratefully acknowledged. We also thank the referees for extremely valuable comments on the original manuscript.

In presenting our approach, we deliberately chose not to present our derivations in a manifestly covariant form to make the presentation as clear as possiblesespecially given the me´nagerie of subscripts and superscripts already in use. In this Appendix we present a short summary of how our approach can be generalized to arbitrary curvilinear coordinate systems by the use of covariant derivatives and the introduction of the metric tensor gij. Using the standard definitions, we can express the gradient operation as24

(88)

where A,i denotes differentiation of scalar variable A with respect to coordinate ui, and we sum over repeated indices. For vectors and tensors, however, we need to bear in mind that the basis vectors b εj are not constant, and we need to write the derivative b , in this basis as of a vector, W

∂εbi b ∂W ∂Wi ) bε + Wi j j j i ∂u ∂u ∂u

Γijm ) bεm

(90)

∂uj

)

b ∂W ∂Wi ) + WiΓijk bεi j j ∂u ∂u

(

(

)

(

(95)

)

∂ψ b†1ψ ) (-∇ b+b Q W1)ψ ) - i + Wi1ψ bεi ∂u

)

∂Wi ) + WiΓijk ∂uj

(91)

(92)

where the subscripted W;i j denotes the covariant derivative of b with respect to coordinate uj. Having the ith component of W written gradients in terms of covariant derivatives, we write the divergence of a tensor or vector quantity as

b·W b ) W;ii ) 1 ∂ (g1/2Wk) ∇ g1/2 ∂uk

(96)

for the vector charge operators when acting on a scalar function and

b+b b+b b·∇ b + V1)ψ (-∇ W1) · (∇ W1)ψ ) (-∇

(

) -

(

(97)

) )

1 ∂ 1/2 ik ∂ g g + V1 ψ g1/2 ∂ui ∂uk (98)

Likewise, operations involving the vector and tensor components can be computed using

b1 · ψ Q b ) ψ;ii + Wi1ψi

(99)

b†1 · ψ Q b ) -ψ;ii + Wi1ψi

(100)

5 ·ψ b Likewise, we can write the sector 2 Schro¨dinger equation (H b ) using ) E2ψ

(101)

i i ) [-ψ;i,j + W1,j ψi + W1i ψ,ji - W1i ψ,ii + W1j W1i ψi]εFj

(102)

The part in parentheses is commonly written in a condensed notation:

W;ji

(94)

T F† · ψ b1(Q H2ψ b)Q 1 b)

Thus, covarient differentiation of a vector or tensor quantity is given by

(

)

∂ψ b1ψ ) (∇ b+b + Wi1ψ bεi Q W1)ψ ) i ∂u

(89)

This can be written in a compact form by introducing Christoffel symbols:

∂εbi

(

b·∇ bψ ) 1 ∂ g1/2gik ∂ψ ∇2ψ ) ∇ g1/2 ∂ui ∂uk

With these basic forms in hand, we can write the various operators appearing in our approach in a fully generalized form

Appendix: Extension to Arbitrary Coordinate Frames

bA ) ∂A bεi ) A,ibεi ∇ ∂ui

where g is the metric tensor and g1/2 ) (det(gij))1/2. Likewise, the Laplacian operator is written as

(93)

Consequently, our approach produces a full generalization of SUSY-QM both to higher dimensions and to arbitrary curvilinear coordinate frames. References and Notes (1) Cooper, F.; Khare, A.; Sukhatme, U. Phys. Rep. 1995, 251, 267– 385. (2) Bittner, E. R.; Maddox, J. B.; Kouri, D. J. J. Phys. Chem. A 2009, 113, 15276–15280. (3) Kouri, D. J.; Markovich, T.; Maxwell, N.; Bittner, E. R. J. Phys. Chem. A 2009, 113, 15257–15264. (4) Andrianov, A. A.; Borisov, N. V.; Ioffe, M. V. Theor. Math. Phys. (Engl. Transl.) 1984, 61, 1078–1089. (5) Das, A.; Pernice, S. A. Mod. Phys. Lett. 1997, A12, 581–588. (6) Cannata, F.; Ioffe, M. V.; Nishnianidze, D. N. J. Phys. A: Math. Gen. 2002, 35, 1389–1404. (7) Andrianov, A. A.; Borisov, N. V.; Ioffe, M. V. Phys. Lett. B 1986, 181, 141. (8) Andrianov, A. A.; Ioffe, M. V. Phys. Lett. B 1988, 205, 507–510. (9) Andrianov, A.; Ioffe, M. V; Nishnianidze, D. Phys. Lett. A 2002, 201, 103–110.

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Kouri et al. (18) Wales, D. J; Doye, J. J. Phys. Chem. A 1997, 101, 5111–5116. (19) Chakravarty, C. J. Chem. Phys. 1995, 102, 956–962. (20) Schmidt, M.; Kusche, R.; Hippler, T.; Donges, J.; Kronmu¨ller, W.; von Issendorff, B.; Haberland, H. Phys. ReV. Lett. 2001, 86, 1191–1194. (21) Chakravarty, C. J. Chem. Phys. 1995, 103, 10663–10668. (22) Rick, S. W.; Leitner, D. L.; Doll, J. D.; Freeman, D. L.; Frantz, D. D. J. Chem. Phys. 1991, 95, 6658–6667. (23) Franke, G.; Hilf, E. R.; Borrmann, P. J. Chem. Phys. 1993, 98, 3496. (24) Arfken, G. Mathematical Methods for Physicists, 3rd ed.; Academic Press Inc.: New York, 1985.

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