Multidimensional Supersymmetric Quantum Mechanics: A Scalar

Article pubs.acs.org/JPCA

Multidimensional Supersymmetric Quantum Mechanics: A Scalar Hamiltonian Approach to Excited States by the Imaginary Time Propagation Method Chia-Chun Chou* Department of Chemistry, National Tsing Hua University, Hsinchu, 30013, Taiwan, and Department of Chemistry, University of Houston, Houston, Texas 77204, United States

Donald J. Kouri Departments of Chemistry, Mathematics, Mechanical Engineering, and Physics, University of Houston, Houston, Texas 77204, United States ABSTRACT: Supersymmetric quantum mechanics (SUSYQM) is shown to provide a novel approach to the construction of the initial states for the imaginary time propagation method to determine the first and second excited state energies and wave functions for a two-dimensional system. In addition, we show that all calculations are carried out in sector one and none are performed with the tensor sector two Hamiltonian. Through our tensorial approach to multidimensional supersymmetric quantum mechanics, we utilize the correspondence between the eigenstates of the sector one and two Hamiltonians to construct appropriate initial sector one states from sector two states for the imaginary time propagation method. The imaginary time version of the time-dependent Schrödinger equation is integrated to obtain the first and second excited state energies and wave functions using the split operator method for a two-dimensional anharmonic oscillator system and a two-dimensional double well potential. The computational results indicate that we can obtain the first two excited state energies and wave functions even when a quantum system does not exhibit any symmetry. Moreover, instead of dealing with the increasing computational complexity resulting from computations in the tensor sector two Hamiltonian, this study presents a new supersymmetric approach to calculations of accurate excited state energies and wave functions by directly using the scalar sector one Hamiltonian.

I. INTRODUCTION Supersymmetric quantum mechanics (SUSY-QM) has been developed as a fascinating method to study one-dimensional problems. In analogy with the harmonic oscillator Hamiltonian, the factorization of a one-dimensional Hamiltonian can be achieved by introducing so-called charge operators.1,2 For the one-dimensional harmonic oscillator, the charge operators are the usual raising and lowering operators. The SUSY charge operators not only allow the factorization of a one-dimensional Hamiltonian but also form a Lie algebra structure. This structure leads to the generation of isospectral sector Hamiltonians. On infinite domains, the ground state of the sector two one-dimensional Hamiltonian is degenerate with the first excited state of the original Hamiltonian. The SUSY charge operators can be used to convert the ground state wave function of the sector two Hamiltonian into the first excited state wave function of the sector one Hamiltonian. As an analytical approach, SUSY-QM has been applied to the discovery of new exactly solvable potentials, the development of a more accurate WKB approximation, and the improvement of large N expansions and variational methods.3,4 In addition, © 2013 American Chemical Society

several studies have been devoted to the generalization of onedimensional SUSY-QM to multidimensional systems. Previous attempts to generalize SUSY-QM to treat more than one spatial dimension and more than one particle generally have involved introducing additional spin-like degrees of freedom.5−10 In contrast with these earlier studies, we provided a generalization of SUSY-QM to treat any number of dimensions and distinguishable particles using a tensorial operator approach.11 For one-dimensional systems, several studies have been devoted to the use of SUSY-QM as a computational tool for calculating accurate excited state energies and wave functions. Using the isospectral property of the SUSY sector Hamiltonians, we applied the variational Monte Carlo scheme2 and the Rayleigh−Ritz variational method1 to the SUSY sector Hamiltonians to obtain higher accuracy and more rapid convergence for excited state energies and wave functions for the original Hamiltonian. In addition, Kar and Bhattacharyya Received: January 30, 2013 Revised: March 25, 2013 Published: March 26, 2013 3449

dx.doi.org/10.1021/jp401068w | J. Phys. Chem. A 2013, 117, 3449−3457

The Journal of Physical Chemistry A

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article,16 we presented a new formulation of the vectorial approach to multidimensional SUSY-QM. For a multidimensional system with N particles, the Hamiltonian of the system is given by

showed that bound states of a one-dimensional Hamiltonian can be obtained through quantum adiabatic switching between the ground states of pairs of SUSY partner Hamiltonians.12 For multidimensional systems, we employed the Rayleigh− Ritz variational method to calculate eigenvalues and eigenfunctions for the tensor sector two Hamiltonian by diagonalizing the sector two Hamiltonian in an approximate truncated basis.11 The computational results for two-dimensional anharmonic oscillator systems demonstrated that the SUSY variational method only requires a small number of basis functions to achieve accuracy for the calculation of excited state energies and wave functions of the original Hamiltonian.11 In addition, we applied the quantum adiabatic switching method to the sector two Hamiltonian to determine the excited states.13 However, there is a significant increase in the computational complexity compared to the scalar sector one Hamiltonian, and the computational effort in sector two scales badly with system dimensionality.14,15 Therefore, it is important to find a way to circumvent the poor scaling of the sector two computations. In the current study, we show that the SUSY-QM formalism provides a novel approach to the construction of the initial sector one states for the imaginary time propagation method to determine the first and second excited state energies and wave functions. When a quantum system exhibits symmetry, it is usually straightforward to construct an initial state that is orthogonal to the ground state for determining excited states of different symmetry from the ground state. On the other hand, if the system lacks symmetry, there does not appear to be a general method for generating trial functions that are orthogonal to the ground state. However, through the correspondence between the eigenstates of the sector one and two Hamiltonians, we can construct appropriate initial sector one states from sector two states for the imaginary time propagation method. The imaginary time version of the timedependent Schrödinger equation is integrated to obtain the first and second excited state energies and wave functions. This computational method is applied to a two-dimensional anharmonic oscillator system and a two-dimensional double well potential. The computational results indicate that we can obtain the first two excited state energies and wave functions even for a two-dimensional quantum system without any symmetry. Therefore, combining the imaginary time propagation method for the sector one Hamiltonian with the structure of the degeneracies between the sector one and two Hamiltonians completely avoids the tensor sector two computations. The organization of the remainder of this study is as follows. In section II, we review the vectorial approach to multidimensional SUSY-QM. Spurious states for the sector two Hamiltonian are briefly discussed. In section III, we show how to construct the appropriate initial states to determine the first and second excited state energies and wave functions by the imaginary time propagation method. In section IV, the methodology is applied to a two-dimensional anharmonic oscillator system and a two-dimensional double well potential. In section V, we summarize our results and conclude with some comments.

H1 = −

ℏ2 2m

N

∑ ∇i2 + V1

(1)

i=1

{ψ(1) n }

where one has a complete set of orthogonal eigenstates and energies {E(1) n } for n = 0,1,2,.... For simplicity, we set the masses of the particles to be equal and use units such that ℏ2/ 2m = 1. The kinetic energy operator can be expressed as ⎛ ∇⃗ ⎞ ⎜ 1⎟ N ⎜ ⃗ ⎟ † ∇ −∑ ∇i 2 = ( −∇1⃗ , −∇2⃗ , ..., −∇N⃗ )·⎜ 2 ⎟ ≡ ∇⃗ ·∇⃗ ⎜ ⎟ ⋮ i=1 ⎜ ⎟ ⎜ ⃗ ⎟ ⎝∇N ⎠

(2)

where ∂ ∂ ∂ + k̂ + ĵ ∇m⃗ = i ̂ ∂zm ∂ym ∂xm

for m = 1, ..., N. It is noted that the coordinates of all particles are referenced to a single three-dimensional Cartesian frame. The ground state wave function satisfies the timeindependent Schrödinger equation H1ψ0(1) = E0(0)ψ0(1)

(3)

The nodeless ground state wave function can be written as ψ0(1) = A e−S(u ⃗)

(4)

where A is a normalization constant, u⃗ = (u1⃗ ,u⃗2, ..., u⃗N), and u⃗m = ix̂ m + jŷ m + kẑ m for m = 1, ..., N. Then, the vector superpotential W⃗ can be defined by the exact differential ⎛ du1⃗ ⎞ ⎜ ⎟ ⎜ du 2⃗ ⎟ dS = W⃗ (u ⃗) ·du ⃗ = (W1⃗ , W2⃗ , ..., WN⃗ )·⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝ duN⃗ ⎠

(5)

where the superpotential W⃗ is expressed in terms of a vector. This expression indicates that the ith particle’s superpotential is related to the ground state wave function by Wi⃗ = −∇i⃗ ln ψ0(1)

(6)

Throughout this study, the ground state wave function is assumed to be purely real; hence, the superpotential components are real. It follows from eq 6 that (∇⃗ + W⃗ )ψ(1) 0 = 0. In addition, the SUSY charge operator and its adjoint operator are defined by

II. MULTIDIMENSIONAL SUPERSYMMETRIC QUANTUM MECHANICS A vectorial approach to the SUSY-QM formalism has been developed to treat multidimensional systems involving any number of distinguishable particles.1,2,11 In the companion 3450

⎛ ∇⃗ + W⃗ ⎞ 1 ⎟ ⎜ 1 ⎟ ⋮ Q⃗ = ∇⃗ + W⃗ = ⎜ ⎜⎜ ⎟⎟ ⎝∇N⃗ + WN⃗ ⎠

(7)

† Q⃗ = ( −∇1⃗ + W1⃗ , ..., −∇⃗N + WN⃗ )

(8)



Article 17 E(1) However, for multidimensional SUSY-QM, we showed 0 . in the companion article that there exist normalizable spurious states for the sector two tensor Hamiltonian with energy equal to the ground state of H1.16 These spurious states satisfy the equation Q⃗ †·ψ⃗ sp = 0. It follows that the spurious state satisfies the vector Schrödinger equation for the sector two Hamiltonian

Then, we can factorize the Hamiltonian H1 into the sector one SUSY form in terms of W⃗ †

H1 − E0(1) = Q⃗ ·Q⃗

(9)

Thus, the sector one Hamiltonian (the original scalar Hamiltonian) can be expressed in terms of the SUSY charge operators as

†

H⃡ 2·ψsp⃗ = Q⃗ Q⃗ ·ψsp⃗ + E0(1)ψsp⃗ = E0(1)ψsp⃗

†

H1 = Q⃗ ·Q⃗ + E0(1)

(17)

Thus, the spurious states ψ⃗ sp are eigenstates of H⃡ 2, all with energy equal to the ground state of H1. Figure 1 displays the

(10)

It follows from eq 6 that the SUSY charge operator annihilates the ground state of the system, Q⃗ ψ(1) 0 = 0, and this implies that (1) (1) H1ψ(1) = E ψ as required. 0 0 0 We now construct the sector two Hamiltonian such that it is isospectral with the sector one Hamiltonian H1. For an excited state ψ(1) n (n ≠ 0) in the sector one satisfying the Schrödinger (1) (1) equation H1ψ(1) n = En ψn , we write †

[Q⃗ ·Q⃗ + E0(1)]ψn(1) = En(1)ψn(1)

(11)

We then form the tensor product by operating on the left with Q⃗ so that †

[Q⃗ Q⃗ + E0(1) 1]⃡ ·Q⃗ ψn(1) = En(1)Q⃗ ψn(1)

It follows that Hamiltonian

Q⃗ ψ(1) n

(12)

is an eigenstate of the sector two tensor

Figure 1. For multidimensional SUSY-QM, there are an infinite number of spurious states for the sector two Hamiltonian with energy equal to the ground state energy of the sector one Hamiltonian.

†

H⃡ 2 = Q⃗ Q⃗ + E0(1) 1⃡

(13)

E(1) n .

with energy equal to Hence, for any of the excited states in the sector one Hamiltonian, Q⃗ ψ(1) n generates an eigenstate of the sector two Hamiltonian with the same energy. In particular, Q⃗ ψ(1) 0 cannot give an eigenstate with energy equal to the ground state energy E(1) 0 because the SUSY charge operator annihilates the ground state, Q⃗ ψ(1) 0 = 0. On the other hand, we consider the eigen-equation for the sector two Hamiltonian H⃡ 2·ψλ⃗ (2) = Eλ(2)ψλ⃗ (2)

correspondence between the eigenstates of the sector one and two Hamiltonians. In addition, because spurious states are annihilated by the adjoint charge operator, they do not generate normalizable, physical states for the sector one Hamiltonian. The Hermitian property of the sector two Hamiltonian implies the orthogonality between spurious and physical states. Furthermore, we have developed an explicit method for construction of the spurious states in a specific form for any quantum system. Several specific spurious states were constructed for a two-dimensional anharmonic oscillator system and for the hydrogen atom.16

(14)

(2) (2) T where ψ⃗ (2) = (ψ⃗ (2) is any column vector λ λ1 , ψ⃗ λ2 , ..., ψ⃗ λN ) eigenfunction. Forming the scalar product of the sector two Hamiltonian with Q⃗ †, we obtain

†

†

III. IMAGINARY TIME PROPAGATION METHOD A. Ground States. The imaginary time propagation method has been used as a way to obtain the ground-state energy and wave function.18−23 In quantum mechanics, the wave function is governed by the time-dependent Schrödinger equation

†

Q⃗ ·[Q⃗ Q⃗ + E0(1) 1]⃡ ·ψλ⃗ (2) = Eλ(2)Q⃗ ·ψλ⃗ (2)

(15)

Rearranging this equation yields †

†

†

[Q⃗ ·Q⃗ + E0(1)](Q⃗ ·ψλ⃗ (2)) = Eλ(2)(Q⃗ ·ψλ⃗ (2))

⃗†

·ψ⃗ (2) λ )

E(2) λ

(16)

⃗†

iℏ

·ψ⃗ (2) λ ).

From eq 10, we obtain H1(Q = (Q Thus, Q⃗ †·ψ⃗ (2) λ is an eigenstate of the sector one Hamiltonian H1 with energy E(2) λ (provided it is normalizable). This analysis holds in ⃡ general for any eigenstate ψ⃗ (2) and ψ(1) λ n . Therefore, H2 is isospectral with H1, and the correspondence between the eigenstates of sectors one and two has been established.11,16 Related issues concerning generation of higher sector Hamiltonians have been pointed out in ref 14 and responded to in ref 15. As discussed in detail in the companion article,16 for onedimensional SUSY-QM on an infinite domain, the sector one and two Hamiltonians have identical spectra with the exception of the ground state of the sector one. If the domain of the position is finite, it is generally possible that the ground state of (2) the sector two Hamiltonian satisfies Q⃗ †ψ(2) 0 = 0 with E0 =

∂ ℏ2 2 ψ = Hψ = − ∇ ψ + Vψ 2m ∂t

(18)

where V is the potential energy of the system. Performing a transformation from real time to imaginary time by introducing the new variable τ = it, we obtain the imaginary time version of the time-dependent Schrödinger equation

∂ ψ = −Hψ ∂τ This equation has the formal solution ℏ

ψ (τ ) = e−Hτ / ℏψ (0)

(19)

(20)

where ψ(0) is the initial wave packet. Using the complete set of states formed by the eigensolutions of the time-independent Schrödinger equation Hϕn = Enϕn, we can expand an arbitrary 3451



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where cj and dλ are the expansion coefficients and ψ⃗ (2) j and ψ⃗ sp,λ are the physical and spurious states for the sector two Hamiltonian, respectively. Applying the adjoint charge operator to this vector state, we obtain

initial state into a series expansion in terms of the eigenfunctions of H ψ (0) =

∑ cnϕn

(21)

n

where the expansion coefficients are given by cn = ⟨ϕn|ψ(0)⟩. Substituting this expansion into the formal solution of the imaginary time Schrödinger equation in eq 20 yields ψ (τ ) = e

−Hτ / ℏ

∑ cnϕn = ∑ cn e n

n

−Enτ / ℏ

ϕn

(22)

†

Q⃗ ·ψα⃗ (2) = −

∂ψ0(1) ∂x

+

= −2

∂ψ0(1) ∂x

⎛ 0 ⎞ ψβ⃗ (2) = ⎜⎜ (1)⎟⎟ ⎝ ψ0 ⎠

(26)

(27)

Following the same procedure, we can show that the corresponding sector one state

ψβ(1) = −2

∂ψ0(1) ∂y

(28)

is orthogonal to the ground state of the sector one Hamiltonian. In general, the state ψ(1) β is not orthogonal to the state ψ(1) α (except when there is sufficient symmetry). Thus, in the case of no symmetry, this state will also converge to the first-excited state of the sector one Hamiltonian. In fact, we can directly prove the orthogonality between the ground state ψ0(1) and its partial derivative ∂ψα(1)/∂x by evaluating the inner product ψ0(1)|

∂ψ0(1) ∂x

∬R ψ0(1)

=

1 2 1 = 2 =

(23)

2

∬R

2

∮C

∞

∂ψ0(1) ∂x

dx dy

∂ (1) 2 [ψ ] dx dy ∂x 0

[ψ0(1)]2 dy

(29)

where the ground state wave function is assumed to be real and Green’s theorem has been used to obtain the final expression.24 In this case, the inner product in eq 29 is equal to zero because the integrand tends to zero when x and y go to infinity. Thus, the vanishing inner product indicates that the state ∂ψ(1) 0 /∂x is orthogonal to the ground state. Analogously, the inner product (1) of the ground state ψ(1) 0 and its partial derivative ∂ψ0 /∂y is given by

∞

λ

W1ψ0(1)

where eqs 6 and 8 have been used. This equation indicates that the vector state ψ⃗ (2) is not a spurious state and that the α corresponding sector one state ψ(1) α does not vanish. Thus, it follows from eq 25 that the sector one state ψ(1) α is orthogonal to the ground state of the sector one Hamiltonian. Therefore, we can determine the first-excited state energy and wave function by propagating ψ(1) α in imaginary time. In addition, we construct another sector two vector state from the ground state wave function of the sector one Hamiltonian

∑ cjψ⃗j(2) + ∑ dλψsp⃗ ,λ j=0

(25)

(1) where Q⃗ †·ψ⃗ (2) and Q⃗ †·ψ⃗ sp,λ = 0 have been used. j−1 ∝ ψj As indicated in eq 25, the corresponding sector one state ψ(1) α (1) = Q⃗ †·ψ⃗ (2) α contains no contribution from the ground state ψ0 . We can also show that all the expansion coefficients in eq 25 are not equal to zero by directly applying the adjoint charge operator to the vector state

As presented in our previous study,11 since the tensor sector two Hamiltonian is Hermitian, the eigenstates of the sector two Hamiltonian form a complete set. Thus, as shown in Figure 1, the vector state ψ⃗ (2) α can be expressed in terms of the eigenstates of the sector two Hamiltonian including the physical and spurious states ψα⃗ (2) =

∑ c′ j ψj(1) j=1

The formal solution in eq 22 shows that each eigenfunction decays exponentially to zero at a rate determined by its eigenvalue for large τ. This means that the ground state, which relaxes most slowly, persists longest. After a time τ, the component of eigenfunction ϕn is reduced relative to the ground state by the ratio e−(En‑E0)τ/ℏ. As time progresses, the wave function ψ(τ) converges to the ground state wave function ϕ0 regardless of the choice of the initial wave function, as long as there is a numerically significant overlap between the initial state ψ(0) and the ground state ϕ0. Thus, propagating an arbitrary initial wave function in imaginary time projects onto the ground state. B. First-Excited States. For the imaginary time propagation method, if we want to obtain a given excited state, we have to remove all states from the initial wave packet whose energy is below that of this excited state. For example, in order to determine the first-excited state, we have to calculate the ground state of the modified Hamiltonian H′ = (1 − P0)H(1 − P0), where the projection operator is P0 = |ϕ0⟩⟨ϕ0|. On the other hand, if we choose an initial wave function, which is orthogonal to the ground state wave function, then this state will converge to the first-excited state. For quantum systems exhibiting definite symmetry, we may easily construct an initial wave function, which differs in symmetry from the ground state. However, for quantum systems without any (obvious) symmetry, it is not straightforward to construct an initial wave function that is orthogonal to the ground state wave function. The vectorial approach to multidimensional SUSY-QM provides a novel method for the construction of the initial states for the imaginary time propagation method to determine the excited-state energies and wave functions. As an example, we consider a two-dimensional system with the ground state wave function ψ0(x, y). Here, we do not require that the ground state wave function have any symmetry. Then, we construct a sector two vector state from the ground state wave function of the sector one Hamiltonian ⎛ ψ (1)⎞ ψα⃗ (2) = ⎜⎜ 0 ⎟⎟ ⎝ 0 ⎠

∞

†

Q⃗ ·ψα⃗ (2) =

(24) 3452


The Journal of Physical Chemistry A ψ0(1)|

∂ψ0(1) ∂y

1 = 2

∬R

1 =− 2

2

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ψ0(1)(x , y) = C exp( −2x 2y 2 − x 2 − x −

∂ (1) 2 [ψ ] dx dy ∂y 0

∮C

[ψ0(1)]2 dx

∞

where C is a normalization constant. Substituting this expression into eq 6 gives the corresponding vector superpotential

(30)

W1x(x , y) = −

∂ ln ψ0(1) = 4xy 2 + 2x + 1 ∂x

W1y(x , y) = −

∂ ln ψ0(1) = 4x 2y + 2 2 y + ∂y

(36)

2

(37)

From eq 9, we can construct the potential and the sector one Hamiltonian given by H1 =−∇2 + V1(x , y) ∂2 ∂2 − 2 + (4xy 2 + 2x + 1)2 2 ∂x ∂y + (4x 2y + 2 2 y + 2 )2 − 4(x 2 + y 2 ) − (2 + 2 2 ).

=−

(38)

E(1) 0

The exact ground state energy for H1 is = 0. Figure 2a presents the ground state wave function in eq 35. It is clear that the ground state wave function does not exhibit any

(31)

Then, we construct a new sector two state of the form ⎛ χ2 ⎞ ψ⃗ = ⎜ ⎟ ⎝−χ1⎠

2 y) (35)

Again, since the integrand vanishes at infinity, the state ∂ψ(1) 0 /∂y is orthogonal to the ground state. Furthermore, we can use an arbitrary sector two state to construct an initial state for the first-excited state. As long as it is not a spurious state, eq 25 implies that applying the adjoint charge operator to it yields a sector one state, which is orthogonal to the ground state of the sector one Hamiltonian. C. Second-Excited States. After obtaining the first-excited state of the sector one Hamiltonian, we can employ SUSY-QM to construct an initial state for the imaginary time propagation method to generate the second-excited state. From the firstexcited state ψ(1) 1 of the sector one Hamiltonian, we obtain the sector two ground state using the charge operator

⎛ χ1 ⎞ Q⃗ ψ1(1) = ψ0⃗ (2) = ⎜ ⎟ ⎝ χ2 ⎠

2 y2 −

(32)

It is obvious that this state is orthogonal to the ground state of the sector two Hamiltonian in eq 31. Analogously, this state can be expressed in terms of the eigenstates of the sector two Hamiltonian ∞

ψ⃗ =

∑ cjψ⃗j(2) + ∑ dλψsp⃗ ,λ j=1

λ

(33)

where cj and dλ are the expansion coefficients and ψ⃗ (2) j and ψ⃗ sp,λ are the physical and spurious states for the sector two Hamiltonian, respectively. The sum over physical states in eq 33 starts with the sector two first-excited state because of the orthogonality between the states ψ⃗ and ψ⃗ (2) 0 . Applying the adjoint charge operator to the state gives †

Q⃗ ·ψ⃗ =

Figure 2. Two-dimensional anharmonic oscillator system: (a) the sector one ground state wave function in eq 35; (b) the partial derivative of the ground state wave function with respect to x.

definite symmetry. Thus, there is no obvious way to construct an initial state that is orthogonal to the ground state to generate the first-excited state by the imaginary time propagation method. As discussed in section III.B, the state ∂ψ(1) 0 /∂x shown in Figure 2b is exactly orthogonal to the ground state. Then, we use this state as the initial state to determine the firstexcited state energy and wave function. Substituting the potential energy in eq 38 into eq 19, we obtain the imaginary time Schrödinger equation

∞

∑ c′ j ψj(1) j=2

(34)

(1) ⃗† where Q⃗ †·ψ⃗ (2) j−1 ∝ ψj and Q ·ψ⃗ sp,λ = 0 have been used. Because this expansion indicates that the sector one state Q⃗ †·ψ⃗ contains no contribution from the ground and first-excited states, this state is orthogonal to the first two lowest energy states. Therefore, the sector one state Q⃗ †·ψ⃗ can serve as the initial state for the imaginary time propagation method to produce the second-excited state of the sector one Hamiltonian, even in the absence of symmetry.

∂ ℏ2 2 ∇ ψ − Vψ ψ= (39) ∂τ 2m We employed the split-operator method to integrate this 25 equation using the initial state ∂ψ(1) 0 /∂x from τ = 0 to τ = 2.5. The computational grid extends from x = −4 to x = 4 and from y = −4 to y = 4 with 28 grid points in x and y, and the integration time step was Δτ = 0.0001. Figure 3a shows the time-dependent energy expectation value. As time progresses, the energy quickly converges to a stable plateau with the firstexcited state energy E = 4.36580. To assess the accuracy of the computational result, we compare the result with a Chebyshev ℏ

IV. COMPUTATIONAL RESULTS A. Anharmonic Oscillator. The first example concerns a nonseparable nondegenerate two-dimensional anharmonic oscillator system without any parity symmetry. The ground state energy of the sector one system is zero and the analytical expression of the ground state wave function is given by 3453



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Figure 3. Two-dimensional anharmonic oscillator system: (a) convergence of the energy for the first-excited state; (b) the first-excited state wave function.

polynomial discrete variable representation (DVR) calculation using 60 grid points in x and y (for a total of 3600 basis functions).26 The ITP computational result is in excellent agreement with the DVR result E = 4.36598. Figure 3b displays the first-excited state wave function and this wave function does not exhibit any definite symmetry. As shown in eq 31, applying the charge operator to the resulting first-excited state of the sector one Hamiltonian yields the ground state of the sector two Hamiltonian. We used the fourth-order finite difference formulas to approximate the spatial derivatives in the charge operator. Figure 4 presents the

Figure 5. Sector one t = 0 state constructed from Q⃗ †·ψ⃗ used to generate the second-excited state for the two-dimensional anharmonic oscillator system.

state, the wave function was propagated for a longer period of time to achieve higher accuracy of the first-excited state wave function. As shown in Figure 6a, we propagated the numerically constructed initial state Q⃗ †·ψ⃗ from τ = 0 to τ = 1.5. Actually, this initial state is numerically orthogonal to the ground and first-excited states since the first-excited state is not exact. Thus, if we propagate this initial state for a longer period of time, the accumulated error will lead to the collapse of the propagated wave function into the ground state. B. Two-Dimensional Double Well Potential. As an example where we do not have an exact, analytical expression for the ground state wave function, we consider a twodimensional double well potential given by

Figure 4. Two components of the sector two ground state for the twodimensional anharmonic oscillator system. Note that both components are nodeless.

two components of the sector two ground state wave function ψ⃗ (2) (0). From eq 32, we constructed a new sector two state ψ⃗ , which is orthogonal to the ground state of the sector two Hamiltonian. Then, applying the adjoint charge operator to this state, we obtain the sector one state Q⃗ †·ψ⃗ , which is orthogonal to the ground and first-excited states of the sector one Hamiltonian. As shown in Figure 5, the resulting sector one state does not display any definite symmetry. We propagated the sector one state Q⃗ †·ψ⃗ in imaginary time from τ = 0 to τ = 1.5 to determine the second-excited energy and wave function. As shown in Figure 6a, the energy converges to the second-excited state energy E = 7.0615. The computational result is in excellent agreement with the DVR result E = 7.0667. Figure 6b presents the sector one secondexcited state wave function, and this wave function does not exhibit any definite symmetry. As noticed in Figure 3a, because we need to use the first-excited state of the sector one Hamiltonian to construct the initial state for the second-excited

V (x , y) = 3x 4 − 8x 2 + y 2 − xy

(40)

The potential energy is a sum of a double well potential along the reaction coordinate x and a harmonic oscillator along the bath coordinate y with a coupling term linear in both coordinates. This potential has been used to describe the intramolecular hydrogen transfer process,27,28 and Figure 7 shows contours of the two-dimensional double well potential. We first determined the approximate ground state energy and wave function by integrating the imaginary time Schrödinger equation in eq 39 from τ = 0 to τ = 6. The initial state is a Gaussian wave packet given by ψ (x , y ) = 3454

⎛ 2 ⎞1/2 −(x 2 + y2 ) ⎜ ⎟ e ⎝π ⎠

(41)



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Figure 6. Two-dimensional anharmonic oscillator system: (a) convergence of the energy for the second-excited state; (b) the second-excited state wave function.

From the computational result for the ground state, we calculated the partial derivative of the ground state with respect to x using the fourth-order finite difference formulas to approximate the spatial derivative. Then, we integrated the imaginary time Schrödinger equation from τ = 0 to τ = 6 with the initial state ∂ψ(1) 0 /∂x to determine the first-excited state energy and wave function. Since the ground state is not exact, this initial state is numerically orthogonal to the ground state. As shown in Figure 9, this initial state converges to the first-excited state with energy E = −0.756487, which is in excellent agreement with the DVR result E = −0.756485. Analogous to the case in the anharmonic oscillator system, we followed the same procedure as presented in section III.C to construct the sector one state Q⃗ †·ψ⃗ , which is orthogonal to the ground and first-excited states of the sector one Hamiltonian. We numerically calculated the superpotential in the adjoint charge operator from the computational result for the ground state. Then, we used Q⃗ †·ψ⃗ as the initial state to determine the second-excited energy and wave function. Figure 10 presents the time-dependent energy expectation value and the secondexcited state wave function. As shown in this figure, the initial state Q⃗ †·ψ⃗ converges to the second-excited state with energy E = 0.8101. Again, the computational result is in excellent agreement with the DVR result E = 0.8118. In particular, the initial state is numerically orthogonal to the ground and firstexcited states. Thus, a long time propagation of the initial state will result in the collapse of the propagated wave function into the ground state.

Figure 7. Contours of the two-dimensional double well potential in eq 40.

where the wave packet is centered at the origin. The computational grid extends from x = −4.5 to x = 4.5 and from y = −4.5 to y = 4.5 with 28 grid points in x and y, and the integration time step was Δτ = 0.0001. Figure 8 presents the time-dependent energy expectation value and the ground state wave function. As shown in this figure, the energy rapidly converges to the ground state energy E = −1.335178, and the computational result is in excellent agreement with the DVR result E = −1.335176 using 60 grid points in x and y. Here, in order to achieve higher accuracy of the ground state wave function, we propagated the wave packet for a long period of time.

Figure 8. Two-dimensional double well potential: (a) convergence of the energy for the ground state; (b) the ground state wave function. 3455



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Figure 9. Two-dimensional double well potential: (a) convergence of the energy for the first-excited state; (b) the first-excited state wave function.

Figure 10. Two-dimensional double well potential: (a) convergence of the energy for the second-excited state; (b) the second-excited state wave function.

V. DISCUSSION AND CONCLUSIONS We briefly reviewed our vector approach to multidimensional SUSY-QM. For one-dimensional infinite domain SUSY-QM, the sector one and two Hamiltonians have identical spectra with the exception of the ground state of the sector one. For multidimensional infinite domain SUSY-QM, the correspondence between the eigenstates of the sectors one and two, except for the ground state of the sector one, is established through the intertwining relations between the charge operators and the sector Hamiltonians. However, as presented in the companion article,16 there exist normalizable spurious states for the sector two Hamiltonian with energy equal to the ground state energy of the sector one Hamiltonian. These normalizable spurious states are annihilated by the adjoint charge operator, and hence, these states do not correspond to physical states for the sector one Hamiltonian. The SUSY-QM formalism provides a novel approach to the construction of the initial states for the imaginary time propagation method to determine the first and second excited state energies and wave functions. Through the correspondence between the eigenstates of the sector one and two Hamiltonians, we can construct the appropriate initial states orthogonal to the lowest or the first two lowest energy states from sector two states. In general, if we expand an arbitrary sector two state in terms of eigenstates of the sector two Hamiltonian, we must include both the physical and spurious states. However, because the spurious states are annihilated by the adjoint charge operator, they can never contaminate the sector one states obtained from any sector two eigenstate. We

can construct the appropriate initial states from the sector two states. Thus, propagating these initial states in imaginary time produces the first and second excited states. We integrated the imaginary time Schrödinger equation to determine the first and second excited state energies and wave functions for a two-dimensional anharmonic oscillator system and a two-dimensional double well potential. We used the partial derivative of the ground state wave function with respect to x to obtain the first-excited state energy and wave function. From the resulting first-excited state, we employed the charge operator to obtain the corresponding sector two ground state. Then, we constructed a sector two state that is orthogonal to the sector two ground state. Applying the adjoint charge operator to the sector two state, we obtained the initial state used to determine the second-excited state energy and wave function. The computational results indicate that, even in the absence of symmetry, we can obtain the first two excited state energies and wave functions for a two-dimensional quantum system by the imaginary time propagation method. The current study demonstrates that the SUSY-QM formalism can be employed to construct initial states for the imaginary time propagation method to obtain the excited state energies and wave functions. In addition to the imaginary time propagation method, the SUSY-QM approach can be used for other methods that are specifically suited to the ground state. For example, the variational Monte Carlo technique has been developed as a powerful way to estimate the ground state of a quantum mechanical system. Therefore, application of the 3456



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SUSY-QM approach to the variational Monte Carlo method for excited state energies and wave functions should be explored. The SUSY-QM formalism provides an alternative method to obtain the excited state energies and wave functions. As presented in our previous studies,1,2 for one-dimensional problems, applying the variational method to the sector two Hamiltonian, we obtain higher accuracy and more rapid convergence for excited state energies and wave functions for the original Hamiltonian. In contrast, for multidimensional problems, dealing with the tensor sector Hamiltonian significantly increases the computational complexity.11,13−15 However, because we carry out all computations of excited states with the scalar sector one Hamiltonian, the computational complexity is the same as exists currently for ordinary ground state calculations. Therefore, the current study presents a way to perform SUSY calculations in the sector one scalar structure and circumvents the poor scaling of computations in the tensor sector two Hamiltonian. Furthermore, it may be possible to obtain more excited state energies and wave functions. As indicated in this study, if we can construct a sector two state that is orthogonal to the ground and first-excited states of the sector two Hamiltonian, applying the adjoint charge operator to this state yields an appropriate initial sector one state to be used to obtain the third-excited state of the sector one Hamiltonian. Applications to the determination of higher-excited states and to quantum systems of higher dimension than two deserve further investigation. In addition, we are currently developing several methods to perform SUSY calculations in a sector one structure. Relevant developments and applications will be reported elsewhere in the future.

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AUTHOR INFORMATION

Corresponding Author

*(C.-C.C.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Robert Welch Foundation (grant no. E-0608) for their financial support of this research. REFERENCES

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Multidimensional Supersymmetric Quantum Mechanics: A Scalar

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