Department of Chemistry, University of Houston, Houston, Texas 77204, USA.
J Phys Chem A. 2010 Aug 19;114(32):8202-16. doi: 10.1021/jp103309p.
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 Schrödinger operator can be factored using a 3N-component gradient operator and introducing vector "charge" operators: Q(1) and Q(1)(dagger). Using these operators, we can write the original (scalar) Hamiltonian as H(1) = Q(1)(dagger) x Q(1) + E(0)((1)), where E(0)((1)) is the ground-state energy. The second sector Hamiltonian is a tensor given by H(2) = Q(1)Q(1)(dagger) + E(0)((1)) and is isospectral with H(1). The vector ground state of sector 2, psi(0)((2)), canbe used with the charge operator Q(1)(dagger) to obtain the excited-state wave function of the first sector. In addition, we show that H(2) can also be factored in terms of a sector 2 vector superpotential with components W(2j) = -(partial partial differential ln psi(0j)((2)))/partial partial differentialx(j). Here psi(0j)((2)) is the jth component of psi(0)((2)). Then one obtains charge operators Q(2) and Q(2)(dagger) so that the second sector Hamiltonian can be written as H(2) = Q(2)(dagger)Q(2) + E(0)((2)). This allows us to define a third sector Hamiltonian which is a scalar, H(3) = Q(2) x Q(2)(dagger) + E(0)((2)). This prescription continues with the sector 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.
我们在这里提出了一种新的方法，将超对称量子力学推广到处理多粒子和多维系统。我们通过在正交超空间中引入向量超势来实现这一点。在三维中有 N 个可区分粒子的情况下，这导致了一个具有 3N 个正交分量的向量超势。原始的标量薛定谔算符可以通过使用 3N 分量梯度算符和引入向量“电荷”算符 Q(1) 和 Q(1)(dagger) 来进行因式分解。使用这些算符，我们可以将原始（标量）哈密顿量写为 H(1) = Q(1)(dagger) x Q(1) + E(0)((1))，其中 E(0)((1)) 是基态能量。第二部分哈密顿量是一个张量，由 H(2) = Q(1)Q(1)(dagger) + E(0)((1)) 给出，与 H(1) 是等谱的。第二部分的向量基态 psi(0)((2))，可以与电荷算符 Q(1)(dagger) 一起使用，以获得第一部分的激发态波函数。此外，我们还表明 H(2) 也可以根据具有分量 W(2j) = -(∂∂ln psi(0j)((2)))/∂∂xj 的第二部分向量超势进行因式分解。这里 psi(0j)((2)) 是 psi(0)((2)) 的第 j 个分量。然后得到电荷算符 Q(2) 和 Q(2)(dagger)，使得第二部分哈密顿量可以表示为 H(2) = Q(2)(dagger)Q(2) + E(0)((2))。这允许我们定义第三个部分的哈密顿量，它是一个标量，H(3) = Q(2) x Q(2)(dagger) + E(0)((2))。这种方法可以继续使用，其中部分哈密顿量在标量和张量形式之间交替，这两种形式都可以通过变分法来处理，以获得标量和张量部分的近似解。我们用一对可分离的一维谐振子的例子和一个不可分离的二维非谐振子的例子（或等效地，一对耦合的一维谐振子）来演示这种方法。我们考虑了简并和非简并的情况。我们还在附录中提出了对任意曲线坐标系的推广。
