Supersymmetric Quantum Mechanics in Multiple Dimensions Applied to Variational Monte Carlo - A Proof of Principle Study

We present a new approach to variational monte carlo using our N-Dimensional generalization of Supersymmetric Quantum Mechanics. We do this by introducing a {\em 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 Schr\"odinger operator can be factored into vector ``charge'' operators: $\vec Q_{1}$ and $\vec Q_{1}^{\dagger}$. Using these operators, we can write the original (scalar) Hamiltonian as $H_{1} = \vec Q_{1}^{\dagger}\cdot \vec Q_{1} + E_{0}^{(1)}$. The second sector Hamiltonian is a tensor given by $H_{2} = \vec Q_{1}\vec Q_{1}^{\dagger} + E_{0}^{(1)}$ and is isospectral with $H_{1}$. The vector ground state of sector two, $\vec\psi_{0}^{(2)}$, can be used with the charge operator $\vec Q_{1}^{\dagger}$ to obtain the excited state wave functions of the first sector. We demonstrate the approach with examples of a pair of separable 1D harmonic oscillators and the example of a non-separable 2D anharmonic oscillator (or equivalently a pair of coupled 1D oscillators).