Quantum mechanics of particles trapped in a Lamé circle or Lamé sphere shaped potential well

Abstract

Ground state and 1st excited state energies and wave functions were calculated for systems of one or two electrons in a 2D and 3D potential well having a shape intermediate between a circle and a square or a sphere and a cube. One way to define such a potential well is with a step potential and a bounding surface of form $|x|^q+|y|^q+|z|^q = |r|^q$, which converts from a sphere to a cube when $q$ increases from $2$ to infinity. This kind of geometrical object is called a Lam\'{e} surface. The calculations were done either with implicit finite difference time stepping in the direction of negative imaginary time axis or with quantum diffusion Monte Carlo. The results demonstrate how the volume and depth of the potential well affect the $E_0$ more than the shape parameter $q$ does. Functions of two and three parameters were found to be sufficient for fitting an empirical graph to the ground state energy data points as a function of well depth $V_0$ or exponent $q$. The ground state and first excited state energy of one particle in a potential well of this type appeared to be very closely approximated with an exponential function depending on $q$, when the well depth and area or volume was kept constant while changing the value of $q$. The model is potentially useful for describing quantum dots that deviate from simple geometric shapes, or for demonstrating methods of computational quantum mechanics to undergraduate students.