Example Exact Solutions of the Time-independent Gross-Pitaevskii and Schrödinger Equations

Abstract

A prescription is given to obtain some exact results for certain external potentials $V\left(\vec{r}\right)$ of the time-independent Gross-Pitaevskii and Schrödinger equations. The study motivation is the ability to program $V\left(\vec{r}\right)$ experimentally in Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given $V\left(\vec{r}\right)$, we ask which $V\left(\vec{r}\right)$ will have a given pdf (probability density function) $P\left(\vec{r}\right)$. Several examples in 1D and 2D are presented for well-known pdfs and for the hydrogen atom in momentum space.

References in corpus (7)

• Ultralight scalars as cosmological dark matter
• Dynamics of collapsing and exploding Bose-Einstein condensates
• Rigorous Derivation of the Gross-Pitaevskii Equation
• Precision measurements of s-wave scattering lengths in a two-component Bose-Einstein condensate
• Quantum Search with General Nonlinearities
• Logical inference approach to relativistic quantum mechanics: derivation of the Klein-Gordon equation
• Advances in atomtronics