A well-known result of quantum mechanics is that for a single particle in one dimension in a bounding potential $V(x)$ that goes to $+\infty$ as $x \to \pm \infty$, the energy eigenfunctions are discrete and the $n$th eigenfunction has exactly $n-1$ nodes at which $\psi(x) = 0$. (Moreover, we can say more - for example, between any two consecutive nodes in the $n$th eigenfunction, there exists a node in the $(n+1)$th eigenfunction.)
Do any similar results apply for single particles in higher than one dimension, or for multiparticle systems (for which the wave function is defined on configuration space rather than real space)? If not, is there an explicit example of a higher-dimensional or multiparticle system whose ground state wavefunction has a node?
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