# The ground state of arbitrary Potential Function

How can one say that the number of nodes in the ground state must be nodeless . And how one can ensure that, when one gets up in the energy spectrum, for consecutive States the difference of number of nodes must be one. Any qualitative or mathematical explain will do.

• If you're talking about a confined particle, the modes are standing waves. The quantization condition just requires that the wave function vanish at the boundaries. The lowest such mode is a half wavelength that looks somewhat like an inverted parabola, and has no nodes. The excited states are just harmonics. See en.wikipedia.org/wiki/Harmonic. Commented Sep 1, 2016 at 13:29
• @KyleArean-Raines I guess the OP wants to know, if it is mathematically ensured that every possible potential V and the according Hamiltonian for, in this case probably time independent bound case, has a solution with no nodes except for the boundary. Next step would be to show that going to the next solution with the next higher energy increases the nodes by 1. This is only that simple in 1D though. Commented Sep 1, 2016 at 14:15
• @mikuszefski ah yes. Missed the "arbitrary" part. Interesting question. Commented Sep 1, 2016 at 14:18
• I have thought about this myself. I suspect (but have not demonstrated) that if you play around with potential functions that have multiple regions of attraction (separated by repulsive regions) that you would find cases where the lowest energy state is not nodeless. Commented Sep 1, 2016 at 14:30
• One usually expects the ground state of a double-well potential to possess one node, so I don't think the conjecture about nodeless ground states is true. It's fairly easy to see why (for a double-well potential): the wave function should vanish in regions where the potential is large to minimise potential energy. Combining this with the constraint of minimising curvature (kinetic energy) leads to a ground state that is negative under parity and therefore possesses a node. Commented Sep 1, 2016 at 14:43