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Wavelet

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I have calculated various eigenstate wavefunctions for a one-dimensional system of a particle in a potential. The potential is an even function.

All the wavefunctions have become either even or odd functions which I understand why. The ground-state wavefunction has always been even, is this always the case and if so why? If not does anyone know of a system with odd ground-state wavefunction?

Thanks in advance,

Wavelet