Why do all three quantum models only consider positive quantum numbers?

[xmflea]

Why conceptually, does the ground state for the particle in the box model correspond to n=1 while for the harmonic oscillator and the particle on a ring model it is n=0? For all three models, why don't we consider negative quantum numbers?


attempt:...the particle in a box and particle on ring use the quantum number n in the equation to solve the time independent diff-eq. and n=1 and n=1 correspond to ground state levels...

 

[kuruman]

Take the one-dimensional harmonic oscillator. Conventionally, the "zero of energy" is the bottom of the parabolic potential, i.e. the potential is zero at x = 0. If the energy is negative, it is lower than the potential everywhere in space. If you use the Schrödinger equation to find the second derivative of the wavefunction, you will see that the second derivative is positive everywhere in space. Now try drawing a few wavefunctions for, say, the ground state that (a) have a positive second derivative everywhere and (b) are normalizable.

In general, it is impossible to have energies that are lower than the potential everywhere in space. Quantum mechanics allows particles to be in "classically forbidden" regions, but not if the entire space available to the particle is classically forbidden.