# Curvature and quantization

1. Sep 1, 2009

### noblegas

1. The problem statement, all variables and given/known data

In a streamlined model for the low energy states of an ammonia atom, (NH3), imagine that a nitrogen atom moves in one dimension in the potential V(x) sketched in figure I.1(found in Peebles textbook on p.86); The potential has two minima, one on each side of the triangle defined by three hydrogen atoms, and a relatively high peak between the minima, at the plane of the three hydrogen atoms. Thus in classical physics the nitrogen atom in the ground state would sit in one of the minima. Work is required to pull the nitrogen atom away from the molecule , and in classical physics work is required to push the nitrogen atom into the plane of the three hydrogen atoms at x=0. Sketched shapes of the waves functions $$\varphi$$0 and $$\varphi$$1 for the ground and first excited states of motion of the nitrogen atom.

The Energies of E_0 and E_1 of the ground state and first excited states in this system are very nearly equal. Explain how this is to be understood.

2. Relevant equations

-h-bar^2/2m*d^2/dx^2*$$\varphi$$+V(x)*$$\varphi$$=E*$$\varphi$$

C(x)=1/($$\varphi$$)*d^2/dx^2*$$\varphi$$(V(x)-E)
3. The attempt at a solution

In order to sketch the graph properly, I think I would need to know the values of phi and C.i think particle resides at the bottom of the potential well, so phi>0 and if V>E, then C is greater than zero; how would i determine C and phi for the first excited state;