Re: why the observed line spectrum provides evidence for discrete levels

Peter,

The observed line spectrum of hydrogen gas can be precisely predicted 
through quantum mechanical methods, and thus provides empirical support 
for the theory.  It is also a very satisfying explanation for an easily 
observable phenomena.

The electrons in hydrogen gas can exist only in certain discrete states, 
because of their wave nature.  These states have different energy levels.  
As an electric potential is applied to the gas, the gas can become excited, 
and the electrons get kicked up to higher, unstable, energy states.  When 
the electrons fall to a lower energy state, the energy is carried off by a 
single photon.  This photon has a wavelength determined by the energy 
difference between the two states of the electron.  Since there are 
discrete energy states, there are specific frequencies of light emitted.

You can check out any sufficently advanced freshman physics text for more 
information about quantum mechanics.