What features of the electronic energies of the hydrogen atom explains why the emission spectrum consists of?

Question

discrete wavelength rather than a continuum wavelength.

Answer 1

When an electron in an excited state falls down to a lower energy level and emits light, the light is of a specific wavelength because the energy drop is a specific amount (e.g. from n=3 to n=1). It's like going down a staircase: you can only go down in specific amounts (i.e. steps); you can't be on a "half-step."