Question

an electron in a hydrogen atom in its ground state absorbs twice is ionization energy what is the wavelength of the emitted electron

Answer 1

The wavelength of the emitted electron would be the same as when it absorb the ionization energy the left out energy would be utilized in providing kinetic energy to the electron. Formula for Ionization Energy:
KE= Energy Supplied-IE=2IE-IE=IE

Where as:
Wavelength= Plank's Constant/ Momentum

Answer 2

Kinetic Energy(free electron)= Energy Supplied - IE = 2IE-IE = IE
 (The idea is similar to Einstein’s idea of photoelectric effect)

KE= momentum*momentum/2*mass of Electron

Momentum= Sqrt(2*KE*mass of an electron)

By De Broglie's Formula, any classical object has an associated classical wavelength by:

Wavelength= Planck’s constant/ Momentum

You know momentum from 3 and Planck’s constant is a fundamental constant, hence plug and chug and you have the wavelength of the associated free electron!