Consider an electron, a proton, and an alpha particle (a helium nucleus), each trapped separately in identical boxes. Which particle has the longest wavelength when the system is in the ground state?

electron

Explanation:

The energy for particle in a box is given by:

for the system in ground state, n =1

Since, the boxes are identical, L is same for all the three particles.

h is Planck's constant.

Since energy is inversely proportional to wavelength, therefore

Thus, electron has the longest wavelength.

diagram is missing, but i will try to explain your doubt.

answer:

amplitude is the maximum displacement or distance moved on a point of a vibrating body or wave measured from its equilibrium position.it is equal to half the length of vibration path.