Question

Consider the hydrogen atom to be a proton embedded in a cavity of radius a not whose charge is neutralized by the addition of an electron to the cavity in vacuum infinitely slowly estimate the average total energy

Answer 1

Work done while bringing an electron infinitely slowly from infinity to proton of radius a0 is given as followsW = - e2/4πε0.a0NOTE : This work done is equal to the total energy of an electron in its ground state in the hydrogen atom. At this stage, the electron is not moving and do not possess any K.E., so this total energy is equal to the potential energy.T.E. = P.E + K.E. = e2/4πε0.a0 ….(i)In order the electron to be captured by proton to form a ground state hydrogen atom it should also attainK.E. = e2/8πε0a0(it is given that magnitude of K.E. is half the magnitude of P.E. Note that P.E. is –ve and K.E is +ve)∴ T.E. = P.E + K.E. = - e2/4πε0a0 + e2/8πε0a0or T.E. = - e2/8πε0a0P.E. = 2 * T.E. = 2 * -e2/8π ∈0 a0 or P.E. = -e2/8π ∈0 a0