Consider the hydrogen atom to be a proton embedded in acavity fo radius ` a_0` (Bohr 's radius ) , whos echarge ius neutralized by the addition of an elerctron to the cavity in vacuum , infinitiely slowly.
(a) Estimate the average of total energy of an elerctron in its ground state in a hydrogen atom as the work done in the above neutralization process , Also , If the halt the magnitude fo the averge potential enrgy find the average potential nergy .
(b) Also derve the wavelength of the elertron when it is ` a_0` from the proton . How does this compare with the wavelength of an elerctron in the ground statBohr's orbit ?

Explanation

(A) Work obtained in the neutralization process is given by
` W=- int_(prop) ^(a_0) F.da =- int_(prop) ^(a_0) ( 1^(-) e^2)/(4 pi varepsilon_0 .a_0^2) . Da_0`
`W=- e^2/(4 pi varepsilon_0 .a_0)` .
This work is to be called as potential energy , However in doing so , one shoul note the this energy is simply lot during the process fo attraction in between proton and electron . As reported in the problem at this condition , the electron simply possesses potential energy and its kinetic energy at this condition is zero.
Thus , ` TE. = P.E. + K.E.=P.E. = - e^2/( 2 pi varepsilon_0a_0) ` ..(1)
Now inorder , the elctron to be captured by the proton to form a group state hydrogen atom it should also attain kinetic energy ` e^2/( 8 pi varepsilon_0 a_0)` ( as it is half of the potential energy given in equestion ). Thus , the total energy of the electron if it attains the ground state in H-atom .
` T.E = P. E. + K.E . =- e^2/(4 pi varepsilon_0 a_0) + e^2/( 8 pi varepsilon_0a_0)= e^2/( 8 pi varepsilon_0 a_0)`
(a) The wavelength of electron when it is simply at a distance ` a_0` from proton can be given as :
` lambda = h/( mu) = h/p`
Also , ` k.e. = 1/2 mu^@ = p^2/(2n) ( :. p = mu)`
Thus , ` lambda = h/( sqrt ( 2m (k.E.0)`
Since , K.E. =0 at this stutation thus ` lambda = prop`
Aso when electron is at a distance ` a_0` in Bohr 's orbit of H-atom
` lambda = h/( sqrt ( 2 m (K.E))) = h/(sqrt ( (2me^2)/(2a_0 . 4 pivarepsilon_0)))`
` =h/(sqrt ( e^2 m)/( 4 pi varepsilon _0a_(0)))`.