Question

An electron at rest is released far away from a proton, toward which it moves. (a) Show that the de Broglie wavelength of the electron is proportional to`sqrt (r)`, where r is the distance of the electron from the proton.?

Answer 1

An electron of charge `(-e)` and mass `m_e` is released far away from a proton and starts moving towards it. Let `r` be distance between the two.
Electric Potential energy of electron when it is located at a distance `=r` from proton is given by the expression

`PE(r)=(ke(-e))/r`
where `k` is Coulomb's Constant

Initial total energy of electron `=PE+KE=0+0=0`

Total energy of electron at distance `r` from proton `=PE(r)+KE(r)=-(ke^2)/r+KE(r)`

Using Law of Conservation of energy
`-(ke^2)/r+KE(r)=0`
`=>KE(r)=(ke^2)/r`

Writing kinetic energy of electron in terms of momentum `vecp` of electron

`|vecp|^2/(2m_e)=(ke^2)/r`
`=>|vecp|=sqrt((2ke^2 m_e)/r)` .....(1)

de Broglie wavelength `lambda` of electron is given as

`λ = h/|vecp|`
where `h` is Planck's constant

Using (1) we get

`λ = h/(sqrt((2ke^2 m_e)/r))`
`λ = h/(sqrt(2ke^2 m_e))sqrtr`
`λ = propsqrtr`