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.?
1 Answer
An electron of charge
Electric Potential energy of electron when it is located at a distance
#PE(r)=(ke(-e))/r#
where#k# is Coulomb's Constant
Initial total energy of electron
Total energy of electron at distance
Using Law of Conservation of energy
Writing kinetic energy of electron in terms of momentum
#|vecp|^2/(2m_e)=(ke^2)/r#
#=>|vecp|=sqrt((2ke^2 m_e)/r)# .....(1)
de Broglie wavelength
#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#