Instead of angular momentum quantization a student posits that energy is quantized as ` E=- E_0 //n (E_0 gt 0)` and n is a positive interger . Which of the following options is correct ?

Options

A. The radius of the electron orbit is` r prop sqrt(n)`
B. The speed of the electron is ` v prop sqrt(n)`
C. The angular speed of the electron is ` omega prop 1//n`
D. The angular momentum of the electron is ` prop sqrt( n)`

Answer

D

Explanation

` F_e = (mv^2 )/( r) implies ( mv^2)/( r) = (k (Ze)(e ))/( r^2)`
` 1/2 mv^2 = (Kze^2 )/( 2r) ….(i)` ( kinetic energy)
Potential energy ` =( Kq_1 q_2 )/(r ) = (K (Ze)(-e))/( r) ….(ii)`
Total energy `= KE +PE = (-Kze^2)/(2r)=-(E_0)/( n)`
` therefore r prop n`
as kinetic energy ` = (KZe^2)/(2r) implies KE prop (1)/(n)`
or ` v^2 prop (1)/(n) implies v^2 prop (1)/( sqrt(n))`
`L = mv r`
` L prop vr `
` L porp ( 1)/( sqrt(n)) (n) prop sqrt(n)`
` implies L prop sqrt(n)`