Question

According to the Bohr-Sommerfeld postulate the periodic motion of a particle in a potential field must satisfy the following quantization rule:

where q and p are generalized coordinate and momentum of the particle , n are integers. Making use of this rule, find the permitted values of energy for a particle of mass m moving

(a) in a unidimensional rectangular potential well of width l with infinitely high walls;

(b) along a circle of radius r;

(c) in a unidimensional potential field U = αx²/2, where a is α positive constant;

(d) along a round orbit in a central field, where the potential energy of the particle is equal to U = —α/r (α is a positive constant).

Answer 1

(a) If we measure energy from the bottom of the well, then V(x) = 0 inside the walls. Then 

(d) It is required to find the energy levels of the circular orbait for the rotential U(r)=α/r 

In a circular orbit, the particle only has tangential velocity and the quantization condition