Problem 6.45

According to the Bohr-Sommerfeld postulate the periodic motion of a particle in a potential field must satisfy the following quantization rule: \[ \oint p d q=2 \pi \hbar n \] 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 \(U=\alpha x^{2} / 2,\) where \(\alpha\) is a positive constant; (d) along a round orbit in a central field, where the potential energy of the particle is equal to \(U=-\alpha / r(\alpha\) is a positive constant).