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).