A block of mass m is oscillating on a smooth level ground by a spring of force c | TestHub

Passage / Comprehension

Bohr and Sommerfeld postulated the periodic motion of a particle moving in a conservative force field. They stated that the momentum $p$ and displacement $s$ are related by the following quantization rule:

$\oint p \cdot d s = nh$

Where $\oint$ denotes the integral over a single periodic closed path. For example, for an electron moving in a circular path in a hydrogen atom:

$\oint p d s = nh$

Since $p$ is constant in a circular path, $m v \times 2 π r = nh$ , which is Bohr's quantization principle for the Bohr atom. Now, we apply this condition to other periodic motions.

A block of mass $m$ is oscillating on a smooth level ground by a spring of force constant k . In ground state, its amplitude is $A_{0}$ . It is excited to $2^{nd}$ excited state by a photon of energy equal to $α kA_{0}^{2}$ , where $α$ is a constant(assume complete absorption). Find $α$ ?

Answer:

Solution:

$mv (2 π r) = nh$ with $m v = B_{0} q r$

$\Rightarrow r \propto n^{1/2} \Rightarrow \frac{r _{5}}{r _{3}} = \frac{5}{3} \approx 1.291$

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