Modern Physics Ques 40
Passage
When a particle is restricted to move along $x$-axis between $x=0$ and $x=a$, where $a$ is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends $x=0$ and $x=a$. The wavelength of this standing wave is related to the linear momentum $p$ of the particle according to the de-Broglie relation. The energy of the particle of mass $m$ is related to its linear momentum as $E=\frac{p^{2}}{2 m}$. Thus, the energy of the particle can be denoted by a quantum number $n$ taking values $1,2,3, \ldots(n=1$, called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line $x=0$ to $x=a$. [Take $h=6.6 \times 10^{-34} Js$ and $e=1.6 \times 10^{-19} C$ ]
- The allowed energy for the particle for a particular value of $n$ is proportional to
(a) $a^{-2}$
(b) $a^{-3 / 2}$
(c) $a^{-1}$
(d) $a^{2}$
(2009)
Show Answer
Answer:
Correct Answer: 40.(a)
Solution:
Formula:
$ \begin{aligned} & \quad a=\frac{n \lambda}{2} \\ & \therefore \quad =\frac{2 a}{n}=\frac{h}{p}=\frac{h}{\sqrt{2 E m}} \quad …….(i)\\ & \text { or } \quad \sqrt{E} \propto \frac{1}{a} \Rightarrow \quad \therefore E \propto \frac{1}{a^{2}} \end{aligned} $
BETA
