JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Modern Physics

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 $E=\frac{p^2}{2m}$. Thus, the energy of the particle can be denoted by a quantum number ' $n$ ' taking values $1,2,3,\dots (n=1$, 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 $h=6.6\times 10^{-34}\mathrm{J}\mathrm{s}$ and $e=1.6\times 10^{-19}\mathrm{C}$.

If the mass of the particle is $m=1.0\times 10^{-30}\mathrm{kg}$ and $a=6.6\mathrm{nm}$, the energy of the particle in its ground state is closest to