When a particle is restricted to move along \( x \)-axis between \( x=0 \) and \( x=a \), where ...
0When 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 \( E=p^{2} / 2 \mathrm{~m} \). 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 \( h=6.6 \times 10^{-34} \mathrm{~J} \) s and \( e=1.6 \times 10^{-19} \mathrm{C} \).
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) \( a^{2} \)
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