When a particle is restricted to move along -axis between and

PHYSICS · Single correct · CBSE · Physics · Atoms · Atoms

Problem

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, ... (

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

x = a

. Take

h = 6.6 \times 10^{-34} \text{ Js}

and

e = 1.6 \times 10^{-19} \text{ C}

. The allowed energy for the particle for a particular value of

n

is proportional to (1)

a^{-2}

(2)

a^{-3/2}

(3)

a^{-1}

(4)

a^2

Related theory

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