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 = p²/2m. 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 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 × 10^–34 J-s and e = 1.6 × 10^–19 C.
1. 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
2. If the mass of the particle is m = 1.0 × 10^–30 kg and a = 6.6 nm, the energy of the particle
in its ground state is closest to -
(A) 0.8 meV (B) 8 meV (C) 80 meV (D) 800 meV
3. The speed of the particle, that can take discrete values, is proportional to -
(A) n^(–3/2) (B) n^(–1) (C) n^(1/2) (D) n
Source: JEE 2009 paper 1
4. A piece of wire is bent in the shape of a parabola y = kx² (y-axis vertical) with a bead of mass m on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with a constant acceleration a. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the y-axis is –
a. a/gk b. a/2gk c. a/4gk d. 2a/gk
5.Under the influence of the Coulomb field of charge +Q, a charge –q is moving around it in an elliptical
orbital. Find out the correct statement(s).
(A) The angular momentum of the charge – q is constant
(B) The linear momentum of the charge –q is constant
(C) The angular velocity of the charge –q is constant
(D) The linear speed of the charge –q is constant
Source: JEE 2009 paper 2