1000 Solved Problems in Modern Physics

138 3 Quantum Mechanics – II

3.8 (a) State and explain Laporte rule for light emission.

(b) What are metastable states?

3.9 Show that the eigen values of a hermitian operatorQare real

3.10 The state of a free particle is described by the following wave function
(Fig. 3.1)
ψ(x)=0forx<− 3 a
=cfor− 3 a<x<a
=0forx>a

(a) Determinecusing the normalization condition

(b) Find the probability of finding the particle in the interval [0,a]

Fig. 3.1Uniform distribution

ofψ

3.11 In Problem 3.10,
(a) Computeandσ^2
(b) Calculate the momentum probability density.

3.12 Particle is described by the wavefunction

ψ= 0 x< 0

=

√

2 e−x/Lx≥ 0

whereL=1 nm. Calculate the probability of finding the particle in the region

x≥1nm.

3.2.2 Schrodinger Equation. .........................

3.13 The radial Schrodinger equation, in atomic units, for an electron in a hydrogen

atom for which the orbital angular momentum quantum number,l=0, is

((d^2 /dr^2 )+(2/r)+(2E))F(r)= 0 ,

whereEis the total energy.