1000 Solved Problems in Modern Physics

122 2 Quantum Mechanics – I

μ=m 1 / 2 = 0. 5 mp

N 0 /N 1 =(1/3) exp

(

^2 /μr^2 kT

)

=(1/3) exp

(

^2 c^2 /μc^2 r^2 kT

)

=

1

3

exp

[ (

197. 3 × 10 −^15

) 2

0. 5 × 938 ×

(

1. 06 × 10 −^10

) 2

×

(

1. 38 × 10 −^23 / 1. 6 × 10 −^13

)

× 300

]

= 0. 445

2.62ω=

√

k

μ

μ=

mNm 0

mN+m 0

= 7 .466 amu

ω=

(

1550

7. 466 × 1. 67 × 10 −^27

) 1 / 2

= 3. 526 × 1014 s−^1

E=ω=

1. 055 × 10 −^34 × 3. 526 × 1014

1. 6 × 10 −^19

= 0. 2325

λ=

1 , 241

0. 2325

nm= 5 ,337 nm= 5. 34 μm

This wavelength corresponds to Infrared region.

2.63 (i)λ=

c

ν

=

3 × 108

1. 153 × 1011

= 2. 6 × 10 −^3 m= 2 .6mm

It lies in the microwave part of electromagnetic spectrum.

(ii) E(eV)=

1241

λ(nm)

=

1241

2. 6 × 106 (nm)

= 0 .000477 eV

(iii)μ=

mcm 0

mc+m 0

=

12 × 16

12 + 16

= 6. 857 u

(iv) E 1 =

1(1+1)^2

2 μr^2

=

^2

μr^2

E 0 = 0

ΔE=E 1 −E 0 =

^2

μr^2

r=

(

^2 c^2

μc^2 ΔE

) 1 / 2

=

(

197. 32 (MeV−fm)^2

6. 857 × 931. 5 × 477 × 10 −^12

) 1 / 2

= 0. 113 × 10 −^9 m

= 1. 13 A ̊

2.64 EJ=

J(J+1)^2

2 Io

Io=μr^2 =

(

Mp

2

)

r^2