1000 Solved Problems in Modern Physics

(Romina) #1

2.1 Basic Concepts and Formulae 89


whereμis the reduced mass given by


μ=

memp
me+mp

(2.17)

andZ=1 for H-atom.
The radius of the smallest orbit, called the Bohr radius,


a 0 =r 1 =

ε 0 h^2
πme^2

= 0. 529 A ̊ (2.18)

Orbital velocity in thenth orbit


vn=

ze^2
2 εonh

(2.19)

Fine structure constant


α=

v
c

=

e^2
2 εohc

=

1

137

(2.20)

αis a measure of the electromagnetic interaction
Kinetic Energy of electron


Kn=

1

2

mvn^2 =

ze^2
8 πεorn

(2.21)

Potential energy of electron


Un=−

ze^2
4 πεorn

(2.22)

Total Energy of electron


En=Kn+Un=−

ze^2
8 πεorn

=−

mz^2 e^4
8 ε^2 oh^2 n^2

=

− 13. 6

n^2

(ev) (2.23)

ν ̃if=

1

λif

=R

(

1

n^2 f


1

n^2 i

)

(2.24)

Mesic atom


A negatively charged muon or pion when captured by the nucleus forms a bound
system called mesic atom before absorption or decay system.


i. rn ∝ 1 /μ; therefore radii are shrunk by a factor of∼200 for muonic atom
compared to H-atom.
ii. En ∝ μ; therefore energy levels are spaced 200 times greater than those of
H-atom. X-rays are emitted instead of visible light, when the muon cascades
down.

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