From Atoms to Lasers Lecture Notes (PHY2062)

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Lecture 1.1 The Rydberg Formula
Rydberg Constant
The Bohr model was formed from the work done by people looking at emission lines of
atoms

The formulas contained integers that related to the quantisation of the energy levels
within the atom

1 1 1
νˉ = = −R ( ′2 − 2 )
λ n n

νˉ = wavenumber n′ and n = integers

hcR
En = −
n2
n = principal quantum number




hcR hcR
hcνˉ = En′ − En = − + 2
n′2 n

1 1
Rydberg Formula νˉ = −R ( − )
n′2 n2


ER = hcR

me e4
R∞ = 2 3
8ϵ0 h c


Lecture 1.1 The Rydberg Formula 1

, Correspondence Principal
Becomes a continuum when the lines get
closer together




ω
ν=
2π
1 1
ν = −Rc ( ′2
− 2
) where n′ = n + 1
n n

1 1
ν = −Rc ( 2
− 2)
(n + 1) n

1
( − 1)
Rc
ν=−
n2 n12 (n + 1)2



Lecture 1.1 The Rydberg Formula 2

, 1
( − 1)
Rc
ν=−
n2 1 + n2 + 1
n2

Rc 1 − 1 − n2 − n12
ν=− 2 ( ) For n ≫ 1
n 1 + n2 + n12

Rc − n2 2Rc
ν=− 2 ( )= 3
n 1 n

Rutherford Model
1 1
ω 4πRc
E(ω) = − (e4 me ω 2 ) 3 ν= ⇒ω=
2(4πϵ0 )
2
3 2π n3

1
e4 me (4π)2 R2 c2 3
1
( )
Rhc
E(n) = − = −
2
2(4πϵ0 ) 3 n6 n2

1 e4 me (4π)2 R2 c2 R3 h3 c3
⋅ =
8(4πϵ0 )2 n6 n6

1 e4 me (4π)2 R2 c2 R3 h3 c
⋅ =
8(4π ϵ0 )2 n6 n6

e4 me
R= 2 3
8ϵ0 h c

R∞ = 1.097371 × 107 m−1 RH = 1.0967758 × 107 m−1

Good agreement but not quite the same




Lecture 1.1 The Rydberg Formula 3

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Lecture 1.2 Balmer Series
A well-known line spectra of H that is visible to the naked eye




1 1 1
νˉ = = −RH ( ′2 − )
λ n 4




Lecture 1.2 Balmer Series 1