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Fundamentals of
Quantum Mechanics

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© The scientific sentence. 2010

Spectral line emissions series



0. Rydberg formula



Eni = E = - 13.6 (Z2/ni2) eV
Enf = E = - 13.6 (Z2/nf2) eV

ΔE = Enf - Eni = 
- 13.6 (Z2/nf2) + 13.6 (Z2/ni2) = 
13.6 Z2 (1/ni2  - 1/nf2) 

ΔE = h ν = hc/λ
Therefore:
hc/λ = 13.6 Z2 (1/ni2  - 1/nf2)
Or:

1/λ = (13.6/hc)Z2 (1/ni2  - 1/nf2)

h = 6.62606957(29)×10-34
c = 3.0  x 108 m/s
13.6 eV = 13.6 . 1.6 x 10-19 J
(13.6/hc) = 13.6 . 1.6 x 10-19/6.63 x 10-343.0  x 108 = 
1.10 x 10 +34 - 8 - 19 = 1.10 x 107 (1/m)

1/λ = 1.10 x 107 Z2 (1/ni2  - 1/nf2 ) (1/m)

Using the Rydberg constant: 
Ry = 1 Rydberg = 1.0974 x 107 m-1, we have:

1/λ = Ry Z2 (1/ni2  - 1/nf2 ) 


Therefore:
λ = 0.91 x 10- 7/Z2[1/(1/ni2  - 1/nf2 )] (m)

λ = 91.1 /Z2(1/ni2  - 1/nf2 )] (nm)


For any layer n, its energy En is less than 
the one of its next En+1. Then if the transition 
goes from ni = n+1 toward nf = n , the enrgiy difference ΔE is 
GIVEN, and still NEGATIVE. Thus, in order 
to have a positive wavelength, we just solve 
the following formula by taking the positive value (or 
absolute value)for &lamda;, or invert the order of nf 
and ni, so:


λ = (91.1/Z2) /(1/nf2  - 1/ni2 )] (nm)


This formula is valid for a hydrogen atom 
and hydrogen-like ion(any atomic nucleus with 
one electron). For atom with many electros, we 
use instead Moseley's law

.

1. Lyman series: → nf = 1

 
Lyman series are the series of transitions for an atom 
when one of its electrons goes from a layer n to the layer 
n = 1 . The number n is the principal quantum number 
(corresponding to a certain level of enrgy of an electron 
in the atom.

The transitions:
from n = 2 to n = 1 is called Lyman-alpha, 
from n = 3 to n = 1 is Lyman-beta, 
4 to 1 is Lyman-gamma, etc. 

All these transitions are in the range of 
the ultraviolet.

We have the Rydberg formula:
1/λ (n) = Ry(1/12 - 1/n2)
Ry = 1 Rydberg = 1.0974 x 107 m-1
or:

λ = (1/Ry) . 1/(1 - 1/n2) 

= 91.1 /(1 - 1/n2) nm

Lyman-alpha:
λ (2) = 91.1/(1 - 1/22) = 121 nm

Lyman-beta:
λ = 91.1/(1 - 1/32) =  102 nm

Lyman-gamma:
λ = 91.1/(1 - 1/42) = 97 nm

2. Balmer series: → nf = 2

 
Balmer series are the series of transitions for an atom 
when one of its electrons goes from a layer n to the layer 
n = 2.

The transitions:
from n = 3 to n = 2 is called Balmer-alpha, 
from n = 4 to n = 2 is Balmer-beta, 
5 to 2 is Balmer-gamma,
6 to 2 is Balmer-delta, etc. 

All these transitions are in the range of 
the visible. For n>7 →2 the 
spectrum is the ultraviolet.

We have the Rydberg formula:

λ = (1/Ry) . 1/(1/22 - 1/ni2) 

= 91.1 /(1/22 - 1/n2) nm

Balmer-alpha:
λ (2) = 91.1/(1/4 - 1/32) = 656 nm (red)

Balmer-beta:
λ = 91.1/(1/4 - 1/42) =  486 nm (cyan)

Balmer-gamma:
λ = 91.1/(1/4 - 1/52) = 434 nm (blue)

Balmer-delta:
λ = 91.1/(1/4 - 1/62) = 410 nm (violet)

Recall:
Color 		Wavelength
violet 	 	380–450 nm
blue 	 	450–475 nm
cyan 		476–495 nm
green 		495–570 nm
yellow 	 	570–590 nm
orange 	 	590–620 nm
red 	 	620–750 nm

3. Paschen series: → nf = 3

 
Paschen series are the series of transitions for an atom 
when one of its electrons goes from a layer n to the layer 
n = 3.

The transitions:
from n = 4 to n = 3 is called Paschen-alpha, 
from n = 5 to n = 3 is Paschen-beta, 
6 to 3 is Paschen-gamma, etc. 

All these transitions are in the band of 
the infrared. 

We have the Rydberg formula:

λ = (1/Ry) . 1/(1/32 - 1/ni2) 

= 91.1 /(1/32 - 1/n2) nm

Paschen-alpha:
λ (3) = 91.1/(1/9 - 1/42) = 1874 nm 

Paschen-beta:
λ = 91.1/(1/9 - 1/52) =  1281 nm 

Paschen-gamma:
λ = 91.1/(1/9 - 1/62) = 1093 nm 


4. Brackett series:→ nf = 4

 
Brackett series are the series of transitions for an atom 
when one of its electrons goes from a layer n to the layer 
n = 4.

The transitions:
from n = 5 to n = 4 is called Brackett-alpha, 
from n = 6 to n = 4 is Brackett-beta, etc. 

All these transitions are in the band of 
 

We have the Rydberg formula:

λ = (1/Ry) . 1/(1/42 - 1/ni2) 

= 91.1 /(1/42 - 1/n2) nm

Paschen-alpha:
λ (4) = 91.1/(1/16 - 1/52) = 4050 nm 

Paschen-beta:
λ = 91.1/(1/16 - 1/62) =  2624 nm 

  


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