- The document discusses Bohr's model of the hydrogen atom, which proposed that electrons orbit the nucleus in fixed, quantized energy levels. - Bohr postulated that the angular momentum of the electron is quantized and that the electron's energy is proportional to 1/n^2, where n is a positive integer called the principal quantum number. - Bohr's model was able to explain the observed spectral lines of hydrogen and provided formulas to calculate the wavelength and frequency of photons emitted during transitions between energy levels.

1. Atomic Physics

2. Atomic Spectra of Gases

3. Balmer Series for Hydrogen
H 2 2
1 1 1
2
3,4,5,...
R
n
n

 
 
 
 

7 1
H 1.0973732 10 m
R 
 

4. Lyman Series
H 2 2
H 2 2
H 2 2
1 1 1
2,3,4,... Lyman series
1
1 1 1
4,5,6,... Paschen series
3
1 1 1
5,6,7,... Brackett series
4
R n
n
R n
n
R n
n



 
  
 
 
 
  
 
 
 
  
 
 

5. Early Models of the Atom

6. Early Models of the Atom

7. Rutherford’s Experiment

8. Rutherford’s Model of the Atom

9. Rutherford’s Model of the Atom
ER
E
K U T
   

10. Bohr’s Model of the Hydrogen Atom
Niels Bohr

11. Assumptions of Bohr’s Theory
increase in orbital energy energy of absorbed photon
i f
E E hf

 
angular momentum is quantized:
1,2,3,...
e
m vr n n
 

12. Bohr’s Model: Total Energy of the Atom
2
2
1
2
E e e
e
E K U m v k
r
   
2 2 2
2
2
e e e
e
k e m v k e
v
r m r
r
  
2
2
1
2 2
  e
e
k e
K m v
r
2
2
e
k e
E
r
 
2
1 2
e e
k q q k e
U
r r
  

13. Radii of Allowed Orbits
2
2 2 2 2
2
2 2 2
1,2,3,...
e
n
e
e e e
k e
n n
v r n
m r
m r m k e
    
2
2
1,2,3,... and e
e
e
k e
m vr n n v
m r
  

14. Bohr Radius and Energy Levels
2
0 2
0.0529 nm
 
e e
a
m k e
 
2 2
0 0.0529 nm
1,2,3,...
n
r n a n
n
 

2
2
0
1
1,2,3,...
2
 
  
 
 
e
n
k e
E n
a n
2
13.606 eV
1,2,3,...
  
n
E n
n

15. Energy-Level Diagram
2
2
2
1,2,3,...
8
n
h
E n n
mL
 
 
 
 
2
13.606 eV
1,2,3,...
  
n
E n
n

16. Frequency and Wavelength of Photon
2
2 2
0
1 1
2
 

  
 
 
 
i f e
f i
E E k e
f
h a h n n
2
2 2
0
1 1 1
2

 
  
 
 
 
e
f i
k e
f
c a hc n n
2 2
1 1 1

 
 
 
 
 
H
f i
R
n n
2
2
0
1
and
2
e
i f n
k e
E E hf E
a n
 
     
 

17. The Bohr Model and Hydrogen-Like Atoms
 
2 2
2 0
2
0
and 1,2,3,...
2
e
n n
a k e Z
r n E n
Z a n
 
   
 
 

18. Bohr’s Correspondence Principle
Quantum physics agrees
with classical physics when
the difference between
quantized levels becomes
vanishingly small.
Niels Bohr

19. A hydrogen atom is in its ground state. Incident on the atom is a
photon having an energy of 10.5 eV. What is the result?
(a) The atom is excited to a higher allowed state.
(b) The atom is ionized.
(c) The photon passes by the atom without interaction.
Quick Quiz 41.1

20. A hydrogen atom is in its ground state. Incident on the atom is a
photon having an energy of 10.5 eV. What is the result?
(a) The atom is excited to a higher allowed state.
(b) The atom is ionized.
(c) The photon passes by the atom without interaction.
Quick Quiz 41.1

21. A hydrogen atom makes a transition from the n = 3 level to the n = 2
level. It then makes a transition from the n = 2 level to the n = 1
level. Which transition results in emission of the longer-wavelength
photon?
(a) the first transition
(b) the second transition
(c) neither transition because the wavelengths are the same for
both
Quick Quiz 41.2

22. A hydrogen atom makes a transition from the n = 3 level to the n = 2
level. It then makes a transition from the n = 2 level to the n = 1
level. Which transition results in emission of the longer-wavelength
photon?
(a) the first transition
(b) the second transition
(c) neither transition because the wavelengths are the same for
both
Quick Quiz 41.2

23. Example 41.1: A Wave Function for a Particle
(A) The electron in a hydrogen atom makes a transition from a higher energy
level to the ground level (n = 1). Find the wavelength and frequency of the
emitted photon if the higher level is n = 2.
 
H
H 2 2
7 1
H
7
3
1 1 1
4
1 2
4 4
3 3 1.097 10 m
1.22 10 m 122 nm

 

 
  
 
 
 

  
R
R
R

24. Example 41.1: A Wave Function for a Particle
8
15
7
3.00 10 m/s
2.47 10 Hz
1.22 10 m
 

   

c
f

25. (B) Suppose the atom is initially in the higher level corresponding to n = 5.
What is the wavelength of the photon emitted when the atom drops from n = 5
to n = 1?
H H H
2 2 2 2
1 1 1 1 1
0.96
1 5
f i
R R R
n n

   
    
   
   
 
  
8
7 1
H
1 1
9.5 10 m 95.0 nm
0.96 0.96 1.097 10 m
R
 

    

Example 41.1: A Wave Function for a Particle

26. Example 41.1: A Wave Function for a Particle
(C) What is the radius of the electron orbit for a hydrogen atom for which n = 5?
   
2
5 5 0.0529 nm 1.32 nm
r  

27. Example 41.1: A Wave Function for a Particle
(D) How fast is the electron moving in a hydrogen atom for which n = 5?
  
  
2
9 2 2 19
2
31 9
5
8.99 10 N m /C 1.602 10 C
9.11 10 kg 1.32 10 m
4.38 10 m/s
e
e
k e
v
m r

 
  
 
 
 

28. What if radiation from the hydrogen atom in part (B) is treated classically? What
is the wavelength of radiation emitted by the atom in the n = 5 level?
1
2
 
v
f
T r
 
5
13
9
4.38 10 m/s
5.27 10 Hz
2 2 1.32 10 m
v
f
r
  

   

3
6
13
3.00 10 m/s
5.70 10 m
5.27 10 Hz
c
f
 

   

Example 41.1: A Wave Function for a Particle

29. The Quantum Model of the Hydrogen Atom
 
2
E e
e
U r k
r
 
2 2 2 2
2 2 2
2
U E
m x y z
  
 
 
  
    
 
  
 
       
,
    
 
r R r f g

30. The Quantum Model of the Hydrogen Atom
values of n: integers from 1 to 
Once n set  values of : integers from 0 to n  1
Once  set  values of m: integers from  to 
2
2 2
0
1 13.606 eV
1,2,3,...
2
 
    
 
 
e
n
k e
E n
a n n

31. Shells and Subshells

32. Quantum Numbers for Hydrogen

33. How many possible subshells are there for the n = 4
level of hydrogen?
(a) 5
(b) 4
(c) 3
(d) 2
(e) 1
Quick Quiz 41.3

34. How many possible subshells are there for the n = 4
level of hydrogen?
(a) 5
(b) 4
(c) 3
(d) 2
(e) 1
Quick Quiz 41.3

35. When the principal quantum number is n = 5, how many
different values of  are possible?
Quick Quiz 41.4 Part I

36. When the principal quantum number is n = 5, how many
different values of  are possible?
five
Quick Quiz 41.4 Part I

37. When the principal quantum number is n = 5, how
many different values of m are possible?
Quick Quiz 41.4 Part II

38. When the principal quantum number is n = 5, how
many different values of m are possible?
nine
Quick Quiz 41.4 Part II

39. Example 41.2: The n = 2 Level of Hydrogen
For a hydrogen atom, determine the allowed states corresponding to the
principal quantum number n = 2 and calculate the energies of these states.
0 0
1 1,0, or 1
  
   
m
m
2 2
13.606 eV
3.401 eV
2
   
E

40. The Wave Functions for Hydrogen
  0
/
1
3
0
1



 r a
s r e
a
0
2 2 /
1 3
0
1



 
  
 
r a
s e
a
 
2 2 2
4
  
 
P r dr dV r dr
 
2
2
4 

P r r
  0
2
2 /
1 3
0
4 
 
  
 
r a
s
r
P r e
a
2
dV


41. The Wave Functions for Hydrogen

42. Example 41.3: The Ground State of Hydrogen
(A) Calculate the most probable value of r for an electron in the ground state of
the hydrogen atom.
   
 
 
0
0 0
0 0
0
2
2 /
1
3
0
2 / 2 /
2 2
2 / 2 /
2
0
2 /
0
4
0
0
2 2/ 0
2 1 / 0

 
 

 
 
 
 
 
 
 
 
 
  
 
 
 
r a
s
r a r a
r a r a
r a
dP d r
e
dr dr a
d d
e r r e
dr dr
re r a e
r r a e
0
0
1 0
   
r
r a
a

43. Example 41.3: The Ground State of Hydrogen
(B) Calculate the probability that the electron in the ground state of hydrogen
will be found outside the Bohr radius.
  0
0 0
2 /
2
1 3
0
4
 

 
 
r a
s
a a
P P r dr r e dr
a
2
2
0 0
3 2 2
0
4 1
2 2 2
 
 
   
 
   
   
 
z z
za a
P e dz z e dz
a
 
2
2
1
2 2
2


    z
P z z e
  2 2
1
0 4 4 2 5 0.677 or 67.7%
2
P eh e
 
 
      
 
 

44. Example 41.3: The Ground State of Hydrogen

45. What if you were asked for the average value of r for the electron in the ground
state rather than the most probable value?
  0
0
2
2 /
avg 3
0 0
0
2 /
3
3 0
0
4
4
 



 
    
 
 
  
 
 

r a
r a
r
r r rP r dr r e dr
a
r e dr
a
 
avg 0
3 4
0 0
4 3! 3
2
2/
 
 
 
 
  
  
r a
a a
Example 41.3: The Ground State of Hydrogen

46. The Wave Functions for Hydrogen
  0
3/2
/2
2
0 0
1 1
2
4 2



   
 
   
   
r a
s
r
r e
a a
2
13.606 eV
3.401 eV
4
E

  
 
0
3/2
/2
2
0 0
1 1
sin
8
r a i
p
r
e e
a a

 

 
   
    
   

47. The Orbital Quantum Number 
 
1
0, 1, 2, ..., 1
L
n
 
 
e
L m vr

1,2,3,...
e
m vr n n
 

48. The Orbital Magnetic Quantum Number m

z
L m

49. Vector Model and Space Quantization
 
cos
1
  

z
L m
L

z
L m

50. Zeeman Effect

51. Example 41.4: Space Quantization for Hydrogen
Consider the hydrogen atom in the  = 3 state. Calculate the magnitude
of , the allowed values of Lz, and the corresponding angles  that
makes with the z axis.
   
1 3 3 1 2 3
    
3 , 2 , ,0, ,2 ,3
   
z
L
3 2
cos 0.866 cos 0.577
2 3 2 3
1 0
cos 0.289 cos 0
2 3 2 3
 
 
 
     

    
,54.7 ,90.0 ,107 ,125 ,150
       
L L

52. Example 41.4: Space Quantization for Hydrogen
What if the value of  is an arbitrary integer? For an arbitrary value of , how
many values of m are allowed?

53. The Spin Magnetic Quantum Number ms

54. The Spin Magnetic Quantum Number ms

55. The Spin Magnetic Quantum Number ms
 
3
1
2
  
S s s
1
2
  
z s
S m
spin  
e
e
m
S
μ
spin,
2
 
z
e
e
m
μ
24
B 9.27 10 J/T
 
 

56. Quantum States for n = 2

57. The Exclusion Principle and the Periodic Table
No two electrons can ever be in the same quantum
state; therefore, no two electrons in the same atom
can have the same set of quantum numbers.

58. The Exclusion Principle and the Periodic Table
When an atom has orbitals of equal energy, the order in which they are filled by
electrons is such that a maximum number of electrons have unpaired spins.

59. The Exclusion Principle and the Periodic Table

60. The Exclusion Principle and the Periodic Table

61. More on Atomic Spectra: Visible and X-Ray
1
0, 1
m
  
  

62. Bohr Theory: First Approximation in Quantum Theory
  2
2 2
2 2
0
13.6 eV
2
 
   
 
 
e
n
Z
k e Z
E
a n n
  2
eff
2
13.6 eV
 
n
Z
E
n

63. X-Ray Spectra

64. X-Ray Spectra

65. Characteristic X-Rays

66. X-Ray Spectra
 
2
L 2
13.6 eV
1
2
  
E Z
 
2
K 13.6 eV
 
E Z

67. X-Ray Spectra
 
2
K 13.6 eV
 
E Z

68. In an x-ray tube, as you increase the energy of the electrons striking
the metal target, the wavelengths of the characteristic x-rays
(a) increase.
(b) decrease.
(c) remain constant.
Quick Quiz 41.5

69. In an x-ray tube, as you increase the energy of the electrons striking
the metal target, the wavelengths of the characteristic x-rays
(a) increase.
(b) decrease.
(c) remain constant.
Quick Quiz 41.5

70. True or False: It is possible for an x-ray spectrum to show the
continuous spectrum of x-rays without the presence of the
characteristic x-rays.
Quick Quiz 41.6

71. True or False: It is possible for an x-ray spectrum to show the
continuous spectrum of x-rays without the presence of the
characteristic x-rays.
Quick Quiz 41.6

72. Example 41.5: Estimating the Energy of an X-Ray
Estimate the energy of the characteristic x-ray emitted from a tungsten target
when an electron drops from an M shell (n = 3 state) to a vacancy in the K shell
(n = 1 state). The atomic number for tungsten is Z = 74.
   
2 4
K 74 13.6 eV 7.4 10 eV
    
E
  
 
2
3
M 2
13.6 eV 74 9
6.4 10 eV
3

    
E
 
3 4
M K
4
6.4 10 eV 7.4 10 eV
6.8 10 eV 68 keV
       
  
hf E E

73. Spontaneous and Stimulated Transitions

74. Spontaneous Emission

75. Stimulated Emission

76. Lasers
Properties:
• coherent
• monochromatic
• small angle of
divergence

77. Lasers

78. Conditions for Stimulated Emission

79. Lasers for Stimulated Emission

80. Lasers: Applications