Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 16
There are three p orbitals for each value of the principal
quantum number
- When Ɩ ≠0, the hydrogen atomic wave functions are
not spherically symmetric; they depend on 𝛉 and 𝛗.
- When Ɩ =1, m =0 or ±1 , there are three p orbitals for
each value of n.
- First when m =0
1
 3 2
Y 10 ( ,  )  
 cos 
 4 
𝛉
cos𝛉
0º
1
20º
0.94
40º
0.77
60º
0.50
(1)
- Because Y 10 ( ,  ) turns out to be independent of ∅.
- When Ɩ =1 states with m≠0 are Y 11 ( ,  ) ,and Y 11 ( ,  )
1
 3 2
Y 11 ( ,  )    sin  e  i 
 8 
(2)
1
Y 1
1
 3 2
( ,  )    sin  e i 
 8 
[1]
(3)
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 16
The probability densities associated with Y 11 ( ,  ) and Y 11 ( ,  )
are the same because
2
Y 11 ( ,  ) 
3
sin 2 
8
(4)
and
2
Y 11 ( ,  ) 
3
sin 2 
8
(5)
1
 3 2
 px  
 sin  cos 
 4 
(6)
1
 3 2
py  
 sin  sin 
 4 
The three functions p x ,
py
and
pz
(7)
are often used as the
angular part of hydrogen atom
- For the Ɩ = 2 case, m = 0, ± 1, and ± 2 and so there are
five d orbitals
1
d z 2 Y
0
2
 5 2
2

 (3cos   1)
 16 
1
d xz
 15  2
Y 21  
 sin  cos  cos 
 4 
[2]
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 16
1
d yz Y
1
2
 15  2

 sin  cos  sin 
 4 
1
d x 2  y 2 Y
2
2
 15  2
2

 sin  cos 2
 16 
1
d xy Y
2
2
 15  2 2

 sin  sin 2
 16 
[3]
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 16
[4]