Physics 451
Quantum mechanics I
Fall 2012
Nov 5, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• HW #17 Tuesday Nov 6 by 7pm
• HW #18 Thursday Nov 8 by 7pm
Quantum mechanics
The hydrogen atom
What is the density of probability of the electron?
Phys 451
The hydrogen atom
  r, ,   R  r  Y  , 
The angular equation (same)
1 1  
Y 
1  2Y 
 l (l  1)

 sin 
 2
2 
Y  sin   
  sin   
Angular function (same)
Yl m  ,   Aeim Pl m  cos 
l
Azimutal quantum number
m
Magnetic quantum number
l 0
m l
Phys 451
The hydrogen atom
  r, ,   R  r  Y  , 
2
d 2u 
l (l  1) 

  V (r ) 
 u  Eu
2
2
2m dr 
2m r 
2
The radial equation
u  rR (r )
Coulomb’s law:
e 2 1
V (r ) 
4 0 r
2
d 2u  e2
l (l  1) 



 u  Eu
2
2
2m dr  4 0 r 2m r 
2
Phys 451
The hydrogen atom
  kr
The radial equation
k
2mE
0 
me2
2 0 2 k
d 2u  0 l (l  1) 
 1  
u
2
2
d

 

Asymptotic behaviors
 
2
d u
u
2
d
u
Ae  
 0
d 2u l (l  1)

u
2
2
d

u
B  l 1
Phys 451
The hydrogen atom
Peeling off the asymptotic behaviors
l 1  
u   e v(  )
d 2v
dv
 2  2(l  1   )
  0  2(l  1)  v  0
d
d

Power expansion
Recursion formula:
v(  )   c j  j
j 0
2( j  l  1)  0
c j 1 
cj
( j  1)( j  2l  2)
2
cj
j 1
Phys 451
The hydrogen atom
u   l 1e  v(  )

v(  )   c j  j
j 0
The series must terminate
2( j  l  1)  0
0
( j  1)( j  2l  2)
2( jmax  l  1)  0
Principal quantum number
n  jmax  l  1 
0
2
Quantum mechanics
Quiz 23a
For a given quantum number n,
how many values of l can exist?
A. one
B. two
C. n
D. n -1
E. An infinity
Quantum mechanics
The hydrogen atom
Principal quantum number
0  2n
Quantization of the energy
Bohr 1913
 m  e2 2  1
 2
En    2 

 2  4 0   n


2
Ground state: “binding energy”
m  e2 
E1   2 
  13.6eV
2  4 0 
Quantum mechanics
The hydrogen atom
 me2  1
k 
2 
4

0

n
Bohr radius
k~
4 0 2
10
a

0.529

10
m
2
me
1
k
na
1
dis tan ce
Quantum mechanics
The hydrogen atom
Energies levels
Stationary states
1
kn 
na
E1
En  2
n
nlm r ,  ,    Rnl (r )Yl m ( ,  )
n: principal quantum number
l: azimuthal quantum number
l  n 1
m: magnetic quantum number
m l
n 1
Degeneracy of nth energy level:
 2l  1
l 0
Quantum mechanics
Quiz 23b
What is the degeneracy of the 5th energy band
of the hydrogen atom?
A. 5
B. 9
C. 11
D. 25
E. 50
Quantum mechanics
The hydrogen atom
Spectroscopy
Energies levels
E1
En  2
n
0
E
E4
Energy transition
Paschen
E3
E2
 1

hc
1
E 
 E1  2  2 
n


n
i
f


Balmer
 1
1 

R 2  2
n


 f ni 
1
Rydberg constant
E1
Lyman