Physics 451
Quantum mechanics I
Fall 2012
Nov 7, 2012
Karine Chesnel
Quantum mechanics
Homework this week:
• HW #18 Friday Nov 9 by 7pm
Pb 4.10, 4.11, 4.12, 4.13
Quantum mechanics
The hydrogen atom
What is the density of probability of the electron?
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
E1
En  2
n
1
kn 
na
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  n
l 0
2
Quantum mechanics
Quiz 24a
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 13.6eV
En  2 
n
n2
0
E
E4
Energy transition
Paschen
 1
1 

E 
 E1 2  2
n


 i nf 
E3
E2
hc
Balmer
 1

1
 R 2  2 
n


n
f
i


1
Pb 4.16
Pb 4.17
Rydberg constant
R  1.097 107 m1
E1
Lyman
Quantum mechanics
Quiz 24b
What is the wavelength of the electromagnetic radiation
emitted by electrons transiting from the 7th to the 5th band
in the hydrogen atom?
A. 465 nm
B. 87.5 x 10-8 m
C. 4.65 mm
D. 87.5 x10-7 m
E. 4.65 x 10-8 m
R  1.097 107 m1
Quantum mechanics
The hydrogen atom
  r, ,   R  r  Y  , 
Coulomb’s law:
Solution to the
radial equation
e 2 1
V (r ) 
4 0 r
1
l 1
R(r )  kr  e  kr v(kr)
r
with
k
 2mE


v(  )   c j  j
j 0
2( j  l  1  n)
c j 1 
cj
 j  1 j  2l  2
Pb 4.10
4.11
Quantum mechanics
The hydrogen atom

v(  )   c j  j
j 0
2( j  l  1  n)
c j 1 
cj
 j  1 j  2l  2
Equivalent to associated Laguerre polynomials
p
v(  )  L2nll11 (2  )
p d 
p
Lq  p ( x)   1   Lq ( x)
 dx 
q

x d 
Lq ( x)  e   e  x x q
 dx 

Pb 4.12
Quantum mechanics
The hydrogen atom
 nlm  r, ,   Rnl  r  Yml  , 
Radial wave functions
Spherical harmonics
(table 4.7)
(table 4.3)
Laguerre polynomials
Legendre polynomials
OR

Power series expansion
with recursion formula
v(  )   c j  j
j 0
c j 1 
2( j  l  1  n)
cj
 j  1 j  2l  2
Quantum mechanics
French mathematicians
• Edmond Laguerre
1834 – 1886
• Adrien-Marie Legendre
1752 – 1833
Quantum mechanics
The hydrogen atom
How to find the stationary states?
nlm r ,  ,    Rnl (r )Yl m ( ,  )
Step1: determine the principal quantum number n
1
kn 
na
Step 2: set the azimuthal quantum number l (0, 1, …n-1)
Step 3: Calculate the coefficients cj in terms of c0
(from the recursion formula, at a given l and n)
Step 4: Build the radial function Rnl(r) and normalize it (value of c0)
Step 5: Multiply by the spherical harmonics Yl ( ,  ) (tables)
and obtain 2l +1 functions nlm for given (n,l)
m
(Step 6): Eventually, include the time factor:

 (r , t )  nlm (r ,  ,  )e  iEnt / 
Quantum mechanics
The hydrogen atom
Representation of
nlm r , ,  
Quantum mechanics
The hydrogen atom
Representation of
nlm r , ,  
Bohr radius
4 0 2
10
a

0.529

10
m
2
me
Quantum mechanics
The hydrogen atom
 nlm  r, ,   Rnl  r  Yml  , 
Expectation values
r   r R r 2dr
2
r
2
  r R r dr
2
2
2
Pb 4.13
x   d  d sin   r cos  sin  R r 2dr
2
Most probable values
Pb 4.14
 r 
2
d  r2
2
2
max
dr
0
Quantum mechanics
The hydrogen atom
 nlm  r, ,   Rnl  r  Yml  , 
Expectation values for potential
 e 2  2 2
V  
 R r dr
 4 0 r 
Pb 4.15