Physics 451
Quantum mechanics I
Fall 2012
Oct 31, 2012
Karine Chesnel
Phys 451
Announcements
Homework this week:
• HW #16 Friday Nov 2 by 7pm
Phys 451
Position- momentum
in 3 dimensions
Pb 4.1
 xi , p j   i  ij
 xi , x j    pi , p j   0
 i p 
i
2
d
1
r 
p
dt
m
Phys 451
Schrödinger equation
in 3 dimensions
z
y
x
Each stationary
state verifies

2
2m
 2 n  V (r ) n  En n
Phys 451
Schroedinger equation
in cartesian coordinates
  2 n  2 n  2 n 

 2   V ( x, y, z ) n  En n
 2 
2
2m  x
y
z 
2
Pb 4.2
Infinite cubical well:
V=0 inside a box
Separation of variables
 n ( x, y, z)  f n1 ( x) gn 2 ( y)hn3 ( z)
Phys 451
Schrödinger equation in
spherical coordinates
z

r

y
  r, ,   R  r  Y  , 
x
The radial equation
1 d  2 dR  2mr 2
r
  2 V (r )  E   k
R dr  dr 
The angular equation
1 1  
Y 
1  2Y 
 k

 sin 
 2
2 
Y  sin   
  sin   
Phys 451
The angular equation
z
r

  r, ,   R  r  Y  , 
y

Further separation of variables:
  r, ,   R  r     ( )
x
2
1
 
 
1


2
sin 
sin


k
sin



 cst


2

 
 
 
 equation:
1 d 2
2

cst


m
 d 2
 equation:
d
sin 
d
    eim
m integer (revolution)
d 

2
2
sin


k
sin


m
0




d 

Phys 451
The angular equation
z

r

k  l (l  1)
y
sin 
x
Solution:
d
d
d 

2
2
sin


l
(
l

1)
sin


m
0




d 

    APl m  cos  

Pl m  x   1  x 2

m /2
m
 d 
  Pl  x 
 dx 
Legendre function
Legendre polynomial
l
Physical condition
l,m integers
l 0
m l
1  d 
Pl ( x)  l   x 2  1
2 l !  dx 


l
Phys 451
The angular equation
z

Legendre function
r
Solution:

x
y
    APl m  cos  

Pl m  x   1  x 2

m /2
m
 d 
  Pl  x 
 dx 
Pl m (cos  ) are polynoms in cos
(multiplied by sin if m is odd)
Phys 451
Spherical harmonics
z


x
r
Yl m  ,   Aeim Pl m  cos 
y
Yl
m
 ,    
Yl  ,  
m
 2l  1  l  m !
4  l  m !
Azimuthal quantum number
Magnetic quantum number
l 0
Simulation: www.falstad.com
eim Pl m  cos  
m l
l
m
Quantum mechanics
Spherical harmonics
z


r
Method to build your spherical harmonics:
y
l
x
1. Legendre polynomial
2. Legendre function
3. Plug in cos

Pl m  x   1  x 2
1  d 
Pl ( x)  l   x 2  1
2 l !  dx 

m /2


l
m
 d 
  Pl  x 
 dx 
    APl m  cos 
4. Normalization factor
Pb 4.3
Phys 451
Quiz 21
Which one of the following quantities
could not physically correspond to a spherical harmonic?
A.
Yl l ( ,  )
 l 1
Y
B. l ( ,  )
C.
Yl l 1 ( ,  )
D.
Y21 ( ,  )
2
Y
E. 3 ( ,  )
Phys 451
The radial equation
z

r

  r, ,   R  r  Y  , 
y
x
The radial equation
Change of variables
u  rR (r )
1 d  2 dR  2mr 2
r
  2 V (r )  E   l (l  1)
R dr  dr 
2
d 2u 
l (l  1) 

  V (r ) 
 u  Eu
2
2
2m dr 
2m r 
2
Form identical to Schrödinger equation !
with an effective potential
l (l  1)
Veff  V (r ) 
2m r 2
2
Centrifugal
term
Phys 451
Normalization
z
   r 
r

y

2
dr
  r, ,   R  r  Y  , 
x
dr  r 2 sin  drd d
Radial part

 R r 
2
r dr  1
2
r 0

or

r 0
u dr  1
2
Angular part

2
 
0
0
Y ( ,  ) sin  d d  1
2
Phys 451
Orthonormality
z

r
  r   r  dr  
*
i
y

j
ij
x
Spherical harmonics are orthogonal
Angular part
2

 
0
0
*
Yl ( ,  )  Yl m' ' ( ,  )sin  d d   ll ' mm '
m
Pb 4.3 and 4.5