Physics 451
Quantum mechanics I
Fall 2012
Nov 2, 2012
Karine Chesnel
Phys 451
Announcements
Homework tonight:
• HW #16 Friday Nov 2 by 7pm
Homework next week:
• HW #17 Tuesday Nov 6 by 7pm
• HW #18 Thursday Nov 8 by 7pm
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
Spherical harmonics are orthogonal
*

 i  r  j  r  dr   ij
y

x
2

 
0
0
Angular part
*
Yl ( ,  )  Yl m' ' ( ,  )sin  d d   ll ' mm '
m
l
Pb 4.6
Also
2
l Pl ( x).Pl ' ( x)dx  2l  1 ll '
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   l (l  1)
R dr  dr 
1 1  
Y 
1  2Y 
 l (l  1)
 sin 
 2
The angular equation 
2 
Y  sin   
  sin   
Phys 451
The radial equation
z

r

  r, ,   R  r  Y  , 
y
x
Change of variables
u  rR (r )
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
The radial equation
Example: infinite spherical well
V=0
V 
Inside the well
Change of variables
u  rR (r )
d 2u  l (l  1) 2m 

 2 E u
2
2
dr
 r

Phys 451
The radial equation
Infinite spherical well
V=0
V 
solution
• For l = 0
d 2u  2m 
   2 E u
2
dr


u  r   A sin(kr )  B cos(kr )
sin(kr )
cos(kr )
R r   A
B
r
r
Boundary condition
n2 2 2
En 
2ma 2
Three quantum numbers: (n, l, m)
here
 nlm (r , ,  )   n00 (r, ,  )
Phys 451
The radial equation
Infinite spherical well
V=0
V 
• If l ≠ 0
d 2u  l (l  1) 2m 

 2 E u
2
2
dr
 r

u  r   ArJl (kr )  BrNl (kr )
solution
Spherical
Bessel function
l
 1 d  sin x
J l ( x)    x  

x
dx

 x
l
Spherical
Neumann function
N l ( x)    x 
Physical condition at r =0
l 1
l
 1 d  cos x


x
dx

 x
B0
Phys 451
The radial equation
Infinite spherical well
V 
V=0
• If l ≠ 0
u  r   ArJ l (kr )
Physical condition at r = a :
1
knl   nl
a
J l (ka)  0
Enl 
2
2ma 2
 nl 2

 nlm  r ,  ,    A nl J l  nl
 a

r  Yl m  ,  

Phys 451
Quiz 22
When a particle is subject to a potential
that depends on the radius only,
which quantum numbers apply to quantize the energy?
A. Only the principal quantum number n
B. Only the azimuthal quantum number l
C. Only the magnetic quantum number m
D. Possibly both numbers (n,l)
E. Possibly all three numbers (n,l,m)
Phys 451
V 
V=0
Pb 4.7: construct n1 ( x)
Spherical well
and
n2 ( x)
show that they blow up at zero
Pb 4.8: case of l = 1
show that
En1 
2ma
n  1/ 2 
2 
V 0
Pb 4.9: Finite spherical well
V=-V0
Find the ground state (l =0)
2
2
2