Bound Pr oblems in the r eal world
Fr om the Schrödinger Equation in 3D to the angular momentum
Schrödinger Equation in 3D
✓
• W e write the time-independent Schrödinger equation
- 2 m r + V ( x, y , z ) ◆ ( x ) = E ( x )
~ 2 2
• in spherical coor dinates
~ 2
— 2 m
1 @
r 2 @ r
r 2 @
✓
@ r
+ 1 @
◆
r 2 si n ✓ @✓
si n ✓ @
✓
@✓
+ 1 @ 2
◆
r 2 si n 2 ✓ 2
( r, ✓ , )
z
θ
r
φ
x
y
= [ E — V ( r )] ( r, ✓ , )
Schr odinger Equation in 3D
• Assumption: V ( r, e , ϕ ) = V ( r )
• By using separation of variables, we find
1) an angular equation
1 @ ✓ si n
@ Y ◆ +
1 @ 2 Y =
( + 1) ( )
si n ✓ @✓
✓ @✓ si n 2 ✓ @ ¢ 2
- l l
Y ✓ , ¢
2) a radial equation
R dr
r
dr
-
~ 2
V - E
1 d ✓ 2 dR ◆ 2 m r 2 (
l
l
) = (
+ 1)
1) Angular Equation: Angular Momentum Operator
• Consider the classical angular momentum and the r elated quantum operator
L ~ ˆ
= ~ r ˆ ⇥
p ~ ˆ =
- i ~ ~ r ˆ ⇥
r ~ ˆ
• In spherical coor dinates we have:
x
∂e ∂ϕ
L = i ~ ✓ sin ϕ ∂ + cot e cos ϕ ∂ ◆ ,
L ˆ = - i ~ ✓ cos ϕ ∂ - cot e sin ϕ ∂ ◆
y
L ˆ z
= - i ~ ∂
∂ϕ
∂e ∂ϕ
~
ˆ
• And the magnitude of the angular momentum
| L | 2
= L ˆ 2
+ L ˆ 2
+ L ˆ 2 is
x
y
z
ˆ 2 2
1 ∂ ✓
∂ ◆
1 ∂ 2
L = _ ~
sin e ∂e
sin e
∂e
+ sin 2 e ∂ ϕ 2
1) Angular Equation
• W e identify the angular equation as the eigenvalue equation for the orbital angular momentum:
2 1 @ ✓ @ Y ◆ 1 @ 2 Y 2
_ ~ si n ✓ @✓
si n ✓
@✓
+ sin 2 ✓ @ ¢ 2
= ~ l ( l + 1) Y ( ✓ , ¢ )
! L 2 Y = ~ 2 l ( l + 1) Y ( ✓ , )
✓
d ✓
✓
d ✓
m
✓
✓
• W e solve the dif fer ential equation by separation of variables, Y ( ✓ , ¢ ) = ⇥ ( ✓ ) < ( ¢ )
d 2
— m
d 2 Φ =
2 Φ (
) si n
d ✓ si n
d ⇥ ◆ = ⇥ 2
- l
l
( + 1) sin 2
⇤ ⇥ ( )
1) Angular Equation
• The normalized angular eigenfunctions ar e then Spherical Harmonic functions
l
Y m ( ✓ , ) =
s (2 l + 1) ( l — m )! P m (cos ✓ ) e im
l
4 ⇡ ( l + m )!
l
• wher e P m (cos ✓ ) ar e Legendr e Polynomials. For example:
0 0 0
P 0 (cos ✓ ) = 1 P 1 (cos ✓ ) = cos ✓ P ± 1 (cos ✓ ) = s i n ✓
Spherical Harmonics
MIT OpenCourseWare http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .