Bound Problems in the real world
From the Schrödinger Equation in 3D to the angular momentum
1
Schrödinger Equation in 3D
• We write the time-independent Schrödinger equation
✓
◆
~
(x) = E (x)
r2 + V (x, y, z)
2m
2
• in spherical coordinates

~
1 @
2m r2 @r
2
✓
2 @
r
@r
◆
1
@
+ 2
r sin ✓ @✓
✓
z
= [E
θ
r
y
φ
@
sin ✓
@✓
◆
x
2
1
@2
+ 2 2
r sin ✓ 2
V (r)] (r, ✓, )
(r, ✓, )
Schrodinger Equation in 3D
• Assumption:
V (r, #,' ) = V (r)
• By using separation of variables, we find
1) an angular equation
✓
◆
1 @
@Y
1 @2Y
sin ✓
+
= -l(l + 1)Y (✓, ¢)
2
2
sin ✓ @✓
@✓
sin ✓ @¢
2) a radial equation
✓
◆
2
1 d
dR
2m
r
r2
(V - E) = l(l + 1)
2
R dr
dr
~
3
1) Angular Equation: Angular Momentum Operator
• Consider the classical angular momentum and the related quantum operator
ˆ ˆ
ˆ
ˆ
ˆ
~
~
L =
~
r
⇥ p~ = -i~r~
⇥ r
• In spherical coordinates ✓
we have:
◆
@
@
Lx = i~ sin '
+ cot # cos '
,
@#
@'
✓
◆
@
@
ˆ
- cot # sin '
Ly = -i~ cos '
@#
@'
@
L̂z = -i~
@'
ˆ2
~
ˆ 2
+ L̂2
+ L̂2
is
• And the magnitude of the angular momentum |L| = L
x
y
z
2
2
L̂ = _~

1 @
sin # @#
✓
@
sin #
@#
4
◆
1
@2
+
2 @'2
sin #
1) Angular Equation
• We identify the angular equation as the eigenvalue equation for the orbital
angular momentum:
_~2

✓
◆
1 @
@Y
1 @2Y
sin ✓
+
sin ✓ @✓
@✓
sin2 ✓ @¢2
!
= ~2 l(l + 1)Y (✓, ¢)
L2 Y = ~2 l(l + 1)Y (✓, )
• We solve the differential equation by separation of variables, Y (✓, ¢) = ⇥(✓)<(¢)
d
d
2
2
=
2
m
( )
d
sin ✓
d✓
✓
d⇥
sin ✓
d✓
5
◆
⇥
⇤
= m - l(l + 1) sin ✓ ⇥(✓)
2
2
1) Angular Equation
• The normalized angular eigenfunctions are then Spherical Harmonic functions
Ylm (✓,
)=
s
(2l + 1) (l m)! m
Pl (cos ✓) eim
4⇡ (l + m)!
m
P
• where l (cos ✓) are Legendre Polynomials. For example:
P00 (cos ✓) = 1
P01 (cos ✓) = cos ✓
6
P0±1 (cos ✓) = sin ✓
Spherical Harmonics
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7
MIT OpenCourseWare
http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
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