1 3-Dim Problems. 2 3-D Problems Separation of Variables – (r ) = R(r) ( ) ( ) – ( )...
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Transcript of 1 3-Dim Problems. 2 3-D Problems Separation of Variables – (r ) = R(r) ( ) ( ) – ( )...
1
3-Dim Problems
2
3-D Problems
• Separation of Variables– (r) = R(r) () ()– () solution– () solution– R(r) equation
• Effective Potentials
3
Separation of Variables
4
Coordinates & Normalization r
spaceall
dV 1
dddrrdV sin2
r
ddxdrr2
0
1
1
2
0
cos'' x
dddrrdV cos2
dddrr2
00
2
0
sin
ddxdrrdV 2
dddrr2
0
1
1
2
0
cos
5
EV
m
p
2
2
r
EV
m2
2
2
2222
22
2
sin
1sin
sin
11
rr
rr rr
2222
22
2 111
srs
srr
r rr
Note derivatives are cleanly separated
VEm2
2 2
rRr
6
VEm2
2 2
VE
m
srs
srr
r rr 22
2222
2
2111
2
22222
2
1211 sr
VEm
ssr
rr rr
2
222
222 211
VEm
ssr
rr
sr rr
RRVE
ms
srr
rsr rr
222
22
22 211
rRr
{a}
{b}
{c}
{d}
7
RRVE
ms
srr
rsr rr
222
22
22 211
222
22
22 211
R
RVEm
ssr
rr
sr rr
LHS = const = RHS
m2
2
222
222 211
mR
RVEm
ssr
rr
sr rr
2
2 m
{e}
{f}
{d}
8
2
2 m
Azimuthal Behavior
mie
2
2 imim ee
22 m
miemiimim eee 2
Intm Re
…, -2, -1, 0, 1, 2, …
Note: 1) EVP 2) Since no V involved only have to do this once forevermore
9
Other Piece
2
222
222 211
mR
RVEm
ssr
rr
sr rr
R
sr
mRVE
ms
srr
r rr 22
2
222
2
211
R
sr
ms
srRVE
mr
r rr 22
2
222
2
121
R
s
ms
sRVE
mrr rr 2
2
222 12
2
2
222 12
sm
ss
R
RVEm
rr rr
{f}
{g}
{h}
{i}
{j}
10
2
2
222 12
sm
ss
R
RVEm
rr rr
LHS = const = RHS
1
1
22
22
R
RVEm
rr rr
2
21
1sm
ss
{j}
{k}
{l}
11
Other Angular Piece (co-lattitude)
2
21
1sm
ss
1
12
2
s
ms
s
Note: 1) EVP 2) Since no V involved only have to do this once forevermore
Solns depend on choice of both l and m
mm Paka
Associated Legendre Polynomials
defer solving til later when we have nicer techniques
{l}
12
Summarizing the Angular Parts So Far
mie mm Por
mimm
m ePm
mY
!
!
4
121,
Since the angular basis functions are the same regardless of the potential chosen.
Define the “spherical harmonics”
John Day @ http://www.cloudman.com/gallery1/gallery1_2.html
http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi
13
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05 0 0.05
-0.050
0.05
-0.2
0
0.2
-0.050
0.05
-0.1-0.05
00.05
0.1-0.1
-0.05
0
0.05
0.1
-0.04-0.02
00.020.04
-0.1-0.05
00.05
0.1
-0.1-0.0500.050.1
-0.1-0.05 00.050.1
-0.4
-0.2
0
0.2
0.4-0.1-0.05 00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1-0.05
00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1 00.1-0.100.1-0.04-0.0200.020.04
-0.1 00.1
http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi
2,mY
14
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05 0 0.05
-0.050
0.05
-0.2
0
0.2
-0.050
0.05
-0.1-0.05
00.05
0.1-0.1
-0.05
0
0.05
0.1
-0.04-0.02
00.020.04
-0.1-0.05
00.05
0.1
-0.1-0.0500.050.1
-0.1-0.05 00.050.1
-0.4
-0.2
0
0.2
0.4-0.1-0.05 00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1-0.05
00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1 00.1-0.100.1-0.04-0.0200.020.04
-0.1 00.1
Comparision of Y11 & Y22
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 20 40 60 80 100120140160180200220240260280300320340360
C0-latitude Angle (deg)
Ab
solu
te S
qu
are
Y11 2̂
Y22 2̂
Comparision of Y11 & Y22
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60 80 100120140160180200220240260280300320340360
C0-latitude Angle (deg)
Val
ues Y11 2̂
Y22 2̂
2,mY
15
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05 0 0.05
-0.050
0.05
-0.2
0
0.2
-0.050
0.05
-0.1-0.05
00.05
0.1-0.1
-0.05
0
0.05
0.1
-0.04-0.02
00.020.04
-0.1-0.05
00.05
0.1
-0.1-0.0500.050.1
-0.1-0.05 00.050.1
-0.4
-0.2
0
0.2
0.4-0.1-0.05 00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1-0.05
00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1 00.1-0.100.1-0.04-0.0200.020.04
-0.1 00.1
2,mY
16
17
4, 3, 04, 3, 14, 3, 24, 3, 34, 2, 04, 2, 14, 2, 24, 1, 04, 1, 14, 0, 0
http://www2.physics.umd.edu/~gcchang/courses/phys402/common/notebooks.html
2,mY
18
Radial Piece
1
22
22
R
RVEm
rr rr
RRVEm
rr rr 12
222
RER
rmVr
rm rr
2
22
2
2 1
2
1
2
effective potential
Note: 1) EVP 2) This has to be solved for every different choice of V(r) 3) Will determine the allowed Etot ‘s
{k}
19
Summary So Far
http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi
RER
rmVr
rm rr
2
22
2
2 1
2
1
2
,mnl YrRr
EV
m2
2
2
2222
22
2
sin
1sin
sin
11
rr
rr rr
20
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05
0
0.05
-0.05 0 0.05
-0.050
0.05
-0.2
0
0.2
-0.050
0.05
-0.1-0.05
00.05
0.1-0.1
-0.05
0
0.05
0.1
-0.04-0.02
00.020.04
-0.1-0.05
00.05
0.1
-0.1-0.0500.050.1
-0.1-0.05 00.050.1
-0.4
-0.2
0
0.2
0.4-0.1-0.05 00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1-0.05
00.050.1
-0.1-0.05 00.050.1
-0.1-0.050
0.050.1
-0.1 00.1-0.100.1-0.04-0.0200.020.04
-0.1 00.1
2,mY
Summary So Far
21
Effective Potential
2
2 1
2 rmrV
Depends on the forces involved
Atomic motion?Nuclear motion?
…
Centripetal Term
22
Bare Coulomb Potential
H-atom
positronium atom
HeLiBeBC***
r
ZeerV
o
))((
4
1
23
1 2 3 4 5 6 7 8rA
-20
-10
10
20
Energy eVEffective Potential: H atom
Atomic Potential ExampleVcoul := -14.42/r1 := 1Vorbital := 3.818 * 1* (1+1) /r^2Veff := Vcoul + VorbitalPlot[ {Vcoul, Vorbital, Veff}, {r, 0.3, 8},PlotStyle ~ {{RGBColor[0, 0,1]}, {RGBColor[0, 1,0]}, {RGBColor[l, 0,0]}},AxesLabel ~ {"r (A)", "Energy (eV)"}]
l = 0
Bound States Etot < 0
Free States Etot > 0
Etot
24
Electron Clouds – dot plots
http://www.uark.edu/misc/julio/orbitals/
Scatter plots of hydrogen-atom wavefunctions
This is a tentative project. The figures that you can link to from this page are made by choosing 2000 points at random, with a probability given by one of the hydrogen atom's wavefunctions. The resulting plots give an idea of the "shape" of the atomic wavefunctions. You can rotate them by clicking and dragging with the mouse; you can also magnify the figure by clicking and dragging vertically while holding down the "shift" key.
The points were generated in Mathematica and the interactive figures were generated using LiveGraphics3D. LiveGraphics3D is an applet (not written by me); for it to work, you need to have java enabled in your browser.
25
Effective Potential: H atom
2 4 6 8rA
-20
-10
10
20
Energy eV
Atomic Potential ExampleVcoul := -14.42/r1 := 1Vorbital := 3.818 * 1* (1+1) /r^2Veff := Vcoul + VorbitalPlot[ {Vcoul, Vorbital, Veff}, {r, 0.3, 8},PlotStyle ~ {{RGBColor[0, 0,1]}, {RGBColor[0, 1,0]}, {RGBColor[l, 0,0]}},AxesLabel ~ {"r (A)", "Energy (eV)"}]
l = 1
Bound States Etot < 0
26
2 4 6 8rA
-20
-10
10
20
Energy eV
2 4 6 8rA
-40
-20
20
40
Energy eV
l = 1
l = 2
Bound States Etot < 0
27
‘Mean’ Nuclear Potential
V(r)
28
Effective Potential: Nuclear Examples
2 4 6 8 10rfm
-40
-20
20
40
60
80
Energy MeV
Vo := -50R := 4a:= 0.67VNcentral = Vo / (l+Exp[(r-R)/a])VNorbital := (197*197/2/940) * 1* (1+1) / r^2VNeff : = VNcentral + VNorbitalPlot[ {VNcentral, VNorbital, VNeff}, {r, 0.3, 10.0},PlotStyle ~ {{RGBColor[0, 0,1]}, {RGBColor[0, 1,0]}, {RGBColor[l, 0,0]}},AxesLabel ~ {"r (fm)", "Energy (MeV)"}]
l = 1
29
2 4 6 8 10rfm
-40
-20
20
40
60
80
Energy MeV
2 4 6 8 10rfm
-40
-20
20
40
60
80
Energy MeV
2 4 6 8 10rfm
-40
-20
20
40
60
80
Energy MeV
l = 1
l = 0
l = 2
Bound States Etot ~< 0
30
2.5 5 7.5 10 12.5 15 17.5 20rfm
-30
-25
-20
-15
-10
-5
5Energy MeVl = 2
free particles Etot > Vtop
quasi-free Vtop > Etot > 0 quasi-bound
a small positive barrier appears“Centripetal barrier”
31
2.5 5 7.5 10 12.5 15 17.5 20rfm
-4
-2
2
4
Energy MeV Application to RadioactiveAlpha Decay
238U = ( 234Th + ) 234Th +
Etot
32
2.5 5 7.5 10 12.5 15 17.5 20rfm
-4
-2
2
4
Energy MeVNeutron-Induced Reactions
Neutrons with l = 0 have NO centripetal barrier and are most efficient for creating nuclear reactions
Etot
33
