collective vibration in general, forbidden – density change! forbidden – cm moves! ok... λ=0...
TRANSCRIPT
Collective Vibration
In general,
,1, *
0 YRR
forbidden – density change!
forbidden – CM moves!
OK...
λ=0
λ=1
λ=2
λ=3
time
Collective Vibration
In general,
,1, *
0 YRR
2
,2
1
CV
λ=2: quadrupole vibration λ=3: octupole vibration
Collective Vibration
classical Hamiltonian
22
2
1
2
1CBE
.)(
),(2
3/1
23/2 corrshell
A
ZNa
A
ZaAaAaZAB ACSV
Binding energy of a nucleus:
20
4
3 RAMB
212
16121
4 0
23/2
Ara
eZAaC
S
S
aV = 15.560 MeV aS = 17.230MeV aC = 0.6970 MeV aA = 23.385 MeV aP = 12.000 MeV
constants:
Collective Vibration
classical Hamiltonian
22
2
1
2
1CBE
221
2
1
CB
H
quantization
22
22
2
1
xCxB
H
energy eigenvalue
2
1
nE
21
22
0wave function 2/2xnnn exHN
Second Quantization
Hamilton operator
2
1
H
rule for boson operators:
creation & annihilation operators increase or decrease the number of phonons in a wave function
ground state (vacuum)
1-phonon state
2-phonon state
´´´´´´
0|
1|0|
2|0|~ ´´
Second Quantization
Hamilton operator
2
1
H
rule for boson operators: ´´´´´´
1-phonon energy:
0||0
2
10||01||1 H
0|1|0
2
10|1|01||1 H
0|1|0
2
10||01||1 H
2
10|1|01||1 H
2
111||1 H
Second Quantization
Hamilton operator
2
1
H
rule for boson operators: ´´´´´´
2-phonon energy:
0||0
2
10||0~2||2 H
2
122~2||2 H
0|1|0
2
10|1|0~2||2 H
0||0
2
10||0~2||2 H
etc.
Second Quantization
Hamilton operator
2
1
H
rule for boson operators: ´´´´´´
2-phonon state: 0||2211
21
2121 mmmm
IM IMmmN
normalization (approximation):
0||01 2 N
22 N
0|1|01 2 N
0||01 2 N
0|111|01 2 N
Second Quantization
Hamilton operator
2
1
H
rule for boson operators: ´´´´´´
2-phonon state: 0||2211
21
2121 mmmm
IM IMmmN
normalization:
0||0||1
22112211
2121
212121212
mmmmmmmm
IMmmIMmmN
0||0||1
222211122111
2121
212121212
mmmmmmmmmm
IMmmIMmmN
2211211221
2121
||1 212121212
mmmmmmmmmmmm
IMmmIMmmN
IMmmIMmmIMmmNmm
|||1 1221212121212
21
21
21
21
1||1 212121212
Imm
IMmmIMmmN 21
12IN
Collective Vibration
Example of vibrational excitations
(E-E ground state)
n=1, = 2, 2+ phonon
n = 2, = 2, J = 0+, 2+, 4+
ħ2
2ħ2
3ħ2 multiple = 2 phononstates, ideally degenerate
3- state?
Second Quantization
Hamilton operator
2
1
H
rule for boson operators: ´´´´´´
2-phonon state: 0||2211
21
2121 mmmm
IM IMmmN
reduced transition probability:
2
22222
220 011
20
4
302;2
mMMm
mm
m
f
iB
ReZEB
2,;
mM f
imMffiEB
*303
0
*
4
3,, mmp R
R
ZdYrmM
22
220
24
302;2
B
ReZEB
2
1~E
Reduced transition probabilities
2-phonon state
22
20
24
3
B
RZQvib
22
12
1;2 if
ifi IEMI
IIIEB
01;2212;2 2222 nnEBnnEB
01;2323;2 2222 nnEBnnEB3-phonon state
Reduced transition probabilities
1-phonon state eQnIEMnI vib 50,021,2 22
22
20
24
3
B
RZQvib
22
12
1;2 if
ifi IEMI
IIIEB
eQnIEMnI vib 181,222,4 22
eQnIEMnI vib 101,222,2 22
eQnIEMnI vib 21,222,0 22
eQnIEMnI vib 392,423,6 22
eQnIEMnI vib 7
902,423,4 22
eQnIEMnI vib 7
992,223,4 22
eQnIEMnI vib 62,423,3 22
eQnIEMnI vib 152,223,3 22
eQnIEMnI vib 7
202,223,2 22
eQnIEMnI vib 72,023,2 22
eQnIEMnI vib 32,023,0 22
2-phonon state
3-phonon state