Folie 1

1
Collective Vibration
In general,
?0
?1
OK...
?2
?3
2
Collective Vibration
In general,
?2 quadrupole vibration
?3 octupole vibration
3
Collective Vibration
classical Hamiltonian
constants
Binding energy of a nucleus
aV 15.560 MeV aS 17.230MeV aC
0.6970 MeV aA 23.385 MeV aP 12.000
MeV
4
Collective Vibration
classical Hamiltonian
quantization
energy eigenvalue
wave function
5
Collective Vibration
classical Hamiltonian
differential equation
6
Second Quantization
Hamilton operator
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
7
Second Quantization
Hamilton operator
rule for boson operators
1-phonon energy
8
Second Quantization
Hamilton operator
rule for boson operators
2-phonon energy
etc.
9
Second Quantization
Hamilton operator
rule for boson operators
2-phonon state
normalization (approximation)
10
Second Quantization
Hamilton operator
rule for boson operators
2-phonon state
normalization
11
Collective Vibration
12
Example of vibrational excitations
multiple ? 2 phonon states, ideally degenerate
13
Second Quantization
Hamilton operator
rule for boson operators
2-phonon state
reduced transition probability
14
Reduced transition probabilities
2-phonon state
3-phonon state
15
Reduced transition probabilities
1-phonon state
2-phonon state
3-phonon state