1. Collective Vibration In general, forbidden – density change! forbidden – CM moves! OK... λ=0 λ=1 λ=2 λ=3 time

2. Collective Vibration In general, λ=2: quadrupole vibrationλ=3: octupole vibration

3. Collective Vibration classical Hamiltonian Binding energy of a nucleus: a V = 15.560 MeV a S = 17.230MeV a C = 0.6970 MeV a A = 23.385 MeV a P = 12.000 MeV constants:

4. Collective Vibration classical Hamiltonian quantization energy eigenvalue 0 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 (E-E ground state) n=1, = 2, 2+ phonon n = 2, = 2, J  = 0+, 2+, 4+ ħ2ħ2 2ħ  2 3ħ  2 multiple = 2 phonon states, ideally degenerate 3- state?

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