1. Open quantum systems

2. Wikipedia:
An open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment. The open quantum system can be viewed as a distinguished part of a larger closed quantum system, the other part being the environment.
INTERDISCIPLINARY
Small quantum systems, whose properties are profoundly affected by environment, i.e., continuum of scattering and decay channels, are intensely studied in various fields of physics: nuclear physics, atomic and molecular physics, nanoscience, quantum optics, etc.
OQS
environment
Example: EM radiation

3. Quasistationary States
(alpha decay)
For the description of a decay, we demand that far from the force center there be only the outgoing wave. The macroscopic equation of decay is
N is a number of radioactive nuclei, i.e., number of particles inside of sphere r=R:

4. relation between decay width and decay probability
We should thus seek a solution of the form
J.J. Thompson, 1884
G. Gamow, 1928
relation between decay width and decay probability
The time dependent equation
can be reduced by the above substitution to the stationary equation
The boundary condition
…takes care of the discrete complex values of E

5. This relation has been checked in numerous precision experiments.
In principle, resonances and decaying particles are different entities. Usually, resonance refers to the energy distribution of the outgoing particles in a scattering process, and it is characterized by its energy and width. A decaying state is described in a time dependent setting by its energy and lifetime. Both concepts are related by:
This relation has been checked in
numerous precision experiments.
See more discussion in
R. de la Madrid,
Nucl. Phys. A812, 13 (2008)
U. Volz et al., Phys. Rev. Lett 76, 2862(1996)

6. A comment on the time scale…
Time Dependent
Schrödinger Equation
Can one calculate G with sufficient accuracy using TDSE?
238U: T1/2=1016 years
256Fm: T1/2=3 hours
For very narrow resonances, explicit time
propagation impossible!

7. Radial Schrödinger equation
How do resonances appear? A square well example
square-well potential
spherical symmetry
l=0 (s-wave)
Radial Schrödinger equation
Region I:
Region II:
Region III:

8. defines “virtual” levels in region I:
In almost all cases |cIII| is much larger than |cI|. We are now interested in those situations where |cIII| is as small as possible.
The condition
defines “virtual” levels in region I:
particle is well localized; very small penetrability through the barrier

9. When c+=0, the penetrability becomes proportional to
This is the semi-classical WKB result:

10. Width of a narrow resonance
open
closed
time-dependent Hamiltonian
…expansion of  in the basis of Ho
As initial conditions, let us assume that at t=0 the system is in the state 0
If the perturbation is weak, in the first order, we obtain:

11. Furthermore, if the time variation of V is slow compared with exp(iwkot), we may
treat the matrix element of V as a constant. In this approximation:
The probability for finding the system in state k at time t if it started from state 0
at time t=0 is:
The total probability to decay to a group of states within some interval labeled
by f equals:

12. Fermi’s golden rule The transition probability per unit time is
Since the function sin(x)/x oscillates very quickly except for x~0, only small region around E0 can contribute to this integral. In this small energy region we may regard the matrix element and the state density to be constant. This finally gives:
Fermi’s golden rule
Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac, who formulated an almost identical equation It is given its name because Fermi called it "Golden Rule No. 2." in 1950.

13. mean lifetime
half-life
transition probability
Fermi’s golden rule
normalized amplitude
uncertainty principle

14. When can we talk about “existence” of an unbound nuclear system?
A typical time associated with the s.p. nucleonic motion
Compute the half-life of:
141Ho proton emitter (G=2･10-20 MeV)
3- state in 10Be at E=10.16 MeV (G=296 keV)
First 2+ state in 6He at E=1.797MeV (G=113 keV)
Hoyle state in 12C at E=7.654 MeV (G=8.5 eV)
8Be ground state (G=5.57 eV)
Baryon N(1440)1/2+ (G=350 MeV)
Discuss the result.

15. Rep. Prog. Phys. 67 (2004) 1187–1232

16. Prog. Part. Nucl. Phys. 59, 432 (2007)

17. 11Li Ikeda Diagram Excitation energy Mass number (7.27) (14.44)
(19.17)
(28.48)
(7.16)
(11.89)
(21.21)
(14.05)
Ikeda Diagram
(4.73)
Excitation energy
(13.93)
(9.32)
11Li
Mass number
sequence of reaction channels 1

18. Beyond the Neutron Drip-Line

19. Tetraneutron??? http://www.cnrs.fr/cw/en/pres/compress/noyau.htm

20. Can Modern Nuclear Hamiltonians Tolerate a Bound Tetraneutron?

21. Viewpoint: Can Four Neutrons Tango?
BUT.....
Kisamori et al.
Viewpoint: Can Four Neutrons Tango?

22. Baryon and meson resonances
Lots of unbound states! N* D*