Width of a narrow resonance. t = H ˆ ψ expansion of ψ in the basis of H o. = E k

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Peregrine Martin
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1 open closed Width of a narrow resonance H(t) = H 0 + V (t) (V << H 0 ) time-dependent Hamiltonian i! ψ t = H ˆ ψ expansion of ψ in the basis of H o H ˆ 0 φ n = E n φ n ψ = c n (t) φ n e ie n t /! n i! dc k dt = c n (t) φ k V φ n e iω knt, ω kn = E k E n n ( ) /! As initial conditions, let us assume that at t=0 the system is in the state φ 0 c n (0) = 1 for n = 0 0 for n 0 If the perturbation is weak, in the first order, we obtain: i! dc k dt = φ k V φ 0 e iω k 0t

2 Furthermore, if the time variation of V is slow compared with exp(iω ko t), we may treat the matrix element of V as a constant. In this approximation: c k (t) = φ k V φ 0 E k E 0 ( 1 e iω k 0t ) The probability for finding the system in state k at time t if it started from state 0 at time t=0 is: c k (t) 2 = 2 φ V φ k 0 E k E 0 2 ( 1 cosω ( ) 2 k 0 t) The total probability to decay to a group of states within some interval labeled by f equals: c k (t) 2 = 2! φ V φ k 0 1 cosω 2 2 k 0 t k f ω k 0 2 ( ) ρ ( E k )de k

3 The transition probability per unit time is W = d dt c k (t) 2 = 2 2 sinω φ! 2 k V φ k 0 t 0 ρ ( E k )de k k f ω k 0 Since the function sin(x)/x oscillates very quickly except for x~0, only small region around E 0 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: W 0 f = 2π! φ 2 f V φ 0 ρ ( E f ) 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 sin x x dx = π

4 E = E 0 i 2 ; = w mean lifetime T 0 = half-life transition probability W 0 f = 1 T 0 f = Γ 0 f! c(e) = normalized amplitude Fermi s golden rule (t) = (0) Z 2 Z 1 0 c(e)e iet/~ de e i(e E 0+i /2)t/~ dt = 2 Γ 0 f = 2π φ f V φ 0 ρ ( E f ) i~ E E 0 + i /2 uncertainty principle

When can we talk about existence of an unbound nuclear system? T 1/ 2 = ln2! Γ,! = 6.58 10 22 MeV sec T s. p. 3 10 22 sec = 3babysec A typical time associated with the s.p. nucleonic motion T >> T 1/2 s.

5 When can we talk about existence of an unbound nuclear system? T 1/ 2 = ln2! Γ,! = MeV sec T s. p sec = 3babysec A typical time associated with the s.p. nucleonic motion T >> T 1/2 s. p. Γ <<1MeV HW: Using NNDC and PDG, find the half-life and width of: 48 Ni ground state 3 - state in 10 Be at E=10.16 MeV First 2 + state in 6 He at E=1.797MeV Hoyle state in 12 C at E=7.654 MeV 8 Be ground state Baryon N(1440)1/2 + Discuss the result. What are the main decay modes?

6 Rep. Prog. Phys. 67 (2004) Reaching the limits of nuclear stability 1195 as such [22]. This statement was supported later by Cerny and Hardy [23]:... lifetimes longer than s, a possible lower limit for the process to be called radioactivity. This definition would be more restrictive than the definition of an element and thus is inappropriate. The International Union of Pure and Applied Chemistry (IUPAC) has published guidelines for the discovery of a chemical element [24]. In addition to other criteria they state that the discovery of a chemical element is the experimental demonstration, beyond reasonable doubt, of the existence of a nuclide with an atomic number Z not identified before, existing for at least s. The justification for this limit is also given: This lifetime is chosen as a reasonable estimate of the time it takes a nucleus to acquire its outer electrons. It is not considered self-evident that talking about an element makes sense if no outer electrons, bearers of the chemical properties, are present. Similarly the definition of a nucleus should be related to the typical timescales of nuclear motion. Nuclear rotation and vibration times are of the order of s which can be considered a characteristic nuclear timescale [22]. The above mentioned definitions of the driplines by Mueller and Sherrill [10] and the Chart of Nuclei [19] can be used as the definition of the existence of a nucleus. If a nucleus lives long compared to s it should be considered a nucleus. Unfortunately this is no sharp clear limit. The most recent editions of the chart of nuclei include unbound nuclei with lifetimes that are of the order of s [19, 25].

7 Prog. Part. Nucl. Phys. 59, 432 (2007) λ n ~ Δ n

8 (7.27) (14.44) (19.17) (28.48) (7.16) (11.89) (21.21) Excitation energy Ikeda Diagram (4.73) (13.93) (14.05) (9.32) 11 Li Mass number sequence of reaction channels

Beyond the Neutron Drip-Line http://www.tandfonline.com/doi/pdf/10.1080/10619127.

9 Beyond the Neutron Drip-Line

Tetraneutron??? PHYSICAL REVIEW C, VOLUME 65, 044006 (2002) Detection of neutron clusters A new approach to the production and detection of bound neutron clusters is presented.

10 Tetraneutron??? PHYSICAL REVIEW C, VOLUME 65, (2002) Detection of neutron clusters A new approach to the production and detection of bound neutron clusters is presented. The technique is based on the breakup of beams of very neutron-rich nuclei and the subsequent detection of the recoiling proton in aliquid scintillator. The method has been tested in the breakup of intermediate energy MeV/nucleon 11 Li, 14 Be, and 15 B beams. Some six events were observed that exhibit the characteristics of amultineutron cluster liberated in the breakup of 14 Be, most probably in the channel 10 Be 4 n. The various backgrounds that may mimic such asignal are discussed in detail.

11 Can Modern Nuclear Hamiltonians Tolerate a Bound Tetraneutron?

BUT... Kisamori et al. http://journals.aps.org/prl/abstract/10.1103/physrevlett.

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

13 Baryon and meson resonances Lots of unbound states! N* Δ*

Open quantum systems

Open quantum systems 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