Weinberg’s Laws of Progress in Theoretical Physics
From: “Asymptotic Realms of Physics” (ed. by Guth, Huang, Jaffe, MIT Press, 1983)
First Law: “The conservation of Information” (You will get nowhere
by churning equations) … garbage in, garbage out…
Second Law: “Do not trust arguments based on the lowest order of
perturbation theory”
Third Law: “You may use any degrees of freedom you like to describe
a physical system, but if you use the wrong ones, you’ll be sorry!”
A remark: physics of open nuclei is demanding !
Interactions
Interactions
Many-body
Correlations
Configuration interaction
•  Mean-field concept often questionable
•  Asymmetry of proton and neutron
Fermi surfaces gives rise to new
couplings
•  New collective modes; polarization
effects
•  Poorly-known spin-isospin
components come into play
•  Long isotopic chains crucial
Open
Channels
Open channels
•  Nuclei are open quantum systems
•  Exotic nuclei have low-energy decay
thresholds
•  Coupling to the continuum important
• Virtual scattering
• Unbound states
• Impact on in-medium Interactions
In mathematics, a rigged Hilbert space (Gel’fand triple, nested Hilbert space,
equipped Hilbert space) is a construction designed to link the distribution and
square-integrable aspects of functional analysis. Such spaces were introduced to
study spectral theory in the broad sense. They can bring together the 'bound
state' (eigenvector) and 'continuous spectrum', in one place.
Mathematical foundations in the 1960s by Gel’fand et al. who combined Hilbert space
with the theory of distributions. Hence, the RHS, rather than the Hilbert space alone, is
the natural mathematical setting of Quantum Mechanics
I. M. Gel’fand and N. J. Vilenkin. Generalized Functions, vol. 4: Some
Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic
Press, New York, 1964.
J.J. Thompson, 1884
G. Gamow, 1928
relation between decay width
and decay probability
A comment on the time scale…
∂ψ ˆ
i
= Hψ
∂t

T1/ 2 = ln2 ,
Γ
Ts. p.€
≈ 3⋅10
−22
€
 = 6.58 ⋅10−22 MeV ⋅ sec
sec = 3babysec.
238U:
T1/2=1016 years
256Fm: T =3 hours
1/2
The usual starting point: one body
Schrödinger equation:
U(r)
at large distances…
R
asymptotically…
r
We are interested in the expectation value:
inner contribution
(r<R)
outer contribution
(r>R)
The inner integral is always finite. The outer integral can be written as:
In the limit of a very weak binding, one can use the asymptotic expressions
for the Hankel functions. This yields:
If pairing is present, this picture changes:
K. Bennaceur et al., Phys. Lett. B496, 154 (2000)
B(N + 1,Z) + B(N −1,Z)
2
B(N,Z + 1) + B(N,Z −1)
Δ p = B(N,Z ) −
2
Δ n = B(N,Z ) −
for odd-N
for even-N
The single-particle field characterized by λ, determined by the p-h component
of the effective interaction, and the pairing field Δ determined by the pairing
part of the effective interaction are equally important when S1n is small.
If pairing is present, the
picture of halo changes:
K. Bennaceur et al., Phys.
Lett. B496, 154 (2000)
Pairing Antihalo Effect
•  square-well potential
•  spherical symmetry
•  l=0 (s-wave)
Radial Schrödinger equation
2M
χ ''+ 2 (E − V ) χ = 0

( χ = ϕr)
Region I:
χ I = Asin pr,
Region II:
χ II = c +e
€
Region III:
q(r−a )
2ME
p = 2

2
+ c−e
−q(r−a )
2M(Vb − E)
, q =
2
χ III = c1e ip(r−b ) + c 2e−ip(r−b )
2
In almost all cases |χIII| is much larger than |χI|. We are now interested
in those situations where |χIII| is as small as possible.!
The condition
p
c + = 0 ⇒ tan( pa) = −
q
defines “virtual” levels in region I:
particle is well localized; very small
penetrability through the barrier
open
closed
H(t) = H 0 + V (t) (V << H 0 )
time-dependent Hamiltonian
∂ψ ˆ
= Hψ
…expansion of ψ in the basis of Ho
∂t
Hˆ 0φ n = E n φ n ⇒ ψ = ∑ c n (t)φ n e−iE n t / 
i
€
n
i
dc k
= ∑ c n (t) φ k V φ n e iω kn t , ω kn = ( E k − E n ) /
dt
n
€
As initial conditions, let us assume that at t=0 the system is in the state φ0. That is,
€
1 for n = 0
c n (0) =
0 for n ≠ 0
If the perturbation is weak, in the first order,
we obtain: € i dc k = φ k V φ 0 e iω t
k0
dt
Furthermore, if the time variation of V is slow compared with exp(iωkot), we may
treat the matrix element of V as a constant. In this approximation:
c k (t) =
φk V φ0
1− e iω k 0 t )
(
Ek − E0
The probability for finding the system in state k at time t if it started from state 0
€ t=0 is:
at time
2
2
c k (t) = 2
φk V φ0
(Ek − E0)
2
(1− cosω k 0 t )
€ total probability to decay to a group of states within some interval labeled
The
by f equals:
2
∑ c k (t) =
k∈ f
€
2
2
∫
φk V φ0
ω
2
k0
2
(1− cosω k 0 t )ρ( E k ) dE k
The transition probability per unit time is
W =
€
d
2
2
c
(t)
=
∑ k
dt k ∈ f
2
2
sin ω k 0 t
ρ( E k ) dE k
ωk0
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:
W0 → f
2
2π
=
φ f V φ 0 ρ( E f )

€
∞
sin x
∫ x dx = π
−∞
€
∫
φk V φ0
Fermi’s
golden rule

T1/ 2 = ln2 ,
Γ
−22
Ts. p. ≈ 3⋅10
 = 6.58 ⋅10
−22
sec = 3babysec.
T1/ 2 >> Ts.p.
€
€
Γ << 1MeV
€
€
MeV ⋅ sec
A typical time
associated with
the s.p. motion
Baumann et al.,
ENAM’08
Extrapolates
very well!
the difference of mass excess of projectile and target
?
€
V ( r) = U ( r ) + W ( r)
W˜ = W + V0
eigenstate
of T+U with
E=E0
eigenstate
~
of T+W with
E=E0
Isolated Breit-Wigner resonance; Fermi’s golden rule
Two-body case, 1-neutron halo
Bertulani and Baur, Nucl. Phys. A480, 615 (1988) Nagarajan, Lenzi, ViCuri, Eur. Phys. J. A24, 63 (2005) 19C Ec
Eb=-Sn
For transiJon: It has maximum at Three-body case, 2-neutron halo
Pushkin, Jonson, and Zhukov, J. Phys. G 22, L95 (1996) 11Li