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What is a resonance?. KEK Lecture (1). K. Kato Hokkaido University Oct. 6, 2010. （ 1 ） What is a resonance ？. The discrete energy state created in the continuum energy region by the interaction, which has an outgoing boundary condition. However, there are several definitions of resonances. - PowerPoint PPT Presentation
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What is a resonance?
K. Kato
Hokkaido University
Oct. 6, 2010
KEK Lecture (1)
（ 1 ） What is a resonance ？
The discrete energy state created in the continuum energy region by the interaction, which has an outgoing boundary condition.
However, there are several definitions of resonances
(i) Resonance cross section
(E) ～ —————１
(E – Er)2 + Γ2/4Breit-Wigner formula
“Quantum Mechanics” by L.I. Schiff
(ii) Phase shift
… If any one of kl is such that the denominator ( f(kl) ) of the expression for tanl,
|tanl| = | g(kl)/f(kl) | ∞ ,
( Sl(k) = e2il(k) ),
is very small, the l-th partial wave is said to be in resonance with the scattering potential.
Then, the resonance: l(k) = π/2 + n π
•Phase shift of 16O + α OCM
“Theoretical Nuclear Physics” by J.M. Blatt and V.F. Weisskopf
(iii) Decaying state
We obtain a quasi-stational state if we postulate that for r>Rc the solution consists of outgoing waves only. This is equivalent to the condition B=0 in
ψ (r) = A eikr + B e-ikr (for r >Rc).
This restriction again singles out certain define solutions which describe the “decaying states” and their eigenvalues.
•Resonance wave function
For the resonance momentum kr=κ–iγ,
ψ(r) = ei κr erγ, (not normalizable (γ>0) ）
G. Gamow, Constitution of atomic nuclei and dioactivity (Oxford U.P., 1931)
A.F.J. Siegert, Phys. Rev. 56 (1939), 750.The physical meaning of a complex energy
E=Er – iΓ/2
can be understood from the time depen-dence of the wave function
ψ(t) = ψ(t=0) exp( － iEt/ ｈ )
and its probability density
| ψ(t)|2 = |ψ(t=0)|2 exp( － Γt/ ２ｈ ).
The lifetime of the resonant state is given by τ = ｈ /Γ ．
4. Poles of S-matrix
The solution φl(r) of the Schrödinger equation;
Satisfying the boundary conditions
,
the solution φl(r) is written as
lll k
r
llrV
dr
d 2222
2
})1(
)(2
{
1),(lim 1
0
rkr l
l
r
})(
)({
2
)(
)},()(),()({2
),(
ikrikr
r
l
ekf
kfe
k
kif
rkfkfrkfkfk
irk
where Jost solutions f±(k, r) is difined as
and Jost functions f±(k)
,1),(lim rkfe ikr
r
)(),(lim)12(0
kfrkfrl l
r
Then the S-matrix is expressed as
The important properties of the Jost functions:
1.
2.
From these properties, we have unitarity of the S-matrix;
.)(
)()1()(
kf
kfkS l
l
),,(),( rkfrkf ),,(),( ** rkfrkf
),()( kfkf ).()( ** kfkf
* * 1.S k S k S k S k
The pole distribution of the S-matrix in the momentum plane
The Riemann surface for the complex energy:
E=k2/2
Ref.
1. J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961), 529-578
2. L. Rosenfeld, Nucl. Phys. 26 (1961), 594-607.
3. J. Humblet, Nucl. Phys. 31 (1962), 544-549.
4. J. Humblet, Nucl. Phys. 50 (1964), 1-16.
5. J. Humblet, Nucl. Phys. 57 (1964), 386-401.
6. J.P. Jeukenne, Nucl. Phys. 58 (1964), 1-9
7. J. Humblet, Nucl. Phys. A151 (1970), 225-242.
8. J. Humblet, Nucl. Phys. A187 (1972), 65-95.
（ 2 ） Many-body resonance states
（１） Two-body problems; easily solved
Single channel systems
Coupled-channel systems
(2) Three-body problems; Faddeev
A=C1+C2+C3
Decay channels of A
A [C1-C2]B+C3, Eth(C3)
[C2-C3]B+C1, Eth(C1)
[C3-C1]B+C2, Eth(C2)
B [C1-C2]R+C3, Eth(C12)
[C2-C3]R+C1, Eth(C23)
[C3-C1]R+C2, Eth(C31)
C C1+C2+C3, Eth(3)
Eth(C3 ） Eth(C2) Eth(C2) Eth(3)
Eth(C3
2 ）Eth(C23 ）
Multi-Riemann sheet
Eth(C31 ）
(3) N-Body problem; more complex
様々な構造をもったクラスター閾値から始まる連続状態がエネルギー軸上に縮退して観測される。
Eigenvalues of H( in the complex energy plane
Complex scaling
U( r rei
k ke-i
U( (r)
=ei3/2 (rei )
H()= U( U(
H
r
ririkrr
R
ee e
EH
)( ik
) tan ( 0
)(
1-rr
) |cos(|
)sin(||
rki
rkikrer
R
r
ri
e
ee
EH
Complex Scaling Method
physical picture of the complex scaling method
Resonance state
.rThe resonance wave function behaves asymptotically as
ikr
rer
)(
When the resonance energy is expressed as
,2
iEE r ,)
2(tan
2
1 1
rr E
the corresponding momentum is
and the asymptotic resonance wave function
,||
||22
r
r
ir
i
ek
eEEik
.)( sin||cos||)|exp(| rrr rkrkiriki
reeer
Diverge!
This asymptotic divergence of the resonance wave function causes difficulties in the resonance calculations.
In the method of complex scaling, a radial coordinate r is transformed as
Then the asymptotic form of the resonance wave function becomes
);(U ,irer .ipep
)sin(||)cos(||
|||| )(
)(
rr
riiri
rkrki
rekireeki
r
ee
eer
Converge!
It is now apparent that when π/2>(θ-θr)>0 the wave function converges asymptotically. This result leads to the conclusion that the resonance parameters (Er, Γ) can be obtained as an eigenvalue of a bound-state type wave function.
This is an important reason why we use the complex scaling method.
Eigenvalue Problem of the Complex Scaled
Hamiltonian • Complex scaling transformation
• Complex Scaled Schoedinger Equation
)re(fe)r(f)(U i2/3i
EH
),(HU)(UH 1 VTH
)(U
ABC Theorem J.Aguilar and J. M. Combes; Commun. Math. Phys. 22 (1971), 269.
E. Balslev and J.M. Combes; Commun. Math. Phys. 22(1971), 280.
i) is an L2-class function:
ii) E is independent on
,u)(ci
ii ||u|| i
)Earg(2
1 res
2/iEE rres
