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.
What is a resonance?
KEK Lecture (1)
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 severaldefinitions of resonances
(i) Resonance cross section
s(E) ～ —————
(E – Er)2 + Γ2/4
(ii) Phase shift
“Quantum Mechanics” by L.I. Schiff
… If any one of klis 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
(iii) Decaying state
“Theoretical Nuclear Physics”by J.M. Blatt and V.F. Weisskopf
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.
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
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
where Jost solutions f±(k, r) is difined as
and Jost functions f±(k)
Then the S-matrix is expressed as
The important properties of the Jost functions:
From these properties, we have unitarity of the S-matrix;
The pole distribution of the S-matrix in the momentum plane
The Riemann surface for the complex energy:
（2） Many-body resonance states
B [C1-C2]R+C3, Eth(C12)
C C1+C2+C3, Eth(3)
Eth(C3） Eth(C2) Eth(C2) Eth(3)
(3) N-Body problem; more complex
Eigenvalues of H(q) in the complex energy plane
U(q) ; r rei q
k ke-i q
Yq= U(q) Y(r)
=ei3/2 q Y(rei q)
H(q)= U(q) H U(q)-1
Complex Scaling Method
physical picture of the complex scaling method
The resonance wave function behaves asymptotically as
When the resonance energy is expressed as
the corresponding momentum is
and the asymptotic resonance wave function
This asymptoticdivergence of the resonance wave function causes difficulties in the resonance calculations.
In the method of complex scaling, a radialcoordinate r is transformed as
Then the asymptotic form of the resonance wave function becomes
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