What is a resonance
1 / 28

What is a resonance? - PowerPoint PPT Presentation

  • Uploaded on

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

PowerPoint Slideshow about ' What is a resonance?' - malcolm-gamble

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
What is a resonance

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 severaldefinitions of resonances

(i) Resonance cross section

s(E) ~ —————

Breit-Wigner formula

(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 tanl,

|tanl| = | g(kl)/f(kl) | ∞ ,

( Sl(k) = e2il(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

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

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;

  • Ref.

  • J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961), 529-578

  • L. Rosenfeld, Nucl. Phys. 26 (1961), 594-607.

  • J. Humblet, Nucl. Phys. 31 (1962), 544-549.

  • J. Humblet, Nucl. Phys. 50 (1964), 1-16.

  • J. Humblet, Nucl. Phys. 57 (1964), 386-401.

  • J.P. Jeukenne, Nucl. Phys. 58 (1964), 1-9

  • J. Humblet, Nucl. Phys. A151 (1970), 225-242.

  • 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)

    Multi-Riemann sheet

    Eth(C3)  Eth(C2) Eth(C2) Eth(3)





    (3) N-Body problem; more complex

    Eigenvalues of H(q) in the complex energy plane

    Complex scaling

    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

    Resonance state

    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

    • Complex scaling transformation

    • Complex Scaled Schoedinger Equation

    • 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) cq is an L2-class function:

    • ii) Eq is independent on q ( )