Quantum Scattering 709

CHAPTER 39. QUANTUM SCATTERING 709

the scattering of a point-particle of (reduced) mass m by a short-range potential V(q), excluding inter alia the Coulomb potential. (The Coulomb potential decays slowly as a function of q so that various asymptotic approximations which apply to general potentials fail for it.) Although we can choose the spatial coordinate frame freely, it is advisable to place its origin somewhere near the geometrical center of the potential. The scattering problem is solved, if a scattering solution Chapter 39 to the time-independent Schr¨odinger equation (36.2) ~2 ∂2 + V(q) φ (q) = Eφ (q) (39.1) −2m ∂q2 ! ~k ~k Quantum scattering can be constructed. Here E is the energy, ~p = ~~k the initial momentum of the particle, and ~k the corresponding wave vector.

When the argument r = q of the wave function is large compared to the typ- | | ical size a of the scattering region, the Schrödinger equation effectively becomes Scattering is easier than gathering. a free particle equation because of the short-range nature of the potential. In the —Irish proverb asymptotic domain r a, the solution φ~(q) of (39.1) can be written as superpo- ≫ k sition of ingoing and outgoing solutions of the free particle Schrödinger equation (A. Wirzba, P. Cvitanovićand N. Whelan) for fixed angular momentum:

o far the trace formulas have been derived assuming that the system under φ(q) = Aφ( )(q) + Bφ(+)(q) , (+ boundary conditions) , consideration is bound. As we shall now see, we are in luck - the semiclas- − S sics of bound systems is all we need to understand the semiclassics for open, ( ) (+) where in 1-dimensional problems φ − (q), φ (q) are the “left,” “right” moving scattering systems as well. We start by a brief review of the quantum theory of plane waves, and in higher-dimensional scattering problems the “incoming,” “out- elastic scattering of a point particle from a (repulsive) potential, and then develop going” radial waves, with the constant matrices A, B fixed by the boundary con- the connection to the standard Gutzwiller theory for bound systems. We do this ditions. What are the boundary conditions? The scatterer can modify only the in two steps - first, a heuristic derivation which helps us understand in what sense outgoing waves (see figure 39.1), since the incoming ones, by definition, have yet density of states is “density,” and then we sketch a general derivation of the cen- to encounter the scattering region. This defines the quantum mechanical scattering tral result of the spectral theory of quantum scattering, the Krein-Friedel-Lloyd matrix, or the S matrix formula. The end result is that we establish a connection between the scattering resonances (both positions and widths) of an open quantum system and the = ( ) + (+) φm(r) φm− (r) S mm′ φm (r) . (39.2) poles of the trace of the Green’s function, which we learned to analyze in earlier ′ chapters. All scattering effects are incorporated in the deviation of S from the unit matrix, the transition matrix T

S = 1 iT . (39.3) 39.1 Density of states − For concreteness, we have specialized to two dimensions, although the final formula is true for arbitrary dimensions. The indices m and m′ are the angular mo- For a scattering problem the density of states (35.16) appear ill defined since for- menta quantum numbers for the incoming and outgoing state of the scattering mulas such as (38.6) involve integration over infinite spatial extent. What we will wave function, labeling the S -matrix elements S mm . More generally, given a set now show is that a quantity that makes sense physically is the difference of two ′ of quantum numbers β, γ, the S matrix is a collection S βγ of transition amplitudes densities - the first with the scatterer present and the second with the scatterer 2 β γ normalized such that S βγ is the probability of the β γ transition. The → | | → absent. total probability that the ingoing state β ends up in some outgoing state must add up to unity In non-relativistic dynamics the relative motion can be separated from the center-of-mass motion. Therefore the elastic scattering of two particles can be 2 S βγ = 1 , (39.4) treated as the scattering of one particle from a static potential V(q). We will study Xγ | |

708 scattering - 29dec2004 ChaosBook.org version15.9, Jun 24 2017 b 711 radial (39.8) (39.7) . This ) does 1 k d ( − m − )) δ - k ( ′ m nd side we al- δ + b ill play no role in values so that . ding order we can on that it vanish at R se of the density of ( m π scattering phase shift, 12) acting from the result 39.5) while the second ChaosBook.org version15.9, Jun 24 2017 erence of Green’s func- ≈ + ﬀ 1 by Taylor expanding. is k n m ( ∆ + . S π . We can now take the limit 4 is an annoying phase factor ! n † + S π/ log π dk dS + = . † d ≈ 2 dk ) 1 S )

− m k π/ ( m χ S ), for the same size container, but this 0 X m Tr k i i − ( G = 0 π π ) 1 1 n d 2 2 − m . k ) . However, not all is well: the area under ( χ ) k so that R Weyl term (38.8), appropriate to 1 ′ m ( / = = δ 12 d S G ) n + ) − k tr ( k ( n ( − ( . ′ m erence” of two bounded refer- 1 π δ Im ff ) + n k k = erence in the eigenvalues of two consecutive states of ( , the dominant behavior is given by the size of the circular k m π 1 ( ff X ′ m R m 2 δ χ + 1 π − The “di ) + dependence has been cancelled away. − n = = ) k is large, the density of states is high, and the phase R R ) ) ( n k k k m angular momentum sbuspace R ( ( ( 39.2: 1 π erence of the two equations we obtain δ 0 0 ect of the change of the phase on state m ff m d d ff δ + ≈ 39. QUANTUM SCATTERING − − ) R + is a common prefactor, and k ) ) 1 ( ence systems, one with andsystem. one without the scattering Figure k k R + since the ( ( . Since m n n term is essentially the 1 ··· d k k d d m . This implies the quantization condition R R The state (39.6) must satisfy the external boundary conditi This is essentially what we want to prove since for the left ha →∞ = scattering - 29dec2004 enclosure with a correction in terms of the derivative of the approximation accurate to order 1 quantization. For large We now ask forfixed the di from the asymptotic expansionwhat of follows. the Hankel functions that w r time without any scatterer, figure 39.2. We also sum over all Taking the di tions, where consideration tends to infinity.from We the regularize free this particle by density subtr of states The The first line follows from the definitionline of follows the phase from shifts the ( unitarity of not change much over such a small interval. Therefore, to lea include the e is the eigenvalue spacing whichstates we within now interpret as the inver ready have the semiclassical theory for the trace of the di CHAPTER R . m 710 (39.6) (39.5) (b) . We will ′ mm its asymptotic problem; free S th angular mo- r m ite cylindrical well me being that a ra- s to some value of re phases. This means the equa- Schrödinger al formula will be true ), and given a potential olution, given in terms ChaosBook.org version15.9, Jun 24 2017 lizing a large matrix in kr ( ) ± ( m is diagonal in the radial basis , the scattering matrix cannot | H S q | ∝ ) ) = gets scattered into an outgoing state q kr ( r θ ) are incoming and outgoing Hankel ( ) ) z , ± im ( − m ( θ , ) ( m e for the action. The . φ ) ∓ ) im ( m ), other than that for large 1 H m e . Angular momentum is still conserved so kr ) kr H χ kr ( = + r ) ( ( ) ) † − − t (a) = ( m φ ( m ) kr , H → ∞ k H SS ( ~ = ) is not radially symmetric and (39.1) has to be ) ( ) / q + R ) m = ) ( m ( kr , where δ q ( E θ S ) V H ( , † ) + + outgo- m im ( 0 0 k S e , φ ( ) H r kr m ( 2 0 kr S ~ ( im S . ) 2 ) cos( θ + k ( m − ( im m H e δ = ) i each eigenstate is of the asymptotically free form 2 k ) spherical waves run- ( e ikr a E ≈ ··· ≈ m ± , = e

39. QUANTUM SCATTERING S 0 ≫ ) , and will later take ) , ∝ k r r r R ( ( ( matrix is unitary: ± 0 erent angular momentum eigenstates, and m m Incoming S ff H φ S G (a) In general, the potential We have already encountered a solution to the 2-dimensional ) we can in principle compute the infinity of matrix elements q ( of the form functions respectively. We now embedof the radius scatterer in a infin that each eigenstate of this (now bound) problem correspond scattering - 29dec2004 For large (39.2) with matrix elements given by mix di The matrix is unitary so inthat a an diagonal incoming basis state all of entries the form are pu solved numerically, by explicit integration,a or specific by basis. diagona To simplifydially things symmetric a scatterer bit, is we centeredfor assume at arbitrary for the asymmetric the origin; potentials. ti thetion Then fin (36.2) the are solutions separable, of form is particle propagation Green’s function (37.48)of is the Hankel a function radial s so the not need much information about where we have used V CHAPTER mentum eigenfunction is proportional to 39.1: spherical waves scattered from an obstacle. Figure ning into an obstacle.ing (b) Superposition of appendix A45 ) ht E ( 713 d unitary (39.14) (39.13) ) can be operator, ) and the E E ( hich is the ( ± d erent classes Q ff + sults of the last n scattering matrix, attering system is a bounded system E nsity th a discrete spec- ing contribution of ttering problems is

hermitian the latter is inserted in the semiclassical the Gutzwiller trace s which will be pre- rators ǫ vides the connection our scattering system i ChaosBook.org version15.9, Jun 24 2017 + loyd sum for the spectral 1 H − i n E E | − ǫ i ǫ i + ǫ − i 1 H 1 H + − n − 1 E E | E spectrum, whereas the spectral density

n − n E E h E scattering wave functions which are not normal- * Im Im erence between these two types of equations is that 0 ﬀ 1 1 π π → lim ǫ 0 0 ) i → → continuous E lim lim ǫ ǫ π 1 − outgoing 2 − − n E , whereas the scattering system is characterized by a . We will see that (39.12) provides the clue. Note that the rig ( = = = δ S H ) n -matrix. How can we reconcile these completely di X E S

39. QUANTUM SCATTERING ≡ − ) n E E ( ( δ d matrix refers to Therefore, the task of constructing the semiclassics of a sc In chapter 37 we linked the spectral density (see (35.16)) of We can now use this result to derive the Krein-Lloyd formula w S completed, if we can find a connection between the spectral de to the trace ofapproximation, the the Green’s function trace of (38.1.1). theformula Green’s (38.11) function in Furthermore, terms is of given aperiodic by smooth orbits. Weyl term and an oscillat scattering matrix via the identity hand side of (39.12) has nearly the structure of (39.14) when into (39.13). The principal di the izable and which have a The manipulations leading to (39.12) are justified if the ope implicit in all of thesented trace in formulas the for subsequent open chapters. quantum system 39.3 Krein-Friedel-Lloyd formula The link between quantum mechanics and semiclassics for sca section and extends the discussion of the opening section. central result of thisbetween chapter. the The trace Krein-Lloyd of formula the pro Green’s function and the poles of the provided by the semiclassical limit of thedensity Krein-Friedel-L which we now derive. This derivation builds on the re linked to trace-class operators. the Hamiltonian refers to a bound systemtrum. with normalizable Furthermore, wave the functions bound wi system is characterized by a operator, the of wave functions, operators and spectra? The trick is to put CHAPTER scattering - 29dec2004 . γ ng 712 (39.9) (39.12) (39.11) (39.10) . number of re rigorous # ) , te labeled ǫ nergy and not V in terms of the 1 ree particle state − † + ) ↔− ǫ Ω ion (more formally i matrix. is the time-ordering ǫ lies even when there − ( interact via forces of ± S T onnection between the − ChaosBook.org version15.9, Jun 24 2017 E ǫ = Ω i − S − 0 rather than the wavenumber as the natural variable in the H 1 E ( E k − + is the free Hamiltonian. In the matrix has the following spectral cult since one cannot appeal to the H 1 0 ~ / S ffi t H = − 0 ) ǫ iH i E e ( ) ~ − ± / , E Q E 1 ! iHt − ) 2 − t − 0 ( 0 ′ e H ~ -matrix is defined as H ( / , , t δ " S ) 0 , we have adapted ) dtH ~ k E / iH E t ( Tr ∞ 0 ( e 1 + − − = iH −∞ is the interaction Hamiltonian and − -matrix is given by # Q →∞ Z 0 lim ) e ) t S i dE S ~ E H E / − = ( ( ∞

− + S 0 iHt − e Q Z H Ω erence of the Green’s function with and without the scatteri d exp . However, as the asymptotic dynamics is free wave dynamics dE † + but must work directly with a non-diagonal ﬀ = k ) T = = m lim E →±∞ V δ t

39. QUANTUM SCATTERING ( † ) = is the full Hamiltonian, whereas = = Ω = S S , and in the distant future they will likewise be free, in a sta E ′ " ± ( β H H S S Ω Tr ciently short range, so that in the remote past they were in a f There are a number of generalizations. This can be done in any Finally, we state without proof that the relation (39.7) app ffi , since this is the more usual exposition. Suppose particles scattering - 29dec2004 such that representation where operator. In stationary scattering theory the labeled by the wavenumber 39.2 Quantum mechanical scattering matrix The results of the previousscattering section indicate matrix that and there the isbetween a trace c the of the di quantum Green’s funct dimensions. It is alsowave more number common to do this as a function of e interaction picture the where k above discussion. is no circular symmetry. The proof is more di center.) We now showmanner. how We will this also connection work in can terms be of derived the in energy a mo phase shifts Møller operators labeled su CHAPTER In the Heisenberg picture the exercise 39.1 η i 715 ) and R (39.18) ; which fi- η i R + . . The periodic ) E . These expres- ( ′ ous periodic or- E d ( η arger than of the . i container. In other ves those spurious riodic orbits. As in e S -axis. In particular, n the interpretation R →∞ . Thus the order of + ver. Second, if the ircase or even delta raction of the empty k Krein-Friedel-Lloyd relation between the f k re continuous to start e scattering region or ernal wall would still R e confining boundary. cattering problems the ormulas the energy ar- ChaosBook.org version15.9, Jun 24 2017 -matrix: ntial is a dispersing bil- imation: Without the n – in addition to the re- ln det S ), we expect an excluded l approximation, they are i π 1 2 = ) R . It replaces the scattering problem by ; η i + ′ E η ( ) R spectral densities in the presence or in the ab- ( (0) L N − , if they were not exponentially suppressed by the e prescription can also be understood by purely phe- 1 is an essential precondition on the suppression of − k must be greater than zero. Otherwise, the exact left ) of a spurious periodic orbit grows linearly with the ) ) η R ′ i R ( R ≫ ( η + ; iL → ∞ ′ L η e smoothed η i R R + = ) E η ( i + N E ( can be replaced by the wavenumber argument m ) are the 2 Krein-Friedel-Lloyd formula η →∞ R

39. QUANTUM SCATTERING lim i √ R ; ) 0 + η R i ( + . The bound E iL erence of two bounded reference billiards of the same radius → lim R + e η ff E is chosen to be real, term because of their Finally, the semiclassical approximation can also help us i The necessity of the ( k η i (0) This is the of the Weyl term contributions forWeyl scattering term problems. appears with In a s negative sign. The reason is the subt liard system (or a finitevolume collection (or of the dispersing sum billiards of excludedwords, volumes) relative the to the Weyl empty termfilled contribution one of and the therefore empty a container negative is net l contribution isscattering - left 29dec2004 o container from the container with the potential. If the pote radius behavior. As the length the unwanted spurious contributions of the container if the formulas (39.17) and (39.18) are evaluated at a finite value o Furthermore, in both versions ofgument the Krein-Friedel-Lloyd f the di sions only make sense for wavenumbers on or above the real + orbits that encounter bothsurvive the the scattering first region limit and the ext potentials, whereas the other does not (see figure 39.2). Her nally will be taken to infinity. The first billiard contains th replaced by a Weyl term (38.10) and an oscillating sum over pe smoothed staircase functions and the determinant of the sence of the scatterers, respectively. In the semiclassica (38.2), the trace formula (39.17) can be integrated to give a d with, since they can be expressed by continuous phase shifts moval of the divergent Weyl term contributions in the limit hand sides (39.18) andfunction (39.17) sums, would respectively, give whereas discontinuous the sta right hand sides a the two limits in (39.18) and (39.17) is essential. term there is nobits reason which why are one there should solelyThe because be subtraction of able the of to introduction the neglect ofperiodic second th spuri orbits (empty) which reference never encounter system the remo scattering regio if CHAPTER nomenological considerations in the semiclassical approx ) + V in π has 714 E (4 R ( / → ∞ d 2 (39.15) (39.16) (39.17) we will R R # π ) → ∞ f the spec- ) over dis- E ne and the . ( R ) R ; S em. Our scat- E . → ∞ ) is subtracted ( E densities d ( 0 can be carried R 0 is allowed. In S ectrum. Still the ) dE al conatiner with replace the spec- d R ) ticity, is broken in η why the Coulomb mplementing this. between incoming easing radius i → 1. y E it → ( η + † η as a delta distribution, ln det ≫ t ) , the limit boundary conditions at ChaosBook.org version15.9, Jun 24 2017 S " R η d spectral density with a ( 1 d n dE √ quadratically, as Tr E i i R R ). By the introduction of the size (radius π π , but then it will turn out (see 1 1 − R 2 2 R ; E η of the box, we have mapped our i = − = same + R ) of the empty reference system and η : i R E ) ( ; ) E E − d η R ( i ) ; − S R η + , only then the limit i ( erence. Furthermore, in the limit 1 = ln n η E + ff ) ( E E R d ( (0) dE ; − d η (0) i the time-dependent behavior of the wave function E Tr d i ) belonging to our box with the enclosed potential ( − η in the energy π − 1 value of R 2 E ) η ; n ( X R E ). It is therefore important that our scattering potential d = ( ; π ¯ R η d ( , 1 i 2 finite n i ) + E R E ) will be unaltered within the box, whereas at the box walls we ≡ . drop out of the di ; ( r ~ ( ) η d 0 i R V ; = + imaginary part η i R →∞ E → ∞

39. QUANTUM SCATTERING lim = R ( r + | 0 d ) R + and with its center at the center of our finite scattering syst in the 3-dimensional case. This problem can easily be cured i finite r E ~ ( ( → 3 lim ectively been altered from an oscillating one to a decaying o R d φ η matrix of the corresponding scattering system: R ff ) of the boxed potential and Let us summarize the relation between the smoothed spectral S R ; η scattering system into a bound system with a spectral densit the energy-eigenfunctions of the box aresimilarly only to normalizable a plane wave.problem So remains we that seem the toand wave recover outgoing functions a waves, continous do whereas sp not thisthe discriminate symmetry, namely scattering the problem. hermi Thetral last density over problem discrete can delta be distributionssmall by finite tackled a imaginary if smoothe part we the limit In this way, for any finite value of the radius recover the scattering system.The smooth But Weyl term some care is required in i was chosen to be short-rangedpotential to requires start special with. care.) (Which explains The hope is that in the lim crete eigenenergies (see figure 39.2). Then all the divergences linked to the incr out, respecting the order of the limiting procedures. First diverges for a spherical 2-dimensional box of radius Note that positive has e or as will choose an infinitely high potential, with the Dirichlet the outside of the box: tering potential scattering - 29dec2004 tral density of an empty reference box of the i practice, one can try to work with a finite value of hermiticity of the Hamiltonian is removed. Finally the limi below) that the scattering system is only recovered if into a finite box asradius in the opening section. We choose a spheric the CHAPTER to be performed for a 1, 717 = valid g 1 v ~ · = s ~ ii † ) S · S th. The proba- ∂ ∂ω S i · will therefore be † x ~ S for which i ead of the general the- ade for wave packets incoming plane wave ore corresponds to the − ChaosBook.org version15.9, Jun 24 2017 /ω tric cavity. Have a look at k ~ cient”), or Beth and Uhlen- edel-Lloyd formula (39.17) r P. Scherer’s thesis [39.15] Re ( ffi = tion to the density of states are = s ~ ) i j S ∂ω ∂ The third volume of Thirring [39.1], and therefore the average delay for ∗ i j 2 | S i j ! j S , of the unitarity relation i j | of the phase X Θ ∂ω i ∂ −

∂/∂ω Re goes as Re ( is , j i = slowness = ω, ii ! ! d i j t i j S g is parallel to the group velocity (consistent with the x ∆ can similarly be found as S v ∂ ∂ω ∂ω 2 | ∂ x . ~ · 1 i j = ) † − i j S | . S ω ω th scattering channel after an injection in the (

) j i S i j ∂ω Θ ω x d ~ − ( − X ∂

· ∂ω Θ s ~ + time delay Krein-Friedel-Lloyd formula. ∂ . The arrival of the wave packet at the position = = Re k ~ = 39. QUANTUM SCATTERING = g x = x ~ v i ) is the group velocity to get i · t 39.1 i j ω t g = k ( ~ ∆ v ~ ∆ d t h τ It helps to start with a toy example or simpliﬁed example inst To show this we introduce the If the scattering matrix is not diagonal, one interprets above). Hence the arrival time becomes where we have used the derivative, where since we may assume as the delay in the for real frequencies. Thisrelated discussion to can partial in waves particular andcorresponding be superpositions to m of free motion. these The like totalsum an Wigner over delay all theref channel delays (39.19). Commentary Remark wave vector the incoming states in channel delayed. This sections 3.6.14 (Levison Theorem) and 3.6.15 (the proof), o (appendix) discusses the Levison Theorem. orem, namely for the radiallythe symmetric book of potential K. in Huang, a chapter symme 10 (on the ”second virial coe beck [39.5], or Friedel [39.7].particular These cases results of for the the correc Krein formula [39.3]. The Krein-Fri scattering - 29dec2004 CHAPTER bility for appearing in channel 716 (39.19) ttractive re- attering system. y the principle of . Thus the packet is t with respect to the dy a plane wave: t -overlapping scatter- e corresponding Weyl ) deﬁned as f the single scatterers, ChaosBook.org version15.9, Jun 24 2017 k t will occur if scattering ( elucidate the connection such formulas for scatter- Θ phase shift . ! Θ ) ) d k k ( )) ( S k + S ( dk d t S ) . k ω ( d † , . ) S log det ( − k + Θ

in the density formula (39.17) is dimensionally time. This ( k x ). ~ ~ Θ Θ d Argdet ( t ∂ k tr · i S dk ∂ ( i i + k ω d ~ e dk Θ − − − d − d = dk t = x ~ ln det ) = = = k · ~ =

39. QUANTUM SCATTERING k φ ( ∂ d ∂ω k ) d ~ dE S ) k ( k = = = ( d x ~ det d φ 0 A related quantity is the total scattering The time delay may be both positive and negative, reflecting a The phase of a wave packet will have the form: so that and can be shown to equalWe now the review total this delay fact. of a wave packet in a sc ing systems, the Wigner delay, defined as suggests another, physically important interpretation of spectively repulsive features of the scatteringbetween system. the To scattering determinant and the time delay we stu The first term is just the group velocity times the given time Here the term in the parenthesis refersis to present. the phase shift tha Thestationary center phase: of the wave packet will be determined b Hence the packet is located at scattering potential is a collectioning of regions, the a Krein-Friedel-Lloyd finite formulas numbercontributions show of are that non completely th independent ofas the long as position these o do not overlap. 39.4 Wigner time delay The term CHAPTER retarded by a length given by the derivative of thescattering - phase 29dec2004 shif , 719 Ann. , 287 C 10 , 7 , Ph.D. A course in , Mat. Sborn. , J. Phys. illations orbit it finds. ich 2554, ISSN eres. II. The den- , 707 (1962). 3 on of the scattering , 207 (1967). , 2242 (1989). ChaosBook.org version15.9, Jun 24 2017 in condensed materials, ulations,” 90 90 , 592 (1971) (North-Holland, Amsterdam, 63 I , 740 (1962); M.Sh. Birman and Nuovo Cim. Ser. 10 Suppl. , 915 (1937). 3 4 , 833 (1993). , 514 (1974). , Sov. Math.-Dokl. 4 On the Theory of Wave Operators and 85 J. Chem. Phys. (John Wiley & Sons, New York (1987). Proc. Phys. Soc. Physica (Springer, New York, 1979). (Springer, Wien Distribution of eigenfrequencies for the wave Solution of the Equation Schrödinger in Terms Upgrade your pinball sim- Ann. Phys. (N.Y.) 3 , 153 (1952); 43 Perturbation Determinants and Formula for Traces Vol. , Sov. Math.-Dokl. Quantum Mechanics, Vol. Quantum mechanics of atoms and molecules On the Trace Formula in Perturbation Theory ,76 (1972). unezs¨neeinesQuantenzustände klassisch chaotischen Billards , 69 (1972), and references therein. St. Petersburg Math. J. Phil. Mag. Statistical Mechanics 69 21 , 597 (1953) ; 33 ulator so that it computes the topological index for each Pinball topological index. Adv. Phys. from a classically chaotic repeller, D.R. Yafaev, Phys. (N.Y.) equation in a finite domain: III. Eigenfrequency density osc Scattering Operators 1961). (N.S.) of Unitary and Self-adjoint Operators of Classical Paths Ann. Phys. (NY) thesis, Univ. Köln (Berichte0366-0885, des Nov. Jülich, 1991). Forschungszentrums Jül sity of single-particle eigenstates, 4661 (1977). mathematical physics 1979). (1958). 39.4. [39.10] R. Balian and C. Bloch, [39.9] P. Lloyd and P.V.Smith, Multiple-scattering theory [39.11] R. Balian and C. Bloch, [39.12] R. Balian and C. Bloch, [39.13] J.S. Faulkner, “Scattering theory and cluster calc [39.3] M.G. Krein, [39.5] E. Beth and G.E. Uhlenbeck, refsScatter - 11aug2005 [39.14] P. Gaspard and S.A. Rice, Semiclassical quantizati [39.15] P. Scherer, References [39.1] W. Thirring, [39.2] A. Messiah, [39.4] M.Sh. Birman and M.G. Krein, [39.6] K. Huang, [39.8] P. Lloyd, Wave propagation through an assembly of sph REFERENCES [39.7] J. Friedel, , 0. ) = kR ) ( k n ( + (1) m M is the Han- H n for the deter- (1) n 1) H k − ( to approximate the + a A ) + kR ( I n − , 79 (1992)) = (1) m 2 H M ) ) ka Chaos ka ( ( R (1) m n J th Bessel function and H n n 718 1) 2 − ( is the channels + n n J , N ] scattering )] weighted m δ N 2 = a /∂ω × n (Hints: note the structure where determinant; or read kel function of theterminant first closest kind. to the Find origin the by zeros solving of det the de- minant of the matrix The full quantum mechanicalcan be version solved of by this finding problem the zeros in ect [39.16]. Page , ab N m ff S ∂ over all 2 M re )( he sum over all ouput aa ab .4] but beware, they are Q ChaosBook.org version15.9, Jun 24 2017 S is the [2 / essor of the more general i he S [39.11, 39.14, 39.15, 39.17, − tering region by a wave packet ed one and therefore a negative iption of scattering. The diagonal Re[( , where = Derive the For a discussion of why the Weyl term ab t /∂ω ∆ S 0, container size ∂ 1 − (Andreas Wirzba) → S Compute the quasi- i η Wigner time delay and the Wigner-Smith time − , which is the trace of the lifetime matrix and is 1 = aa + j Q Q        l a 2 + P p j p t Λ = − W τ 1        l , j , p Y Draw examples for the three types of period = ) ε Weyl term for empty container. Wigner time delay. ( ? of the lifetime matrix Z aa Q 39.2 39.3 →∞ Spurious orbits under thetruction. Krein-Friedel-Lloyd con- orbits under the Krein-Friedel-Lloydthe construction: genuine periodic (a) orbitsspurious of the periodic scattering orbits region, whichsubtraction (b) can of be the removed referenceodic by orbits system, which the cannot (c) be removed spuriousWhat by is this peri- the subtraction. role of the double limit for the two disk problem. Use the geometry b The one-disk scattering wave function. one-disk scattering wave function. Quantum two-disk scattering. classical spectral determinant Also, have a look at pages 15-18 of Wirzba’s talk on the Casimi 39.1. 39.2. 39.3. exerScatter - 11feb2002 Exercises is the Wigner time delay matrix, are interpreted in terms ofincident the time in spent one in the channel. scat channels (both in reflection As and transmission) shown of by Smith [39.26], they are t proportional to the density of states. was derived in refs.39.18]. [39.3, 39.7, The 39.8, original 39.9],mathematicans. papers are see by also Krein refs. and Birman [39.3, 39 by the probabilityof emergingfrom that channel. The sum of t 16 discusses the Beth-UhlenbeckKrein formula formula for [39.5], spherical the cases. predec Remark contribution of the empty container is largernet than contribution of is the left fill over, see ref. [39.15]. Remark EXERCISES delay matrix, are powerful concepts for aelements statistical descr 721 1989). 994). j v and H.-J. Som- ChaosBook.org version15.9, Jun 24 2017 mers, J. Math. Phys. 38, 1918 (1997). [39.35] R. Landauer and Th. Martin, Rev. Mod.[39.36] Phys. 66, E. 217 H. (1 Hauge and J. A. Støveng, Rev. Mod. Phys. 61, 917 ( refsScatter - 11aug2005 References [39.34] For a complete and insightful review see Y. V. Fyodor 720 (Springer, 9712015. / s. Rev. Lett. ett. 77, 3005 U. Smilansky in the limit of r interactions, nal in channel ion of the local , Course CXIX, ur die Beugung 1992). , 374 (1955). uantum Dynamics, em: Sinai’s Billiard me Dtab is defined in ev. A 54, 4022 (1996). orbit approximation in 10a ChaosBook.org version15.9, Jun 24 2017 , 83 (1918). , (Dover, New York, 1964). , 774 (1968); L. Schulman, 1 A 95 , 163 (1981). 131 , 77 (1992). Handbook of Mathematical Functions 2 Z. Naturforschung openfull.ps.gz overheads, 2003). / , 1-116 ( 1999); arXiv:chao-dyn , 705 (1954). , 145 (1955). , 349 (1960). CHAOS 309 9a 98 118 Wirzba / Ann. Phys. (N.Y.) Proc. Roy. Soc. London Ser. ¨ Uber die Greenschen Funktionen des Zylinders und der Phys. Rev. projects / Quantum Mechanics and Semiclassics of Hyperbolic n-Disk , 1558 (1968). Phys. Rev. e and I. C. Percival, J. Phys. B Theorie der Beugung Elektromagnetischer Wellen ff Physics Reports 176 , Z. Naturforschung Proceedings of the Int. School of Physics “Enrico Fermi” 78, 4737 (1997). ref. [39.26] as theb appearance after of the the injection of peak a indelay wave the time packet outgoing tab in sig channel in a. Eq. Ournarrow (1) definit bandwidth coincides pulses, with as the shown definition in of Eq. Dtab (3). (1996). einer ebenen Welle am Zylinder,” and the KKR Method,” Scattering the 2-and 3-disk problems,” Phys. Rev. Kugel,” Berlin 1957); “ with Formulas, Graphs and Mathematical Tables Varena, 23 July - 2 August 1991,(North-Holland, eds Amsterdam, G. Casati, 1993). I. Guarneri and in: ChaosBook.org [39.30] Following the thesis of Eisenbud, the local delay ti [39.31] E. Doron and U. Smilansky, Phys. Rev.[39.32] Lett. 68, G. 1255 Iannaccone, ( Phys. Rev. B 51, 4727[39.33] (1995). V. Gasparian, T. Christen, and M. Phys. Büttiker, R refsScatter - 11aug2005 [39.29] P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, Phy [39.26] F.T. Smith, [39.25] E.P. Wigner, [39.23] A. Wirzba, “Validity of the semiclassical periodic [39.27] A. Wirzba, [39.28] V. A. Gopar, P. A. Mello, and M. Buttiker, Phys. Rev. L [39.20] G.N. Watson, [39.21] M. Abramowitz and I.A. Stegun, [39.22] W. Franz and R. Galle, “Semiasymptotische Reihen f¨ [39.24] M.V. Berry, “Quantizing a Classically Ergodic Syst [39.19] W. Franz, [39.18] A. Norcli [39.16] A. Wirzba, A force from nothing into nothing: Casimi References [39.17] P. Gaspard, Scattering Resonances: Classical and Q