Quantum theory of scattering by a potential. Lecture notes 8 (based on CT, Sec4on 8)

Quantum theory of scattering by a potential Lecture notes 8 (based on CT, Sec4on 8) ì

Introduction Ø In physics, the fundamental interac4ons between par4cles are o?en studied by le@ng these par4cles collide with each other Ø A beam of par4cles (1) is directed onto a target composed of par4cles (2) Ø The proper4es of the par4cles in the final state of the system are studied

Introduction Ø If the par4cles (1) and (2) are composed of more elementary components, these can rearrange to give rise to new par4cles in the final state (rearrangement collisions) Ø At high energies, part of the energy can materialize into new par4cles in the final state Ø The word scajering labels those reac4ons in which the final and ini4al states are composed of the same par4cles (1) and (2)

Scattering by a potential Ø We will consider the elas4c scajering of the incident par4cles (1) by the target par4cles (2) Ø We will work under the following hypotheses: 1. We neglect the spin of par4cles (1) and (2) 2. We neglect the internal structure of par4cles (1) and (2) 3. We neglect mul4ple scajering processes (thin target hypothesis) 4. We neglect possible coherence between the waves scajered by the different par4cles in the target: we consider the scajering of one par4cle in the beam with one par4cle in the target: the flux of detected par4cles is the sum of fluxes scajered by each of the N target par4cles, namely N 4mes the flux scajered by any of them 5. We describe the interac4on through a poten4al V(r 1 - r 2 ): in the center of mass we study the scajering of par4cle of mass μ=m 1 m 2 /(m 1 +m 2 ) by the poten4al V(r)

Definition of scattering cross section Ø Consider a beam of incident par4cles of mass μ along z Ø The poten4al V(r) is localized around the origin O of the coordinate system (center of mass of the par4cles (1) and (2)

Definition of scattering cross section Ø F i is the flux of par4cles in the incident beam: F i = number of par4cles per unit 4me which traverse a unit surface perpendicular to z in the region where z has very nega4ve values Ø We put a detector far from the origin, in the direc4on fixed by the polar angles θ and φ, with an opening facing O and subtending the solid angle dω Ø We count the number dn of par4cles scajered per unit 4me into this solid angle dω Ø dn is propor4onal to F i and dω: dn=f i σ(θ,φ)dω

Definition of scattering cross section Ø σ(θ,φ) is called differen4al scajering cross sec4on Ø It is o?en measured in barns: 1 barn =10-24 cm 2 because this is usually the order of magnitude of the sec4on of a heavy nucleus Ø Interpreta4on: The number of par4cles per unit 4me which reach the detector is equal to the number of par4cles which would cross a surface σ(θ,φ)dω placed perpendicular to z in the incident beam Ø The total scajering cross sec4on is defined as: Ø The detector is placed outside the trajectory of the incident beam: it receives only the scajered par4cles

Stationary scattering states Ø We need to study the 4me evolu4on of the wave packet represen4ng the state of the par4cle Ø The proper4es of this wave packet are known for large nega4ve values of the 4me t, when the par4cle is in the region z<0, not yet affected by the poten4al V(r) Ø The evolu4on of the wave packet can be immediately obtained, if we can express it as a superposi4on of sta4onary states v We need to study the eigenvalue equa4on of the Hamiltonian: with

Stationary scattering states Ø Schroedinger s equa4on describing the evolu4on of the par4cle in a poten4al V(r) is sa4sfied by solu4ons associated with a well- defined energy E (sta4onary states): where is a solu4on of the eigenvalue equa4on: We assume that the poten4al decreases faster than 1/r as rà We consider solu4ons associated with a posi4ve energy E, which is the kine4c energy of the incident par4cle before it reaches the zone of influence of the poten4al: E=P 2 /2μ

Stationary scattering states Ø We define from which we can write the eigenvalue equa4on as: Ø For each value of k, this equa4on can be sa4sfied by infinitely many solu4ons (the eigenvalues of the Hamiltonian are infinitely degenerate) Ø We need to choose the solu4ons corresponding to the physical problem being studied

Asymptotic form of stationary scattering states Ø The sta4onary scajering states are the solu4ons of the previous eigenvalue equa4on, which sa4sfy the condi4ons needed to describe a scajering process Ø For large, nega4ve values of t the incident par4cle is free à its state is represented by a plane wave packet Ø The sta4onary wave func4ons we are looking for, must contain a term of the form e ikz, where k is the constant which appears in the eigenvalue equa4on Ø In the region of influence of V(r), the packet is deeply modified Ø For large posi4ve t, the packet is split into a transmijed wave packet which propagates in the posi4ve z direc4on (having the form e ikz ) and a scajered wave packet

Asymptotic form of stationary scattering states Ø The wave func4on represen4ng the sta4onary scajering state associated with the energy is the superposi4on of the plane wave e ikz and a scajered wave Ø The structure of the scajered wave depends on the poten4al V(r) Ø Its asympto4c form is simple: 1. In a given direc4on (θ,φ), its radial dependence is of the form: e ikr /r: it is an outgoing wave which has the same energy as the incident wave. The factor 1/r results from the fact that there are 3 spa4al dimensions:

Asymptotic form of stationary scattering states 2. Since scajering is not isotropic, the amplitude of the outgoing wave depends on the considered direc4on (θ,φ) Therefore, the wave func4on associated with the sta4onary scajering state has the following asympto4c behavior: In this expression, only the scajering amplitude V(r) depends on Ø The eigenvalue equa4on has only one solu4on, for each value of k, which sa4sfies the above condi4on

Asymptotic form of stationary scattering states Ø In order to study the evolu4on of the wave packet represen4ng the state of the incident par4cle, we need to expand it in terms of eigenstates of the total Hamiltonian H: we consider a wave func4on of the form: where the (real) func4on g(k) has a pronounced peak at k=k 0 Ø This func4on is a solu4on of Schroedinger s equa4on: it correctly describes the 4me evolu4on of the par4cle Ø We have to show that it sa4sfies the right boundary condi4ons

Ø Asympto4cally it behaves as: Asymptotic form of stationary scattering states Ø The posi4on of the maximum of each of these packets can be obtained from the sta4onary phase condi4on; a simple calcula4on gives, for the plane wave packet: with Ø For the scajered packet we have: is the deriva4ve with respect to k of the argument of

Asymptotic form of stationary scattering states Ø For large nega4ve t there is no scajered wave packet: only the plane wave packet exists at those 4mes, and it progresses towards the interac4on region with a group velocity v G Ø For large posi4ve t both packets are present: the first one moves along the posi4ve x axis, the second one diverges in all direc4ons: the asympto4c condi4on well describes the wave packet Ø The spa4al extension Δz is related to the momentum dispersion by

Asymptotic form of stationary scattering states Ø The wave packet moves at a velocity v G towards the origin and it takes a 4me to cross this zone (assuming that the size of the poten4al s zone of influence is negligible compared to Δz) Ø If we call t=0 the 4me at which the center of the incident wave packet reaches the point O, the scajered wave exists only for t>δt/2 Ø At t=0, the most distant part of the scajered wave packet is at a distance of the order of Δz/2 from O

Asymptotic form of stationary scattering states Ø We consider a different problem: a 4me- dependent poten4al obtained by mul4plying V(r) by a func4on f(t) which increases slowly from 0 to 1 between t=- ΔT/2 and t=0 Ø For t<<- ΔT/2, the poten4al is 0 and the state of the par4cle is described by a plane wave Ø At t=- ΔT/2 the plane wave is modified; at 4me t=0 the scajered wave looks like those studied before Ø There is a similarity between these two problems

Asymptotic form of stationary scattering states Ø If Δkà 0, the wave packet tends toward a sta4onary scajering state (g(k)à δ(k- k 0 )) Ø At the same 4me, ΔTà : the turning on of the poten4al becomes infinitely slow (adiaba4c) Ø Therefore, it is possible to describe a sta4onary scajering state as the result of adiaba4cally imposing a scajering poten4al on a free plane wave

Calculation of the cross section Ø To find the cross sec4on, we should study the scajering of an incident wave packet by the poten4al V(r) Ø Easier way: consider the sta4onary scajering states, which describe a probability fluid in steady flow and calculate the cross sec4on from the incident and scajered currents Ø We calculate the contribu4ons of the incident and scajered waves to the probability current in a sta4onary scajered state Ø The current J(r) associated with a wavefunc4on is:

Calculation of the cross section Ø The incident current can be obtained by replacing equa4on for J =e ikz in the Ø J i is directed in the posi4ve z direc4on and its modulus is Ø The scajered wave is expressed in spherical coordinates: we need to calculate the scajered current J d in this coordinate system:

Calculation of the cross section Ø By replacing = in the equa4on for J we get the scajered current in the asympto4c region: Ø Since r is large, the scajered current is prac4cally radial

Calculation of the cross section Ø The incident beam is composed of independent par4cles, all prepared in the same way. Sending a great number of these par4cles amounts to repea4ng the same experiment a great number of 4mes with one par4cle, whose state is always the same Ø If the state is the incident flux F i (number of par4cles of the incident beam which cross a unit surface perpendicular to z per unit 4me) is propor4onal to the flux of the vector J i across the surface:

Calculation of the cross section Ø The number dn of par4cles which strike the opening of the detector per unit 4me is propor4onal to the flux of the vector J d across the surface ds of this opening: Ø dn is independent of r, if r is sufficiently large Ø Replacing all of the above into the defini4on of the differen4al cross sec4on we get: Ø The differen4al cross sec4on is the square of the modulus of the scajering amplitude

Calculation of the cross section Ø We have neglected so far a contribu4on to the current associated with in the asympto4c region: the interference between the plane wave and the scajered wave Ø These interference terms do not appear when we are interested in scajering in direc4ons other than the forward one (θ=0) Ø The wave packet always has a finite lateral spread. A?er the collision we find two wave packets: a plane one and a scajering one moving away from O in all direc4ons Ø The transmijed wave results from the interference between these two packets Ø We place the detector outside the transmijed beam

Calculation of the cross section Ø Such interference cannot be neglected when studying scajering in the forward direc4on Ø The interference is destruc4ve: the transmijed wave packet must have a smaller amplitude than the incident one, because of conserva4on of total probability

Integral scattering equation Ø We now want to show that there exist sta4onary wave func4ons whose asympto4c behavior is of the form Ø We will introduce the integral scajering equa4on, whose solu4ons are exactly these sta4onary scajering state wave func4ons Ø Eigenvalue equa4on for H:

Integral scattering equation Ø Suppose that there exists a func4on G(r) such that: Ø G(r) is called Green s func4on of the operator Δ+k 2 Ø Any func4on which sa4sfies where is a solu4on of obeys the eigenvalue equa4on for H

Integral scattering equation Ø Indeed, if we apply the operator Δ+k 2 to both sides we get which yields: Ø Therefore, we can replace the differen4al eigenvalue equa4on for H by an integral equa4on: Ø Advantage: by choosing and G(r) one can incorporate into the equa4on the correct asympto4c behavior

Integral scattering equation Ø We start by considering Ø This means that (Δ+k 2 )G(r) is zero in any region which does not include the origin (which is the case when G(r)=e ikr /r) Ø Moreover, G(r) must behave like - 1/4πr when rà 0 Ø Indeed the func4ons sa4sfy Ø They are called outgoing and incoming Green s func4ons

Integral scattering equation Ø The desired asympto4c behavior leads us to choose and G(r)=G + (r) =e ikz Ø Indeed, the integral scajering equa4on can be wrijen as: whose solu4ons present the desired asympto4c behavior Ø To show this, we put ourselves at a point M which is very far from the points P of the zone of influence of the poten4al

Integral scattering equation Ø The angle between MO and MP is very small: Ø It follows that, for large r: Ø It follows that the asympto4c behavior of is:

Integral scattering equation Ø By se@ng we get the desired asympto4c behavior

The Born approximation Ø We can write the integral scajering equa4on as: Ø We try to solve it by itera4on Ø We change nota4on: rà r ; r à r and write:

The Born approximation Ø Inser4ng this into the original equa4on we get: Ø The first two terms on the r.h.s. are known, the third one contains the unknown func4on

Ø We can repeat the procedure: The Born approximation

The Born approximation Ø We are construc4ng the Born expansion of the sta4onary scajering wave func4on Ø Each term carries one extra power of the poten4al than the preceding one Ø If the poten4al is weak, each successive term is smaller than the preceding one Ø If we push the expansion far enough, we can get terms of known quan44es just in

The Born approximation Ø Replacing this expansion into the formula for the scajering amplitude, we can get the Born expansion of the scajering amplitude: where K is the scajering wave vector

The Born approximation Ø Therefore, in the Born approxima4on, the scajering cross sec4on is simply related to the Fourier transform of the poten4al Ø For a given θ and φ, the Born cross sec4on varies with k, the energy of the incident beam Ø For a given energy, the cross sec4on varies with θ and φ

Scattering by a central potential Ø In the special case of central poten4al, the orbital angular momentum L of the par4cle is a constant of the mo4on Ø There exist sta4onary states with well- defined angular momentum: eigenstates common to H, L 2 and L z Ø We call the wavefunc4on associated with these states par4al waves Ø The corresponding eigenvalues are

Scattering by a central potential Ø Their angular dependence is given by the spherical harmonics Ø The poten4al V(r) influences only their radial dependence Ø For large r, we expect the wavefunc4ons to be close to the common eigenfunc4ons of H 0, L 2 and L z, where H 0 is the free Hamiltonian Ø These are free spherical waves

Scattering by a central potential Ø We have seen in Quantum Mechanics 1 that these free spherical waves are where j l is a spherical Bessel func4on defined by: Ø The corresponding eigenvalues are

Scattering by a central potential Ø These spherical waves are orthonormal in the extended sense: Ø They form a basis in the state space:

Physical properties of free spherical waves Ø The angular dependence of the free spherical waves is en4rely given by the spherical harmonics Ø It is fixed by the eigenvalues of L 2 and L z, not by the energy Ø Let us consider an infinitesimal solid angle dω 0 about the direc4on Ø When the state of the par4cle is, the probability of finding the par4cle in this solid angle between r and r+dr is propor4onal to:

Physical properties of free spherical waves Ø It can be shown that, when ρ approaches zero, Ø This means that the probability behaves like r 2l+2 near the origin: the larger the l, the more slowly it increases

Physical properties of free spherical waves Ø remains small as long as Ø We may assume therefore that the probability is prac4cally zero for: Therefore a par4cle in the state is unaffected by what happens inside a sphere centered in O and with radius

Physical properties of free spherical waves Ø It can be shown that, for ρà, from which the asympto4c behavior follows: It is the superposi4on of an incoming wave e - ikr /r and an outgoing wave e ikr /r, whose amplitudes differ by a phase difference equal to lπ

Physical properties of free spherical waves Ø We have two dis4nct bases formed by eigenstates of H 0 : the { k>} basis associated with the plane waves and the associated with the free spherical waves Ø Any ket of one basis can be expanded in terms of vectors of the other one Ø In par4cular we consider { 0,0,k>} associated with a plane wave of wave vector k directed along z: Ø It represents a state of well- defined energy and momentum

Physical properties of free spherical waves is independent of φ: therefore, this state is eigenstate of L z with the eigenvalue 0: L z 0,0,k>=0 Ø Using the closure rela4on we can write: Ø Since 0,0,k> and are both eigenstates of H 0, they are orthogonal if k k : their scalar product is propor4onal to δ(k- k ) Ø There are both eigenstates of L z : their scalar product is propor4onal to δ m0

Physical properties of free spherical waves Ø Therefore we can write: Ø The coefficients can be calculated explicitly, so that we obtain: Ø A state of well- defined linear momentum is therefore formed by a superposi4on of states corresponding to all possible angular momenta Ø Analogously, we can write:

Partial waves in the potential V(r) Ø For any central poten4al V(r), the par4al waves wrijen as: can be where u k,l (r) is the solu4on of the radial equa4on: sa4sfying the condi4on:

Partial waves in the potential V(r) Ø This corresponds to a one- dimensional problem in which a par4cle of mass μ is under the influence of the poten4al Ø For large r, the radial equa4on reduces to: whose general solu4on is of the form

Partial waves in the potential V(r) Ø A and B cannot be arbitrary, due to the condi4on at r=0 Ø The asympto4c expression for u k,l (r) represents the superposi4on of an incident plane wave e - ikr and a reflected plane wave e ikr Ø There can be no transmijed wave, therefore the reflected and incident currents must be equal to each other: this implies that A = B Ø Therefore: Ø Or equivalently:

Partial waves in the potential V(r) Ø The real phase β l is determined by imposing con4nuity between the above asympto4c solu4on and the one of the full radial equa4on which goes to zero at the origin Ø In the case of V(r)=0 we have seen that β l =lπ/2 Ø It is convenient to take this value as a point of reference and write:

Partial waves in the potential V(r) Ø We can write the expression for the asympto4c behavior of Ø The par4al wave, like a free spherical wave, is the superposi4on of an incoming and an outgoing wave Ø We can rewrite the above expression to make it more similar to the free case

Partial waves in the potential V(r) Ø We define a new par4al wave by mul4plying by Ø This global phase factor has no physical importance Ø We can choose the constant C such that: Ø Interpreta4on: - ini4ally we have the same incoming wave as in the case of a free par4cle - As it approaches the zone of influence of V(r), it is more and more perturbed by it - When, a?er turning back, it is transformed into an outgoing wave, it has accumulated a phase shi? of 2δ l rela4ve to the free outgoing wave

Partial waves in the potential V(r) Ø A poten4al V(r) which has a finite range r 0, namely has virtually no effect on waves for which since the incoming wave turns back before reaching the zone of influence of V(r) Ø Therefore, for each value of the energy there exists a cri4cal value l M of the angular momentum, which is given by:

Cross section in terms of phase shifts Ø If we know the phase shi?s (which characterize the modifica4on of the asympto4c behavior of sta4onary states with well- defined angular momentum due to the poten4al) we should be able to determine the cross sec4on Ø We have to express the sta4onary scajering state in terms of par4al waves and calculate the scajering amplitude Ø We must find a linear superposi4on of par4al waves with asympto4c behavior:

Cross section in terms of phase shifts Ø Since is an eigenstate of the Hamiltonian H, its expansion involves only par4al waves having the same energy Ø Besides, if the poten4al is central, the problem is symmetric with respect to rota4on around the z axis defined by the incident beam: is independent of φ: its expansion only includes par4al waves for which m=0 à We need to find the coefficients c l

Cross section in terms of phase shifts Ø When V(r)=0, =e ikz and the par4al waves become free spherical waves à we already know this expansion Ø For non- zero V(r), well as a plane wave includes a diverging scajered wave as Ø differs from asympto4cally only by the presence of the outgoing wave, which has the same radial dependence as the scajered wave Ø We therefore expect that:

Cross section in terms of phase shifts Ø Therefore we can write: Ø from which we can read

Cross section in terms of phase shifts Ø The differen4al scajering cross sec4on is given by the formula from which we deduce the total scajering cross sec4on: Since the spherical harmonics are orthonormal, we have:

Cross section in terms of phase shifts Ø The terms resul4ng from interference between waves of different angular momenta disappear from the total cross sec4on Ø For any poten4al, the contribu4on has an upper bound, for a given energy, of is posi4ve and Ø We need to know all the phase shi?s δ l : this method is ajrac4ve only when there is a sufficiently small number of non- zero phase shi?s: this is the case for a finite- range poten4al Ø If one of the δ l =π/2 for E=E 0, the cross sec4on may show a sharp peak at E=E 0. This is called scajering resonance