Lecture5_two-body_scattering

COLLISION THEORY. MEPHI T(7)-40, 2013, LECTURE 5

TWO{PARTICLE SCATTERING

Up to now we dealt with scattering of a particle on a xed force center. This was a one{particle scattering problem. Use of the xed force center approximation is applicable when the mass of a projectile is small as compared to the mass of a scatterer. Our aim in the present lecture is to pass to the case of an arbitrary mass ratio between a projectile and a scatterer. The non{relativistic case will be considered here. We shall generalize the results of preceding lectures and we shall see that the two{particle scattering problem e ectively reduces to just the one{particle scattering problem we dealt with before.

I. SEPARATION OF THE CENTER OF MASS MOTION IN THE TWO{BODY

PROBLEM

Let us pass from the particle momenta p1, p2 and the particle coordinates r1, r2 to other pairs of variables that are more convenient for description of a two{body system. One of these variables is the total momentum operator P,

Another variable convenient to use is the particle relative momentum. To introduce it let us denote

where m1 and m2 are particle masses. The quantity (2) is called the reduced mass. The relative momentum is denoted p and it is p = v where v = v1 v2 is the particle relative velocity. Thus

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We shall see that our two{problem may be reduced to a subsidiary one{body problem. To achieve this let us note the following. The non{relativistic Hamiltonian is of the form

H = T + V where

is the kinetic energy and V is the particle interaction potential. Expressed in terms of the

P and p momenta the kinetic energy is known to be of the form

Now let discus the structure of the particle interaction potential. In the framework of classical mechanics, this potential depends on particle coordinates and momenta. Let us argue in brief that, when expressed in terms of the above introduced variables, the V potential should be independent of R and P and thus should have the form of V (r; p).

Indeed, the Hamiltonian H(r; p) should have the same form in all the inertial reference frames. In addition, it should be translation invariant. The latter follows from the requirement of invariance, for example, of the Hamilton equations of motion at the transformation of particle momenta and particle coordinates between coordinate systems. The translation invariance means that if a shift by a vector a of the origin of a reference frame is performed then the potential energy V as the function of variables pertaining to the two reference frames should be the same. At such a shift one has R ! R + a while the P, p, and r variables remain unchanged, and therefore this is only possible if V (r; R; p; P) is independent of R.

Furthermore, at Galileo transformations the Hamiltonian should transform, up to a constant, as follows, H(r0i; p0i) = H(ri; pi) + V0 P. Here primed and unprimed variables refer, respectively, to a "new" and an "old" reference frame, and V0 is the velocity of the "new" frame with respect to the "old" one. This is the general law of the transformation of energy at Galileo transformations. It can be obtained, for example, from the requirement of invariance of the Hamilton equations of motion or from the corresponding general law of the transformation of relativistic energy. But according to Eq. (7) one already has

T (p0i) = T (pi) + V0 P + const. Therefore, the potential V should be invariant with respect to Galileo transformations. At these transformations, one has P ! P + (m1 + m2)V0 while p and r remain unchanged. This is possible only if V is not only R{independent but also

P{independent. Thus V = V (r; p).

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A quantum mechanical potential may be obtained via replacing the variables r and p with operators. Local potentials V (^r) are frequently employed. Their coordinate representation

matrix elements are of the form hr0jV (^r)jri = V (r) (r r0).

Let us also take into account that as it is easy to check the p operator and the P operator

act on the coordinate representation wave functions, respectively, as

The r operator and the R operator act on the momentum representation wave functions,

respectively, as

ih@@p and ih@@P:

Therefore, the Hamiltonian is such that in the coordinate representation the stationary Schr•odinger equation H = E possesses product type solutions 1(r) 2(R). And in the momentum representation it possesses product type solutions 1(p) 2(P). Such solutions form a complete set.

In more general terms, one can state the following. The quantum mechanical space H of states of a two{particle system is a tensor product of the H1 and H2 spaces pertaining, respectively, to the rst particle and the second particle. This means that any state in H may be represented as an expansion over states of the product form which are taken from

H1 and H2. In particular, one may represent two{body states as expansions of the form

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dp1dp2a(p1; p2)jp1ijp2i

where jp1i and jp2i are states with given momenta from the H1 space and H2 space, respectively. The products jp1ijp2i may also be written as jp1; p2i. Because of the relations (1) and (3) they may be rewritten as jp; Pi. Let us write these states as jpijPi and correspondingly let us introduce another tensor decomposition of the H space of states. This is the decomposition into the internal motion space Hint and the center{of{mass space Hcm which has the same sense as the above mentioned decomposition into the H1 and H2 spaces. The space Hint is spanned by vectors jpi and the space Hcm is spanned by vectors jPi. (The space Hint is also spanned by vectors jri and the space Hcm is also spanned by vectors jRi.)

Our non{relativistic Hamiltonian is a sum of the free{motion P 2=[2(m1 + m2)] contribution. and the p2=(2 ) + V contribution. The rst of them acts in the center of mass space

Hcm and corresponds to a subsidiary one{body problem for a free ctitious particle with

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the mass m1 + m2. The above reasoning suggests that the second contribution acts in the internal motion space Hint and corresponds to a subsidiary one{body problem for anotherctitious particle, that having the mass . It can be shown that such a separation of the two{particle non{relativistic Hamiltonian is valid not only in the case of the class of potentials considered above but also in the case of the most general V interaction compatible with the above mentioned symmetry requirements.

Our nearest task will be to extend the results of preceding lectures, on the time evolution of scattering states and on the cross sections, to the present two body case. In what follows we shall use corresponding continuum spectrum states of the form jPij p+i where p+ is the relative motion state of the form we considered previously in the one{body scattering problem. All the di erences with the one{body case we considered before are related to the presence now of the extra center{of{mass motion. These di erences are rather trivial.

II.THE PROBABILITY AMPLITUDE FOR TWO{PARTICLE SCATTERING

Let a1(p1) and a2(p2) be the wave packets of colliding particles. We shall use the notation

It easy to check that dpdP = dp1dp2. Taking this into account one has the normalization condition

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ja(p; P)j2dpdP = 1: (9)

In the present two{body case the scattering probability amplitude will be denoted b(p0; P0). The expression for a similar amplitude in the case of a particle scattered by a xed center obtained in Lecture 2 was

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b(p0) = dpa(p) (p p0) + i f(p ! p0) (p p0) : (10) 2 hp0

Basing on that expression it is rather obvious that in the two{body scattering case the

scattering probability amplitude is

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b(p0; P0) = dpdPa(p; P) (P P0) (p p0) + i f(p ! p0) (p p0) : (11) 2 hp0

Apart from the simple dependence from the center{of{mass variables it di ers from the one{ body expression (10) only in that while in Eq. (10) p0 and p denote the particle momenta in Eq. (11) p0 and p denote the momenta of the relative motion of two particles.

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Eq. (11) may be obtained e.g. as follows. In the notation of Eq. (8), at t ! 1 the

Normally this can be done up to any accuracy. Let us consider the contribution to the state vector at any time value from a separate term (p) (P)jpijPi. Because in our two{body problem the center{of{mass motion is separated from the internal motion this contribution evolves in time as (1)(t) (2)(t) where (1)(t) emerges from the t ! 1 wave packet

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dp (p)e iEpt=hjpi

in the internal space, and (2)(t) emerges from the t ! 1 wave packet

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dP (P)e iEP t=hjPi

in the center{of{mass space. The latter wave packet moves freely at any time. Then according to the reasoning of Lecture 2 the contribution to the probability amplitude of a term (p) (P)jpPi in the expansion (12) is of the form of Eq. (11) with the replacement a(p; P) with (p) (P). This just leads to Eq. (11).

Quite similar to Lecture 2 we may rewrite the scattering amplitude entering Eq. (11) in terms of the matrix element of the potential (see the transition from Eq. (12) to Eq. (13) in Lecture 2). Namely,

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is de ned in the same way as before but now it is associated with the Hint space.

The {functions entering the probability amplitude (13) may be transformed to a more transparent form. Taking into account that dpdP = dp1dp2 one concludes that

(P P0) (p p0) = (p1 p01) (p2 p02):

Furthermore,

(P P0) (Ep Ep0) = (P P0) (Ep+EP Ep0 EP 0) (p1+p2 p01 p02) (Ep1 +Ep2 Ep01 Ep02 ):

Therefore Eq. (13) manifests the energy{momentum conservation in scattering.

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III.EVALUATION OF CROSS SECTIONS FOR TWO{PARTICLE SCATTERING

Usually cross sections for scattering of the projectile particles and the target particles are of interest. But because of the structure of Eq. (11) it is convenient to calculate rst a subsidiary cross section, namely the cross section for nding after scattering the relative momentum in a given solid angle. This is done in the present section. Cross sections for scattering of the projectile and the target particles will be deduced in the next section from the result of the present section.

Because of the momentum conservation the distribution in the total momentum P0 after scattering is, of course, very narrow and is of no interest. Therefore we shall employ the probability

ZZ 1

integrated not only over p0 as in the one{particle case in Lecture 3, but also over P0. The same reasoning as in Lecture 3 leads in the present case to the de nition of the cross section d ,

where now dNsc is the number of events at which the relative momentum p0 lies in a solid angle d p0. Here, as in Lecture 3, the cross section d is the integral over the probabilities that are of Eq. (14){form but involve the projectile particles shifted in the transverse plain,

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As in Lecture 3, here u is the vector determining the position of the wave packet in the transverse plain. As it was explained in Lecture 3, at the shift of the projectile wave packet by a vector u its momentum representation wave function transforms as follows, a1(p1) ! exp( ip1 u=h)a1(p1). (Here and below the particle number one is chosen to be a projectile.) This gives

This substitution is to be done in Eqs. (11) and (14) when Eq. (16) is used.

The cross section d is calculated according to Eqs. (16), (14), (11), and (17). This calculation is rather similar to that we already performed in the one{body scattering case in Lecture 3. The expression we started with in the one{body scattering case was (see the

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This result is very similar to the result we have obtained in the one{particle scattering case but now p0 is the (average) initial relative momentum and p0 is the nal relative momentum in the p0 direction.

Coming back to Eq. (15), let us note that, in principle, the number Ncons of available constituents may decrease in time due to scatterings occurred before. Eq. (15) is thus applicable only when this is of no importance. If one considers the process during the time interval dt and sets beam = nbeamvbeamdt, where nbeam and vbeam are the concentration of the projectile particles and their velocity, then Eq. (15) is valid at least as the relation for dNsc=dt. In general, balance equations are to be considered that involve velocities of various processes satisfying the relations of the type of Eq. (15).

IV. EXPRESSIONS FOR THE PARTICLE SCATTERING CROSS SECTIONS

In this section we shall express the projectile and the target di erential cross sections in terms of the cross section (20). Let us speak, for example, of the projectile. In the above relation (15) the quantity d was de ned as the cross section for the process in which the nal state relative momentum p0 can be found inside a solid angle d p0. At the

same time, of six variables such as the components of the nal{state particle momenta p01 and p02 only two variables are independent due to momentum and energy conservation in scattering processes. Because of this, the direction of the relative momentum p0 completely determines the direction of nal{state momentum p01 of the projectile. I.e. the solid angle d p0 completely determines the solid angle d 1 pertaining to p01.

Due to this fact, the probability of Eq. (14) is simaltaneously the probability for the projectile to be scattered in the corresponding solid angle d 1. And the cross section d ,

therefore, may be equally viewed as just the cross section for the projectile momentum p01 to be found inside d 1. Similar reasoning applies to the target particle di erential cross section. As a result, the projectile and the target particle di erential cross sections d =d 1 and d =d 2 can be expressed in terms of the cross section (20) as follows,

Thus what we only need is to calculate the kinematical factors d p0=d 1 and d p0=d 2. Note, in addition, that the di erential cross section d =d p0 may itself be interpreted as

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the cross section for scattering of a particle, although not in the laboratory frame we dealt with up to now. Namely, such an interpretation is valid in the center{of{mass reference frame. (One refers to the P = 0 frame as to the center{of{mass frame.) Indeed, rst let us note that the relative momentum p is the same in all the inertial reference frames. Furthermore, in the center{of{mass frame one has p2 = p1. Therefore in the center{of{ mass frame one has p1 = p2 = p and d 1 = d 2 = d p0. And the quantities dNsc and

beam are the same in the laboratory and the center{of{mass reference frames. Therefore the quantity d is also the same in these two frames. As a result, one may conclude that, indeed, d =d p0 = (d =d 1)cm = (d =d 2)cm where the two latter quantities denote the center{of{ mass cross sections for the projectile and the target particle. Therefore the relations (21) may be viewed also as transformations of the cross sections from the center{of{mass frame to the laboratory frame. (As to the total laboratory cross sections, they are obviously equal to the total center{of{mass cross section.)

Let us perform the calculation of the required transformation factors d p0=d 1 and d p0=d 2 from Eqs. (21). Let the incoming particle move along the z axis. One has p01 = p0 + [m1=(m1 + m2)]P, and p02 = p0 + [m2=(m1 + m2)]P. This may be presented in the form

where n01, n02, and np0 are, respectively, the unit vectors in the directions of p01, p02 and p0, the quantity ez is the unit vector along the z axis, and

= [m1=(m1 + m2)]P=p0; = [m2=(m1 + m2)]P=p0:

The latter quantities are in fact constants. Indeed, the total momentum P is conserved so that P equals to the projectile momentum before scattering, i.e. p1. In addition, the absolute value of the relative momentum is conserved because of energy conservation. Therefore p0 = p, and p = [m2=(m1 + m2)]p1. Thus P=p0 = (m1 + m2)=m2. As a result, in Eq. (22)

= m1=m2; = 1:

Let us denote 1 and '1 the polar and azimuthal angles determining the vector n01,

2 and '2 such angles determining the vector n02, and p0 and 'p0 those determining the

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The latter of these equalities may also be written as 2 = ( p0)=2. Furthermore, it is

seen from the geometrical picture provided by Eqs. (22) that '1 = 'p0 and '2 = + 'p0.

Taking into account that d = sin d d' = d cos d' we can then write the kinematical

If the mass of the projectile particle is small compared to the mass of the target particle, i.e. is small, then the kinematical factor entering Eq. (24) is close to unity. And the reduced mass determining the cross section d =d p0 is close to m1. Therefore, in this case the two{body cross section (24) turns to the cross section for scattering of a projectile on axed center we considered before.

To analyze the relationship of Eq. (23) between 1 and p0 one may represent the rst of the relations (22) with the help of the vector diagram of the Figure listed in the next page. (The angle p0 is denoted rel in the Figure.) From this diagram one concludes, in particular, that when < 1 (i.e. m1 < m2) the scattering angles 1 varies between zero and

and there exists one{to{one correspondence between 1 and p0 values. When > 1 (i.e. m1 > m2) allowed 1 values do not reach and there exist two p0 values that correspond to each allowed 1 value. The di erential cross section is a sum of the two corresponding cross sections of Eq. (24) form. When m1 = m2 one has simply 1 = p0=2.

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