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Scattering and Decay of Particles

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All human things are subject to decay, and when fate summons, monarchs must obey. John Dryden (1631-1700) 9 Scattering and Decay of Particles So far we have discussed only free particles. If we want to detect the presence and detailed properties of any of them, it is necessary to perform scattering experiments or to observe their decay products. In this chapter, we shall develop an appropriate quantum mechanical description of such processes. 9.1 Quantum-Mechanical Description We begin by developing the appropriate quantum-mechanical tools for describing the scattering process. 9.1.1 Schr¨odinger Picture Consider a quantum mechanical system whose Schr¨odinger equation Hˆ ψ (t) = i∂ ψ (t) (9.1) | S i t| S i cannot be solved analytically (here we use natural units withh ¯=1). The standard method of deriving information on such a system comes from perturbation the- ory: The Hamilton operator is separated into a time-independent part Hˆ0, whose Schr¨odinger equation Hˆ ψ (t) = i∂ ψ (t) (9.2) 0| S i t| S i is solvable, plus a remainder Vˆ , to be called interaction. The operator Hˆ0 is also called the unperturbed Hamiltonian, and the interaction Vˆ is often called the per- turbation. For the sake of a simple notation we have omitted the hats on top of all operators without danger of confusion. The time-dependent state ψS(t) carries a subscript S to indicate the fact that we are dealing with the Schr¨odinger| i picture. Observables are operators OˆS (p, x, t) whose matrix elements between states yield transition amplitudes. The interaction will be assumed to be of finite range. This excludes, for the moment, the most important interaction potential, the Coulomb 660 9.1 Quantum-Mechanical Description 661 potential. As an example, we may assume the unperturbed Hamiltonian to describe a particle in a Rosen-Morse potential well: const x Vˆ x = V (x)= . (9.3) h | | i cosh2 x | | If the constant is sufficiently negative, the system has one or more discrete bound states. The important Coulomb potential does not fall directly into the class of potentials under consideration. It must first be modified by multiplying it with a very small exponential screening factor. The consequences of an infinite range that would be present in the absence of screening will have to be discussed separately. The interaction Vˆ may in general depend explicitly on time. The results will later be applied to quantum field theory. Then Hˆ0 will be a sum of the second-quantized Hamilton operator of all particles involved, and Vˆ will be some as yet unspecified short-range interaction between them. If the particles are all massive, all forces are of short range. Moreover, the vacuum state is a discrete state which is well separated from all other states by an energy gap. The lowest excited state contains a single particle with the smallest mass at rest. This mass is the energy gap. 9.1.2 Heisenberg Picture The other, equivalent, description of the system is due to Heisenberg. It is based on time-independent states which are equal to the Schr¨odinger states ψ (t) at a | S i certain fixed time t = t0: ψ ψ (0) = Uˆ(t, t )−1 ψ (t) . (9.4) | H i≡| S i 0 | S i For simplicity, we choose t0 = 0. When using these states, the time dependence of the system is carried by time-dependent Heisenberg operators −1 OˆH (t)= Uˆ(t, 0) OˆSUˆ(t, 0). (9.5) Here Uˆ(t, 0) is the unitary time evolution operator introduced in Section 1.6 which governs the motion of the Schr¨odinger state. It has the explicit form [recall (1.269)] t −i dt′ Hˆ (t′) Uˆ(t, 0) = Tˆ e 0 . (9.6) R If Hˆ has no explicit time dependence, which we shall assume from now on, then Uˆ(t, 0) has the explicit form Uˆ(t, 0) = e−iHt/ˆ ¯h and satisfies the unitarity relations Uˆ −1(t, 0) = Uˆ †(t, 0) = Uˆ(0, t). (9.7) Then the relation (9.5) becomes iHt/ˆ ¯h −iHt/ˆ ¯h OˆH (t)= e OˆSe . (9.8) 662 9 Scattering and Decay of Particles 9.1.3 Interaction Picture As far as perturbation theory is concerned it is useful to introduce yet a third description called the Dirac or interaction picture. Its states are related to the previous ones by ˆ ˆ ψ (t) = eiH0t ψ (t) = eiH0tUˆ(t, 0) ψ . (9.9) | I i | S i | H i In the absence of interactions, they become time independent and coincide with the Heisenberg states (9.4). The interactions, however, drive ψI (t) away from ψH (t). When using such states, the observables in the interaction picture are iHˆ0t −iHˆ0t iHˆ0t −iHtˆ iHtˆ −iHˆ0t OˆI (t)= e OˆSe = e e OˆH e e . (9.10) These operators coincide, of course, with those of the Heisenberg picture in the ab- sence of interactions. By definition, the unperturbed Schr¨odinger equation Hˆ ψ(t) 0| i can be solved explicitly implying that the time dependence of the operators OˆI (t), d Oˆ (t) = [Hˆ , Oˆ (t)], (9.11) dt I 0 I is completely known. If the Schr¨odinger operator OˆS is explicitly time-dependent, this becomes d ˙ Oˆ (t) = [Hˆ , Oˆ (t)] + [Oˆ (t)] , (9.12) dt I 0 I S I where ˙ ˆ ˙ ˆ [Oˆ (t)] eiH0tOˆ (t)e−iH0t. (9.13) S I ≡ S 9.1.4 Neumann-Liouville Expansion A state vector in the interaction picture moves according to the following equation of motion: iHˆ0t i∂t ψI (t) = i∂te ψS(t) | i | i ˆ ˆ = Hˆ ψ (t) + eiH0tHeˆ −iH0t ψ (t) − 0| I i | I i = Vˆ (t) ψ (t) , (9.14) I | I i where ˆ ˆ Vˆ (t) eiH0tVˆ e−iH0t (9.15) I ≡ S is the interaction picture of the potential VˆS. It is useful to introduce a unitary time evolution operator also for the interaction picture. It determines the evolution of the states by ψ (t) = Uˆ (t, t ) ψ (t ) . (9.16) | I i I 0 | I 0 i 9.1 Quantum-Mechanical Description 663 Obviously, UˆI (t, t0) satisfies the same composition law as the previously introduced time translation operator Uˆ(t, t0) in the Schr¨odinger picture: ′ ′ UˆI (t, t0)= UˆI (t, t )UˆI (t , t0). (9.17) Using the differential equation (9.14) for the states we see that the operator UˆI (t, t0) satisfies the equation of motion i∂tUˆI (t, t0)= VˆI (t)UˆI (t, t0). (9.18) The equation of motion (9.18) can be integrated, resulting in the Neumann- Liouville expansion [recall (1.200), (1.201)]: t ( i)2 t UˆI (t, t0) = 1 i dt1VˆI (t1)+ − Tˆ dt1dt2VˆI (t1)VˆI (t2)+ ... − Zt0 2! Zt0 t ′ ′ Tˆ exp i dt VˆI (t ) . (9.19) ≡ − Zt0 The expansion holds for t > t0 and respects the initial condition UˆI (t0, t0) = 1. Note that the differential equation (9.18) for UˆI (t, t0) implies that UˆI (t, t0) sat- isfies an integral equation: t ′ ′ ′ UˆI (t, t0)=1 i dt VˆI(t )UˆI (t , t0). (9.20) − Zt0 ′ Indeed, we may solve this equation by iteration, starting with UˆI (t , t0) = 1 which yields the lowest approximation: t UˆI (t, t0) = 1 i dt1VˆI(t1). (9.21) − Zt0 Reinserting this into (9.2), we obtain the second approximation t ( i)2 t UˆI (t, t0) = 1 i dt1VˆI (t1)+ − Tˆ dt1dt2VˆI (t1)VˆI(t2). (9.22) − Zt0 2! Zt0 Continuing this iteration we recover the full Neumann-Liouville expansion (9.19). For t < t0, we may use the relation (9.7) according to which Uˆ(t, t0) is simply the −1 inverse Uˆ (t0, t), and an expansion is again applicable. The time-ordering operator T makes sure that the operators VˆI(ti) in the expansion appear in chronological order, with all later VˆI (ti) standing to the left of earlier ones. If Vˆ and therefore Hˆ has no explicit time dependence, the Schr¨odinger state is governed by the simple exponential time evolution operator: ˆ ψ (t) = Uˆ(t, t ) ψ (t ) = e−iH(t−t0) ψ (t ) . (9.23) | S i 0 | S 0 i | S 0 i Combining this with (9.9), we see that iHˆ0t −iHˆ (t−t0) −iHˆ0t0 UˆI (t, t0)= e e e . (9.24) 664 9 Scattering and Decay of Particles If the potential Vˆ depends explicitly on time, this has to be replaced by [recall (9.6)] t ˆ −i dt′ Hˆ (t′) ˆ iH0t t −iH0t0 UˆI (t, t0) = e Tˆ e 0 e R iHˆ0t −iHˆ0t0 = e Uˆ(t, t0)e . (9.25) For a time-independent Hˆ , this relation shows that UˆI (t, t0) satisfies the same uni- tarity relations (9.7) as Uˆ(t, t0): ˆ −1 ˆ † ˆ UI (t, 0) = UI (t, 0) = UI (0, t). (9.26) The Heisenberg representation (9.5) of an arbitrary operator is obtained from the interaction representation (9.10) via ˆ ˆ ˆ ˆ −1 −iH0t ˆ iH0t ˆ OHˆ (t) = U (t, 0)e OI(t)e U(t, 0) ˆ −1 ˆ ˆ = UI (t, 0)OI(t)UI (t, 0). (9.27) We have observed above that the operator in the interaction picture OˆI(t) has a particularly simple time dependence. Its movement is determined by the unper- turbed equation of motion (9.11). Thus (9.27) establishes a relation between the complicated time dependence of the Heisenberg operator OˆH (t) in the presence of interaction and the simple time dependence of OˆI(t) which reduces to the Heisenberg operator in the absence of an interaction.
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