Validity of scattering theory

Nussenzveig, Causality and Dispersion Relations, p. 182-3:

In general, however, the transient modes are not even approximately orthogonal, so that one cannot ascribe an independent physical meaning to each term in the transient-mode expansion. Transient modes occupy an intermediate position between stationary states and free-particle wave packets, sharing some of the properties of both. It is only for poles located close to the real axis, within the domain of applicability of the exponential law, that concepts taken over from the treatment of stationary states may be employed for an approximate description of the system; we then have approximate orthogonality, and the expansion coefficients may be approximately interpreted in terms of probability amplitudes. For large times, however, free-particle features predominate; the wave packets associated with different poles finally get mixed together as a result of the spreading.

Bohm, Quantum Theory, p. 549:

[to be filled in after requested from library]

Dodd and McCarthy, Scattering of Energy-Time Wave Packets from Many-Body Systems:

The condition for a time-dependent scattering experiment is that the experimental definition of time must be accurate in comparison with the characteristic time of the scattering amplitude.

Newton, Scattering Theory of Waves and Particles, p. 271:

One may now make the connection with the time-dependent theory by forming wave packets out of [time-independent equation] and considering $t \rightarrow \infty$.

Newton, Scattering Theory of Waves and Particles, p. 597:

In the extreme case of a monochromatic beam there is obviously no decay but a steady state.

Newton, Scattering Theory of Waves and Particles, p. 214:

Let us look at the time development of the exact state vector $\Psi (\alpha, t)$ in the light of our knowledge of the properties of the time-independent steady states $\Psi (E, \alpha)$.

Newton, Scattering Theory of Waves and Particles, p. 570:

A slow change of the kinetic energy relative to the wavelength is of course a necessary condition for the concept of a local wavelength to be meaningful. The demand now is that everywhere in the spatial region of interest the local wavelength change slowly. Just as physical optics then allows us to speak of rays and goes over into geometric optics, so quantum physics allows us to speak of trajectories and goes over into classical mechanics. The mathematical reason is the same

Goldberger and Watson, Collision Theory, end of Chapter 3:

We have been able to avoid any discussion of the dynamical behavior of the particles during the time that they interact. Instead we have found that a matrix S… which couples initial and final asymptotic states contains a complete specification of possible experimental results.

Interpreted in this manner, our description is expected to have a more general validity than perhaps the Schroedinger equation which has been used to obtain our results. That is, there seems to be little doubt that the Schroedinger equation (or for that matter Newton's equations) provides an adequate description of the propagation of sufficiently large wave packets for noninteracting particles. Therefore, even if an appropriate Schroedinger equation may not really exist for the evaluation of S, there presumably does exist an S-matrix in terms of which the cross section for any reaction can be obtained.

The Schroedinger equation has received exhaustive experimental verification only for electromagnetic interactions. Here it has provided both detailed and accurate descriptions for a very wide range of phenomena, including atomic and solid state physics. Applications of the Schroedinger equation to nuclear physics have, of course, been exceedingly useful and have provided considerable insight into the mechanisms of nuclear dynamics. On the other hand, these applications have tended to be either phenomenological or semiquantitative. Applications to elementary particle phenomena (other than electrodynamics) have been crude and never quite convincing—except in the sense we have mentioned, that S is a phenomenological quantity subject to certain symmetry relations.

…most efforts to study interactions between elementary systems have been based on quantum field theories, for which the Schroedinger equation provides the dynamical principle.

Goldberger and Watson, Collision Theory, p. 124:

It has… not been necessary to define the term "elementary particle", although we have occasionally used it rather loosely. From a formal point of view it is not clear that the term has a meaning other than as given by somewhat arbitrary definitions. We shall adopt a practical definition, saying that a particle is "elementary" if in a particular application we can ignore its possible composite structure as a bound system of other particles. (Thus in most applications involving electronic excitation of atoms we may consider the atomic nuclei to be "elementary".) From a more formal point of view, there is assigned to each elementary particle a vector space which is spanned by the set of state vectors of that particle. This space is a subspace of that for the physical system being considered and is orthogonal to the subspace assigned to each other elementary particle.

Goldberger and Watson, Collision Theory, p. 230:

It is perhaps useful to recall once again the special sense in which we use the term elementary particle. Because we have assumed that V has no matrix elements which lead to a transition between internal states of the particles, we describe them as "elementary".

Goldberger and Watson, Collision Theory, p. 192:

For describing physical scattering processes we are interested only in the channel eigenstates $\chi_{c;a}$ corresponding to separated, nonoverlapping channel configurations. Although it is obvious that the corresponding $\psi$ are not complete, at a given energy the totality of all channel eigenstates is complete in a well-defined physical sense. In fact the channel configurations are by definition an enumeration at a given energy of all the possible physical situations.

Goldberger and Watson, Collision Theory, p. 118:

we can treat the wave functions for different channels as effectively orthogonal when discussing the final states into which a scattering can go.

Goldberger and Watson, Collision Theory, p. 219-220:

the formalism developed in this chapter (in particular the discussion in which the existence of an unperturbed Hamiltonian was not assumed) may with only trivial modifications be taken over in field theories

page revision: 14, last edited: 10 Jun 2013 22:16