Issues in Preparation of Scattering Superposition

To describe how the required superposition state [Eq. (7.5)] can be constructed in the laboratory requires some introductory remarks. Note first that Eqs. (7.2) to (7.7) and the E, q, m 0) states are understood to be in the center-of-mass coordinate system and describe the relative translational motion as well as the internal state of A and B. In typical A-B scattering, separating out the center-of-mass motion comes about in a straightforward way. That is, let rA and rB denote the laboratory position of A and B and ikA, Mr denote their laboratory momenta. The relative momentum k, relative coordinate r, center-of-mass momentum K and position Rcm are defined as 

V Note that the scattering in Eq. (7.9) occurs at a fixed value of the center-of-mass 

Consider now preparation of the generalized superposition states [Eq. (7.5)] where for simplicity we limit consideration to a superposition of two states. To do so we examine the scattering of A and B, each previously prepared in the laboratory in a superposition state. The wave functions of A and B in the laboratory frame, ij/A and t//B, are of the general form  

As constructed, Eq, (7.14) is composed of four independent noninterfering inci-Jf dent states since each has a different center-of-mass wave vector Ky. However, we can set conditions so that interference, and hence control, is allowed. That is, we require the equality of the center-of-mass motion of two components, plus energy degeneracy 1 

but where the first term allows for control via the interference between the An and A2 terms. The two remaining terms, proportional to An and A22, are uncontrolled satellite contributions. 

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