Coherence and incoherence of inelastically scattered electron waves

Whether the elastically and inelastically scattered electron waves are mutually coherent is a key question for the quantitative evaluation of the contrast of a high resolution transmission electron microscope image and for confirming the energy filtering ability of electron holography. Using a simplified object model composed of only two scattering centers the elastically and inelastically scattered electron waves and their detection are discussed. The only used proof for the main results of this paper is the orthogonality of the object states. The reasoning can be extended straightforward to a general case, e.g. a complicated object as well as complicated interaction Hamiltonians as long as they remain time independent. The results of this paper are: (1) The elastically and inelastically scattered waves which have different energy losses are mutually incoherent. The reason for that is simply that the corresponding excited object states are mutually orthogonal; (2) the inelastically scattered waves which have the same energy loss and are scattered by the same object state are coherent, whereas they are incoherent if they are scattered by different object states though they have the same energy; (3) the coherence degree of the scattered electron waves is proportional to the modulus of the scalar product of the corresponding object states.