In a current effort, we examine the analogies between electromagnetic scattering and quantum mechanical potential scattering of an incident arbitrary shaped beam. Quantum arbitrary shaped beams, introduced previously in a recent paper, have been found to be inadequate to provide a satisfactory analogy to the electromagnetic case. In this paper, we explain the origin of this inadequacy through the use of various examples, and introduce quantum Eigen-arbitrary shaped beams which exhibit a close analogy between electromagnetic scattering and quantum mechanical potential scattering of shaped beam.