BIFACTORIZABLE WAVE-FUNCTIONS

Physical motivation is given for studying properties of bifactorizable (BF) functions, i.e. functions of two variables which can be factored in two different ways. The functional equation which a BF function must satisfy is derived and the form of its solution is shown to be a Gaussian. This also yields the functional equation defining a Gaussian, in analogy to the equation E(x + y) = E(x)E(y) defining the exponential function. Further, the following theorem is proved: if two systems are prepared independently, and their centre of mass is found to be in a pure state, then both systems were prepared in pure states, each of which is a Gaussian in the coordinate representation, and so are the centre of mass and relative coordinate states.