Beam splitters and quantum mechanics

Suppose two incoming photons one coming from the axis x and the other from the y axis, both in the positive direction of this axes. Now suppose we put a beam splitter in the origin, oriented with 45° wrt x axis. The initial state of the photon x is (1,0) and of the photon y is (0,1) After both meet the beam splitter, the photon x states now become (1,i) and the photon y becomes (i,1) (non normalized).

This is what i read, but i am not understand the latter part, how does this change of states occurs? I would guess that the change of phases make the photon x become (1,-1) and the y (-1,1), -1 = i*i = rotation of 180° or, in another word, change of phase of 180°

• You are interested in the case, where at time $t$ only a single photon is split. If both photons arrive at the beam splitter simultaneously, the problem becomes more complex, see quantum optics. – Semoi Dec 8 '20 at 18:57