Beam splitter->ONB->transition amplitudes

[klabautermann]

Homework Statement

a general beam splitter is charcatericed by its transition amplitudes S_xy=<X_in|Y_out> where X,Y=L or R. |L_in> represents a photon which leaves the beam splitter to the right. we are only considering photons with the same energy and polarization and neglect all other state parameters.
the beam splitter is loss-free, what mathematical properties has the matrix S of transition amplitudes?

Homework Equations

The Attempt at a Solution

{|X_in>,|Y_out>} form an orthonormal basis and |Y_out>=$\sum$S_XY|X_in>. S describes the change of basis. S preserves the inner product, therefor S is unitary.

i am reviewing problem sets, because i have quantum mechanics exam tomorrow. i have a question. could i also take for example |Y_in>, |X_in> as basis vectors? why do i know that S preserves the inner product.

thanks!

 

[_coyote_]

Yes, you could take |Yin>,|Xin> as basis vectors. The matrix S preserves the inner product because it is unitary. A unitary matrix preserves the inner product because it preserves the lengths of vectors and angles between them. This means that, for any two vectors |u⟩ and |v⟩, the inner product ⟨u|v⟩ is preserved by the matrix S. This implies that S preserves the inner product in all cases.