Beam Splitter - Commutation relations

[Hazzattack]

Hi guys, why does the following mean B is unitary?

if we have two rotations such that;

b1 = B₁₁a1 + B₁₂a2
b2 = B₂₁a1 + B₂₂a2

and the following commutator results are;

[b1, b1(dagger)] = |B₁₁|^2 + |B₁₂|^2 --> 1

[b2, b2(dagger)] = |B₂₁|^2 + |B₂₂|^2 --> 1

[b1, b2(dagger)] = [B₁₁ B*₂₁] + B₁₂ B*₂₂ --> 0

thus B is unitary.

I'm assuming it's something to do with the probabilities adding to 1, but I'm hoping for a more 'visual' understanding.

Thanks in advance.

 

[jfizzix]

A transformation is unitary if the inner product between any two vectors remains unchanged before and after the transformation.

So if a1 and a2 are orthonormal vectors, and the transformation is unitary, then b1 and b2 will also be orthonormal vectors.

This is also a necessary and sufficient condition, so you can prove the transformation is unitary, if the inner products always remain the same.