Quantum interpretation of three polarizers

Let's describe the polarization state of light as a two-dimensional vector, as illustrated in the figure. Vertically polarized light corresponds to a vector pointing upwards (0, 1), horizontally corresponds to (1,0). We use Dirac notation to represent these vectors, |V> and |H> respectively. An arbitrary vector is written as \(|α〉=cos⁡α |V〉+sin⁡α |H〉 \).

Quantum mechanics explains how to calculate: 1) the probability of transmission of these states through a polarizer, 2) the state at the exit of the polarizer. When the state \(|V>\) passes through the second polarizer, oriented at 45°, we have that: The transmission probability is given by

[math] 〖Prob=|〈V|P_(45°) |V〉|〗^2=1/2 [/math]