Photon stream impinging on half-silvered mirror

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The discussion centers on the behavior of photons when they encounter a half-silvered mirror, a key concept in quantum optics. When a photon represented as [A> interacts with the mirror, it evolves into a superposition state [B> + i[C>, where C denotes the reflected light. The imaginary unit "i" signifies a phase shift of one-quarter wavelength that occurs between the reflected and transmitted beams, crucial for understanding interference patterns in quantum mechanics.

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In the classic experiment, light impinges on half-silvered mirror, the mirror reflects just half the light and transmits the rest.

The photon before encounter with the mirror is [A>, then afterwards it evolves according to U to become [B> + i[C> (where C is the reflected light). The imaginary factor "i" is said to arise because of a net phase shift by a quarter of a wavelength which occurs between reflected and transmitted beams.

I'm confused with this last part- Why, again, is "I" used? I'm unclear about the phase shift by 1/4 wavelength. Can anyone explain this in laymens?
 
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