Normalizing wave functions / superposition

[582153236]

A remote control shot a single photon at a window that has a 50% chance of transmitting and
50% chance of reflecting photons. Once the photon wavepacket hits the glass (without an observer present), it propagates as a superposition of states, one in which it was transmitted and one in which it was reflected. We can write down ψ_photon as being in a superposition of states ψ_transmitted and ψ_reflected as follows:
ψ_photon = c(ψ_transmitted + ψ_reflected)
where c is some normalization constant. Let’s assume ψ_transmitted and
ψ_reflected are a set of orthonormal functions. i.e.,

ψ_m^∗ψ_m dτ = 1

ψ_m^∗ψ_n dτ = 0

Find the normalization constant, c and What does the wavefunction of a single photon become if the detector
of the TV on the opposite side of the window registers the photon?

Any idea how I'm supposed to begin finding the normalization constant? I'm confused since the wave functions aren't explicitly given.

 

[Ray Vickson]

A remote control shot a single photon at a window that has a 50% chance of transmitting and
50% chance of reflecting photons. Once the photon wavepacket hits the glass (without an observer present), it propagates as a superposition of states, one in which it was transmitted and one in which it was reflected. We can write down ψ_photon as being in a superposition of states ψ_transmitted and ψ_reflected as follows:
ψ_photon = c(ψ_transmitted + ψ_reflected)
where c is some normalization constant. Let’s assume ψ_transmitted and
ψ_reflected are a set of orthonormal functions. i.e.,

ψ_m^∗ψ_m dτ = 1

ψ_m^∗ψ_n dτ = 0

Find the normalization constant, c and What does the wavefunction of a single photon become if the detector
of the TV on the opposite side of the window registers the photon?

Any idea how I'm supposed to begin finding the normalization constant? I'm confused since the wave functions aren't explicitly given.

If the inner product of wave functions $\psi_m$ and $\psi_n$ is denoted as $\langle \psi_m | \psi_n \rangle$, you have
$$\langle \psi_r | \psi_r \rangle = \langle \psi_t | \psi_t \rangle = 1\\ \langle \psi_r | \psi_t \rangle = \langle \psi_t | \psi_r \rangle = 0$$
Here, $\psi_r = \psi_{\text{reflected}}$, $\psi_t = \psi_{\text{transmitted}}$, and let $\psi_p = \psi_{\text{photon}}$.
You want $\langle \psi_p | \psi_p \rangle = 1$. Now just use standard properties of $\langle \cdot|\cdot \rangle$.