Homework Help: Normalizing wave functions / superposition

1. Oct 29, 2014

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 ψphoton as being in a superposition of states ψtransmitted and ψreflected as follows:
ψphoton = c(ψtransmitted + ψreflected)
where c is some normalization constant. Let’s assume ψtransmitted and
ψreflected are a set of orthonormal functions. i.e.,

ψmψm dτ = 1

ψmψn 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.

2. Oct 30, 2014

Ray Vickson

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 $\langle \cdot|\cdot \rangle$.