Neutrinos are very low mass, extremely weakly interacting particles that permeate the universe. About a quadrillion will pass through you while you read this problem. There are multiple types, or flavors, of neutrinos, and they can oscillate into one another as time passes. A model for the oscillations of neutrinos is the following. Consider two neutrinos that are also energy eigenstates, \(\left|v_{1}\rightangle\) and \(\left|v_{2}\rightangle\), with energies \(E_{1}\) and \(E_{2}\), respectively. Neutrinos are produced and detected not as energy eigenstates, but as eigenstates of a different Hermitian operator, called the flavor operator. There are two flavors of neutrino, called electron \(\left|v_{e}\rightangle\) and muon \(\left|v_{\mu}\rightangle\) neutrinos, and these can be expressed as a linear combination of \(\left|v_{1}\rightangle\) and \(\left|v_{2}\rightangle\) :

\[\begin{equation*}\left|v_{e}\rightangle=\cos \theta\left|v_{1}\rightangle+\sin \theta\left|v_{2}\rightangle, \quad\left|v_{\mu}\rightangle=-\sin\theta\left|v_{1}\rightangle+\cos\theta\left|v_{2}\rightangle, \tag{3.166}\end{equation*}\]

for some mixing angle \(\theta\). These flavor-basis neutrinos then travel for time \(T\) until they hit a detector where their flavor composition is measured. \({ }^{8}\)

(a) Assume that the initial neutrino flavor is exclusively electron, \(\left|v_{e}\rightangle\). What is the electron-neutrino state after time \(T,\left|v_{e}(T)\rightangle\) ?

(b) After time \(T\), what is the probability for the detector to measure an electron neutrino? What about a muon neutrino? The detector only measures those flavor eigenstates as defined by Eq.(3.166). You should find that the probability of measuring the muon neutrino is

\[\begin{equation*}P_{\mu}=\sin ^{2}(2 \theta) \sin ^{2}\left(\frac{\left(E_{1}-E_{2}\right) T}{2 \hbar}\right) . \tag{3.167}\end{equation*}\]

The flavor basis is orthogonal and complete, so it can be used to express the identity operator.

(c) Describe why this phenomenon is called neutrino oscillations.