Read on Twitter
As tweet number 2 in my neutrino physics series, I want to give a bit of an introduction into a fascinating phenomenon that was theorized in the past century and only fairly recently confirmed experimentally- 𝗻𝗲𝘂𝘁𝗿𝗶𝗻𝗼 𝗼𝘀𝗰𝗶𝗹𝗹𝗮𝘁𝗶𝗼𝗻𝘀! 
Thread complexity:
/
https://twitter.com/AstrophysAdam/status/1294759231770460162
Thread complexity:
/ https://twitter.com/AstrophysAdam/status/1294759231770460162
Before we delve into this, this thread is, for the most part, self-contained; however, if you're interested in a bit of background in neutrino oscillations and particularly how they solved a problem in astrophysics, check out the 1st thread in this series! https://twitter.com/AstrophysAdam/status/1294080436470251520?s=20
So without further ado, let's discuss the very basics of neutrino oscillations!
What are Neutrino Oscillations?
Time-Evolution of a Quantum State
Neutrino Masses
Two-Neutrino Mixing & Oscillations
Three-Neutrino Mixing & Oscillations
What are Neutrino Oscillations?Time-Evolution of a Quantum StateNeutrino MassesTwo-Neutrino Mixing & OscillationsThree-Neutrino Mixing & Oscillations
Neutrino oscillations are, in essence, a manifestation of the quantum mechanics (QM) of mixed states. In 1962, Ziro Maki, Masami Nakagawa, & Shoichi Sakata first developed their theory of neutrino flavor oscillation, which was further expanded upon by Bruno Pontecorvo in 1967.
The theory involves mixing between neutrino mass and flavor states. According to their theory, neutrinos are in their flavor states (they are either electron, muon, or tau neutrinos) when emitted or absorbed, but travel through space in their mass states.
We'll discuss exactly what this means in a bit; the important thing to know now is that the flavor and mass states are not the same. It is impossible to assign a numerical "mass" to, for example, an electron neutrino- instead, the flavors are superpositions of 3 mass states.
(Quick note- this upcoming section is the most difficult part- if you're able to power through it, it'll really help your understanding of the green and blue sections- but if not, don't sweat it, just know that that random exponential term comes from this section!)
There is more to understand about how mass and flavor states make neutrino oscillations happen, but since this is an introduction (there will be at least one more advanced thread 
) we're just going to look at how these states evolve with time!
Time-evolution is an operation in QM which governs how a quantum state... changes with time! Say we have some quantum system in a state ν at some initial time t₀- we could represent this as |ν,t₀>. We may expect this state to change with time- we write this like |ν,t₀ ; t>.
(A short aside- if you're not familiar with this "bra ket" notation, that's ok- |ν,t₀ ; t> just represents that we WANT TO LOOK [after the semicolon] at this system at time t, and as a reference/starting point, the system WAS [before the semicolon] in state ν at time t₀.)
So we need an operation to get the state |ν,t₀ ; t> from our initial state |ν,t₀>. We call this operation time-evolution, and we represent it with the time-evolution operator 𝔘(t, t₀)- In other words, 𝔘(t, t₀) |ν,t₀> = |ν,t₀ ; t>.
Ok, we're getting a bit abstract now, and this is just an introduction. So, let's instead try to work through an equation. If we want to find an equation for this operation, we can use the Schrodinger equation for the operation.
In these equations, i is the imaginary number, ħ is Planck's constant, and H is the Hamiltonian operator- if you're not familiar with what that means, one way you could think of it for now is that it represents the total (kinetic + potential) energy of the state it's acting on.
For neutrinos, we want to use the Hamiltonian for a free particle. When you end up doing all the math, because this specific Hamiltonian is time-independent, this equation yields the following solution for the time-evolution operator. In practice, that H is just energy.
Alright, phew. That was a lot of work, but it was important to do to thoroughly understand this process! As mentioned, neutrino oscillations happen because of mixing of the flavor and mass states of neutrinos. We talked about what flavor states are, but what are mass states?
Neutrinos have 3 mass states- we call these ν₁, ν₂, and ν₃. Each of these mass states DOES have an actual, numerical mass, but since quantum mechanics is very strange and particles don't act like billiard balls, a neutrino is a superposition of all 3 of these states.
1 neutrino has 3 masses! Oscillations, as we'll see in a bit, depend on these three states having different (particularly non-zero) masses- however, neutrino oscillations are sensitive to the difference of the squares of these masses- so we don't know their exact values yet!
We can build a neutrino mass hierarchy out of experimental data from oscillations, because that tells us these differences in squared masses which may indicate which state is heaviest and which is lightest- but that's for a different thread. (
https://neutrinos.fnal.gov/mysteries/mass-ordering/)
Now, after all this background knowledge, we're actually going to CALCULATE these neutrino oscillations! There are three neutrino flavor states that oscillate, but we'll start out with two until we feel comfortable with the mathematics- electron and muon neutrinos (νₑ and νᵤ)
We can think about the mass & flavor states as sorta like 2 rotated coordinate systems. Let's say a neutrino is propagating as a mass state ν₁- in the FLAVOR coordinate system, vector decomposition could tell us that the neutrino is a bit of electron, muon, and tau.
First, the 2D case of electron and muon states (with mass states ν₁ and ν₂). Because the neutrinos propagate as mass states, we use this "vector decomposition" to represent the mass states as linear combinations of the flavor states (think of these as orthogonal vectors)
Let's say we start off with an electron neutrino- so at time t=t₀, we have νₑ=1 and νᵤ=0. This means we can rewrite those equations as simply ν₁(t₀) = -sin(θ) and ν₂(t₀) = cos(θ), where θ is the "mixing angle". When we apply time evolution from orange section, we get...
Great, we're ALMOST there! Let's say we want to see the probability that the neutrino has oscillated from an electron neutrino to a muon neutrino after time t- we can solve for νᵤ to figure this out, as the square of νᵤ is equal to that probability.
When we FINALLY explicitly solve for the probability of this transition, we see that it oscillates sinusoidally in time- and THAT is what neutrino oscillations are! The flavor states are just oscillating back and forth due to the difference in squares of the neutrino masses!
In 3 dimensions, we need to use a "mixing matrix" rather than just 1 mixing angle. This matrix is sometimes called the PMNS matrix (Pontecorvo–Maki–Nakagawa–Sakata matrix). This matrix necessitates 3 angles (θ₁₂, θ₂₃, and θ₁₃) and a phase factor 𝛿 (see 2nd green tweet)
In this matrix, c₁₂ represents cos(θ₁₂), s means sin, etc. The full system of equations defining the flavor states is represented as the matrix product of this mixing matrix times the mass states. (Pic from http://math.ucr.edu/home/baez/diary/october_2014.html)
This thread was more advanced than I usually do, so if you understood it to the end, BIG congratulations (especially if you're not a graduate physics student- this truly is a higher level concept!) Thank you for reading along to the end :)
Stay tuned for more entries in this neutrino series! I'm planning another, more advanced thread on neutrino oscillations, a thread on the see-saw mechanism that tries to explain why neutrino masses are so small, and a thread on neutrinoless double beta decay. 
Vector representation of neutrino states img from https://www.slideserve.com/rhona/1-neutrino-oscillation, mixing matrix imgs from Wikipedia "Neutrino Oscillation", others mine. Consulted Intro to Elementary Particles (Griffiths), Modern QM (Sakurai), Neutrinos in High Energy and Astroparticle Physics (Valle)
