A minimal example in vacuum, step-by-step

A minimal probability calculation. The simplest complete calculation: the three-flavor vacuum probabilities for a 1-GeV neutrino over 1 300 km. The oscillation parameters—here and in the rest of Usage and examples— take their defaults, the NuFIT 6.1 best fit with Super-Kamiokande atmospheric data in normal ordering [EGonzalezGarciaM+24].

import magnus.oscprob as oscprob
import magnus.globaldefs as gd

# Natural units: energies in eV, lengths in eV^-1
E = 1.0*gd.UNIT_GEV
L = 1300.0*gd.UNIT_KM

P = oscprob.osc_prob_3nu_vacuum(E, L)

print('Pme = %.5f, Pmm = %.5f, Pmt = %.5f'
 % (P[gd.NUMU][gd.NUE],
 P[gd.NUMU][gd.NUMU],
 P[gd.NUMU][gd.NUTAU]))

# Pme = 0.03127, Pmm = 0.39231, Pmt = 0.57643

Listing A minimal probability calculation is a complete calculation, built in four steps.

Import the two modules a typical calculation needs. oscprob holds the probability functions; globaldefs holds the physical constants, the unit conversions, and the flavor indices.
```
import magnus.oscprob as oscprob
import magnus.globaldefs as gd
```
Give the energy and the baseline in natural units. Every energy crossing the interface is in eV and every length in eV\(^{-1}\) (Conventions). The factors UNIT_GEV and UNIT_KM convert from the units a user thinks in.
```
E = 1.0*gd.UNIT_GEV
L = 1300.0*gd.UNIT_KM
```
Call one wrapper. Its name fixes the flavor count and the environment. The oscillation parameters default to the NuFIT 6.1 best fit in normal ordering; any of them may be overridden by name, e.g., dCP=0.0. Nothing in the call selects an engine or a numerical setting: Magνs chooses the engine and, where the profile varies, the slab count. [They can still be fixed via arguments (Engines and dispatch and Methodology).]
```
P = oscprob.osc_prob_3nu_vacuum(E, L)
```
Read the matrix. The call returns every flavor pair at once rather than one number, indexed by the constants NUE, NUMU, and NUTAU.
```
P[gd.NUMU][gd.NUE]
```

The returned matrix is indexed initial flavor first, \(P[\alpha][\beta] = P_{\nu_\alpha \to \nu_\beta}\). Each row and each column sums to one, because the evolution is unitary. The corresponding evolution operator element, which the call returns when asked (Returning the evolution operator), is \(\mathbb{U}_{\beta\alpha}\), so \(\lvert \mathbb{U}\rvert^2\) comes out indexed the other way:

# P is indexed initial flavor first
P[gd.NUMU][gd.NUE]           # 0.03127

# The amplitude is indexed the other way
P, U = oscprob.osc_prob_3nu_vacuum(
    E, L, return_evolution_operator=True)
abs(U[gd.NUE][gd.NUMU])**2   # 0.03127
abs(U[gd.NUMU][gd.NUE])**2   # 0.00854

In vacuum, the Hamiltonian does not depend on position, so the Magnus series stops at its first term: the evolution operator is a single exponential, exact to round-off. Therefore, Listing A minimal probability calculation exercises none of the method of Methodology. Most of the examples that follow are the same four steps with a different wrapper. Once the profile varies, the refinement of Methodology engages, at the default rtol \(=\) atol \(= 10^{-3}\) unless another tolerance is requested.