---
title: "Neutrino Matter Resonance Interactive Lab | Sandbox Physics"
description: "When a neutrino passes through matter of changing density, does its quantum state keep up with the change? It keeps up if the density changes slowly enough."
canonical_url: "https://sandboxphysics.com/labs/neutrino-matter-resonance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/neutrino-matter-resonance/"
last_modified: "2026-10-27"
---

Neutrinos · D07 · A changing quantum compass

# Neutrino Matter Resonance

**When a neutrino passes through matter of changing density, does its quantum state keep up with the change?** It keeps up if the density changes slowly enough. The neutrino then stays in the local matter eigenstate as that state turns, and can come out almost entirely as the other flavor, with nothing absorbed. Passing through the resonance does not by itself guarantee this: if the density changes too fast, the state falls behind.

Send one coherent neutrino state through a region where the electron density changes. Follow the state on the Bloch sphere, compare it with evolution in vacuum and find out when slow, smooth conversion breaks down.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model has two active flavors, a prescribed one-dimensional density profile and coherent forward scattering. The sphere is a diagram of the quantum state, not a particle orbit or a detector record. Absorption, three-flavor corrections, averaging over production points in the Sun, Earth structure, interactions and instrument response are left out. Each midpoint step of the numerical evolution is unitary. The change after refining the steps is a diagnostic, not a rigorous error bound. Sources and conventions are listed below.

One state through a changing medium

## Conversion without losing probability.

### Coherent electron flavor probability

Compare vacuum

Mint: numerical matter evolution. Dashed: the same entrance state propagated in vacuum. Gold marker: resonance crossing, when present. This is a calculation, not repeated measurements of a travelling particle.

### Density and the resonance condition

The horizontal axis is distance through an illustrative one-dimensional profile. It is not solar radius or a measured Earth model.

### Instantaneous branch populations

Gold: upper branch population. Rose: local mixing strength. Constant branch population indicates following in an instantaneous basis; it need not imply constant flavor.

### After leaving the medium

Instantly set the potential to zero at exit, project onto vacuum mass states, then average away their relative phase. This average is a separate operation; it is not the coherent endpoint probability.

- **Computed vacuum phase average**: $0$
- **Adiabatic reference: slow, vacuum exit**: $0$

### Check the calculation

- **Maximum normalization drift**: $0$
- **Maximum step refinement difference**: $0$

Exact unitary exponentials at interval midpoints. Compare 4,800 with 9,600 steps at 1,201 common positions. The difference diagnoses convergence; it is not an error guarantee. CSV retains all 9,601 positions. Dense chart columns show the sampled minimum and maximum within each pixel.

Conventions, model boundaries & sources

Electron flavor is the north pole; the other active flavor is the south pole. Bloch coordinates encode amplitudes, not real-space directions. Gold is the normalized trace-free Hamiltonian axis. The two-flavor splitting is positive. Antineutrinos reverse the charged-current potential; this real two-flavor model has no CP phase. Entrance eigenstates are recomputed when the medium changes. Vacuum comparisons always use the identical prepared entrance state.

Density means mass density times electrons per nucleon. The decreasing profile follows a shifted exponential over four decades and reaches zero exactly at exit. Solar-like means the shape and illustrative parameters, not a solar density fit. Constant density is abruptly removed at exit; its adiabatic vacuum-exit reference is inapplicable. No absorption or detector sampling is modeled.

- [PDG 2024 · §14.5](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf)
- [Blennow & Smirnov · Neutrino Propagation in Matter](https://arxiv.org/abs/1306.2903)

[Read the MSW story →](https://sandboxphysics.com/stories/neutrinos/solar-matter-msw-effect/) [Return to vacuum propagation →](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)

Physics tutorial

## The medium changes the basis, not the total probability

**Background** Coherent forward scattering adds an extra potential for the electron flavor. When the density changes, the axis of the local Hamiltonian turns, and the quantum state either follows it or falls behind.

**Why it matters** The same mechanism is behind flavor conversion in the Sun and matter effects in the Earth. Here a prescribed two-flavor density profile shows it on its own, without a detector or a realistic density model.

Start with the essentials

- **Focus question**: If a neutrino enters in an eigenstate that is almost pure electron flavor, can it leave almost entirely as the other flavor without anything being absorbed?
- **One-sentence intuition**: When the density changes slowly, the state stays in the same local eigenstate while that eigenstate’s flavor content changes. Passing through resonance alone does not guarantee conversion.

### Core mathematical model

#### Two-flavor evolution equation

$$
i\hbar c\,\frac{d\psi}{dx}=\left[\frac{\Delta m^2}{4E}\begin{pmatrix}-\cos 2\theta&\sin 2\theta\\\sin 2\theta&\cos 2\theta\end{pmatrix}+\frac{V}{2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}\right]\psi
$$

The first component is the electron flavor and the second is the other active flavor. A phase common to both is removed. Internally, distance is in kilometres, energy in eV and the positive mass-squared splitting in eV squared.

#### Matter potential from electrons

$$
V=\sqrt{2}G_F N_e\simeq 7.6324\times10^{-14}\,\mathrm{eV}\,\frac{\rho Y_e}{\mathrm{g\,cm^{-3}}}
$$

The density factor is the mass density times the number of electrons per nucleon. For antineutrinos the potential changes sign. Neutral-current terms are the same for both active flavors, so they do not change this two-flavor evolution.

#### Resonance, where local mixing is largest

$$
2EV=\Delta m^2\cos 2\theta,\qquad \sin^2 2\theta_m=\frac{(\Delta m^2\sin 2\theta)^2}{(\Delta m^2\cos 2\theta-2EV)^2+(\Delta m^2\sin 2\theta)^2}
$$

With a positive splitting and a mixing angle below 45 degrees, neutrinos meet the resonance at a positive density. There the gold axis is perpendicular to the flavor axis. With zero mixing the levels cross exactly, and no off-diagonal term converts one flavor into the other.

#### Flavor probability averaged after exit

$$
\overline{P}_e=p_1\cos^2\theta+p_2\sin^2\theta,\qquad p_1+p_2=1
$$

The exiting state is projected onto the vacuum mass states, and their relative phase is dropped. This differs from the coherent electron probability at the last position. The separate adiabatic reference assumes the state follows slowly all the way out to vacuum.

### Common difficulties

#### A Bloch sphere is not a neutrino orbit

**Typical misconception** The neutrino physically spirals around a sphere inside the Sun.

**Better mental model** The sphere represents a normalized two-component quantum state. Its coordinates are the coherences between the two amplitudes and the difference between the flavor probabilities, not positions in space.

#### Resonance is not absorption

**Typical misconception** Electron neutrinos disappear because matter blocks them.

**Better mental model** The model is Hermitian, so the total probability of the active flavors is conserved. Matter changes the coherent evolution; the model includes no absorption.

### Run the experiment

1. 01

   #### Watch the compass turn

   Choose Solar-like gradient and replay it. Drag the position across the gold resonance marker.

   **What to observe:** As the density falls, the gold evolution axis turns. The state stays normalized while its electron probability changes.

2. 02

   #### Give the state time to follow

   Compare Slow crossing with Sudden crossing. Both start in the same upper eigenstate and cover the same density range.

   **What to observe:** A longer path spreads the same density change over more kilometres, so it changes more slowly per kilometre. In the slow case the state almost stays on the upper branch; in the sudden case it does not.

3. 03

   #### Hold the density fixed

   Choose Constant resonance. Inspect the probability and branch curves.

   **What to observe:** The local mixing is nearly maximal, and coherent flavor oscillations continue. The upper-branch population stays constant because the Hamiltonian does not change. The exit is abrupt, so the slow-exit reference does not apply.

4. 04

   #### Reverse the potential

   Return to Solar-like gradient, select Antineutrino, then Propagate.

   **What to observe:** For the positive splitting and angle used here, the resonance at positive density disappears. This two-flavor model has no CP phase, so a difference caused by matter is not a discovery of CP violation.

5. 05

   #### Check the numerical accuracy

   Look at the normalization and step-refinement diagnostics, then export the whole trajectory.

   **What to observe:** Every step is unitary, so the normalization holds, but splitting the density into steps can still change the trajectory. The step comparison checks this one numerical approximation, not the physics that was left out.

Physical conclusion

Density turns both the evolution axis and the local eigenstates. Whether the state follows them depends on the dynamics, and averaging over phases is a separate physical assumption.

## Related experiments and articles

Continue exploring

### How flavor change is predicted and tested

Propagation through vacuum and matter predicts a flavor probability. How do solar, atmospheric, reactor and beam experiments test it?

1. [Lab **Neutrino Flavor Propagation** Start with coherent vacuum propagation and ideal flavor trials.](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)
2. [You are here **Neutrino Matter Resonance** Follow a rotating matter Hamiltonian and compare slow with sudden conversion.](https://sandboxphysics.com/labs/neutrino-matter-resonance/)
3. [Lab **Solar Neutrino Flavor Observatory** Separate electron flavor from total active solar flux with three reaction responses.](https://sandboxphysics.com/labs/solar-neutrino-flux/)
4. [Lab **Atmospheric Neutrino Zenith Experiment** Turn Earth-crossing baselines into direction-dependent event counts.](https://sandboxphysics.com/labs/atmospheric-neutrino-zenith/)
5. [Lab **Reactor Near–Far Spectra** Fit the same near and far counts while testing calibration and normalization.](https://sandboxphysics.com/labs/reactor-near-far-spectra/)
6. [Lab **Long-Baseline CP Observatory** Compare both accelerator beam polarities on frozen spectra and separate CP phase, matter and response assumptions.](https://sandboxphysics.com/labs/long-baseline-cp/)

Neutrinos · Reading path

### What happened to the Sun’s neutrinos?

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why does passing through the Sun change a neutrino’s flavor?**](https://sandboxphysics.com/stories/neutrinos/solar-matter-msw-effect/)
2. [Lab **Solar Neutrino Flavor Observatory**](https://sandboxphysics.com/labs/solar-neutrino-flux/)

Neutrinos · Reading path

### Understand flavor change and mass

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Can the Earth help tell which neutrino mass comes first?**](https://sandboxphysics.com/stories/neutrinos/earth-matter-neutrino-mass-ordering/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)