---
title: "How a Field Becomes Quantized Interactive Lab | Sandbox Physics"
description: "How can a continuous field produce discrete particle excitations? The mode fixes the spatial pattern and frequency."
canonical_url: "https://sandboxphysics.com/labs/field-modes/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/field-modes/"
last_modified: "2026-10-27"
---

Quantum Lab 04 · One shape, many energy levels

# Field Modes and Quantization

**How can a continuous field produce discrete particle excitations?** The mode fixes the spatial pattern and frequency. Its occupation counts the quanta added in equal energy steps. Even at zero occupation the oscillator has a finite spread, and a number state keeps that probability distribution unchanged in time.

Drag a fixed-end field mode and choose its quantum occupation. Compare wavelength, energy and the stationary oscillator probability density.

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

Turquoise previews the basis; the number state has stationary spread and zero mean field.

## The state has a shape of its own

1 quantum

### Coordinate probability

Exact density for repeated preparations of this number state.

$q=Q_k\sqrt{\omega_1/\hbar}\quad\longrightarrow\quad p_{k,n}(q)$

### Energy ladder

Drag the gold level; arrow keys work too.

$E/(\hbar\omega_1)=k(n+1/2)$

What the picture represents

This ideal massless field has fixed endpoints and one selected mode in a number state. Turquoise shows the spatial basis with unit peak amplitude. The stationary violet band shows one spatial RMS width from the selected mode at fixed schematic gain; other modes are excluded. This spread is not a confidence interval.

$$
u_k(x)=\sin(k\pi x/L),\qquad\omega_k=k\omega_1,\quad\omega_1=\pi c/L
$$

$$
\langle Q_k\rangle_n=0,\qquad\langle q^2\rangle_n=(n+1/2)/k
$$

A number state changes only its overall quantum phase in time, so every probability density shown here stays stationary. The coordinate is mass-normalized; the readouts use the fundamental mode as a fixed frequency and energy reference.

$$
X=\sqrt{k}\,q,\quad p_{k,n}(q)=\frac{\sqrt{k}}{\sqrt{\pi}\,2^n n!}H_n(X)^2e^{-X^2}
$$

[MIT 8.04 · Lecture 8, stationary states and normalized oscillator functions](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/808334d09369e3a726f6f20d82315c20_MIT8_04S13_Lec08.pdf) · [NIST · Hermite orthogonality](https://dlmf.nist.gov/18.3)

[Continue with the derivation of a quantized mode →](https://sandboxphysics.com/learn/field-theory/quantizing-a-field/)

Physics tutorial

## A mode chooses the pattern; occupation chooses its energy

**Background** Fixed endpoints select standing-wave shapes. Each shape behaves as one harmonic oscillator, with a separate number of quanta occupying it.

**Why it matters** Changing the spatial pattern and adding a quantum are different operations. Keeping their roles separate makes the vacuum and its finite spread easier to understand.

Start with the essentials

- **Focus question**: What changes when you choose another mode, and what changes when you add a quantum?
- **One-sentence intuition**: The mode fixes its wavelength and frequency; the occupation sets its oscillator state and stationary probability distribution.

### Core mathematical model

#### The fixed-end spatial modes

$$
u_k(x)=\sin(k\pi x/L),\qquad\omega_k=k\omega_1,\quad\lambda_k=2L/k
$$

The integer $k$ counts half-wavelengths across the interval. The turquoise curve has unit peak amplitude and shows this basis shape.

#### Energy with one common reference

$$
E_{k,n}=\hbar\omega_1 k(n+1/2)
$$

Each extra quantum adds energy $\hbar\omega_k$. The readout keeps the fundamental frequency fixed as its reference when the mode changes.

#### A number state has zero mean and finite width

$$
q=Q_k\sqrt{\omega_1/\hbar},\qquad\langle q\rangle_n=0,\quad\langle q^2\rangle_n=\frac{n+1/2}{k}
$$

The mass-normalized oscillator coordinate has a stationary distribution. Its nonzero RMS width remains even in the vacuum.

### Common difficulties

#### Two different integers

**Typical misconception** A mode with three half-waves must contain three particles.

**Better mental model** The mode number labels its spatial basis. Its occupation can independently be zero, one or more quanta.

#### A basis curve is not a number-state trajectory

**Typical misconception** The turquoise line shows how the vacuum moves through time.

**Better mental model** The line shows the spatial function used to expand the field. A number state has zero mean field and a time-independent quadrature probability density.

### Run the experiment

1. 01

   #### Keep the occupation and change the pattern

   Choose One quantum, two lobes. Keep the occupation at one and change Spatial mode from one to three.

   **What to observe:** Two interior nodes appear. Frequency and energy triple, wavelength falls to one third, and the quadrature RMS falls by the square root of three.

2. 02

   #### Look at the vacuum distribution

   Choose Vacuum still has width and inspect the violet probability curve.

   **What to observe:** The distribution is a Gaussian with finite width. The energy remains half a fundamental quantum.

3. 03

   #### Climb the energy ladder

   Drag the gold level upward. Keep the spatial mode fixed.

   **What to observe:** The spatial nodes stay fixed. Energy rises in equal steps, and the quadrature density develops more structure and a larger variance.

Physical conclusion

A free-field particle is one excitation of a spatial mode; the vacuum retains a finite quadrature spread.

## Related experiments and articles

Fields, scales and critical phenomena · Reading path

### From field modes to critical phenomena

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

1. [Article **How a chain of oscillators becomes a field**](https://sandboxphysics.com/learn/field-theory/from-oscillators-to-fields/)
2. [Article **Symmetry tells us what flows**](https://sandboxphysics.com/learn/field-theory/symmetry-and-causality/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

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