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.

Frequency

\omega_k/\omega_1

$1$

Wavelength

\lambda/L

$2$

Energy

E/(\hbar\omega_1)

$1.5$

Coordinate RMS

\sqrt{\langle q^2\rangle}

$1.22$

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 · NIST · Hermite orthogonality

Continue with the derivation of a quantized mode →

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 misconceptionA mode with three half-waves must contain three particles.

Better mental modelThe 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 misconceptionThe turquoise line shows how the vacuum moves through time.

Better mental modelThe 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

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.
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.
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.