Q010 · Eigenstate / superposition / displacement

Quantum Harmonic Oscillator

Prepare an energy eigenstate, mix adjacent levels coherently, or displace the ground state. Connect nodes, energy occupations and position records without inventing a hidden particle orbit.

Model mean energy $\text{—}$

Model uncertainty product $\text{—}$

FIRST, MAKE A PREDICTION

If the probability cloud stays still, is its energy zero?

Replay the ground state, then mix two levels or displace it. Compare the cloud with the energy occupations and independently sampled positions.

Predict, compare, then collect evidence.

Measurement time — each sample uses a fresh preparation

PreparationTrap angular frequency

\omega

1

Lower energy level

n

0

Population of the next level

w

0.5

Observable to sample

Time and measurement basis preserve records. Preparation changes clear them. Position and momentum are measured on different preparations.

State details and sampling

Relative phase

\varphi

0^{\circ}

Initial displacement

a

3

Trials per time and basis

N

400

Sampling seed

s

1984

Dimensionless position, momentum and time. The lesson defines the scale conversion. Sampling uses a reproducible seed and a 30,000-record limit.

Ideal one-dimensional harmonic potential. Hermite eigenfunctions and exact displaced-ground-state evolution; changing frequency prepares a new state, not a sudden quench. Energy bars describe a basis; the position density describes space.

Recorded distributions and time series

SIMULATED INDEPENDENT PREPARATIONS

Build a distribution, one event at a time

Blue bars are recorded frequencies; the gray curve is a separate model density. Time-series dots show sample means with one standard error; orange dots show sample standard deviations. These are not confidence intervals or particle paths.

Current sample mean and spread

\text{—}

Current measured uncertainty product

\text{—}

Grid norm and boundary mass

\text{—}

Model occupation beyond level 32

\text{—}

Hermite eigenstates and displaced-state evolution are analytic. Position and Fourier momentum are sampled on a 2,048-point interval of length 64. The occupation display groups higher levels; its tail does not truncate the evolved displaced state.

Model sources: MIT 8.04 · Coherent-state evolution

Start with the essentials

Focus question: Does a stationary density mean no kinetic energy?
One-sentence intuition: A stationary density can have finite momentum spread and finite energy.

Core mathematical model

Units

X=x/\ell,\quad P=p\ell/\hbar,\quad \tau=\hbar t/(m\ell^2)

Choose a length scale and particle mass. The displayed numbers use these dimensionless variables; the code uses unit mass and reduced Planck constant.

Harmonic Hamiltonian

H=\tfrac12P^2+\tfrac12\omega^2X^2,\quad E_n=\omega(n+\tfrac12)

The trap is exactly quadratic at every displayed amplitude. Frequency changes reprepare the state. Equal spacing and nonzero ground energy belong to this ideal Hamiltonian.

Energy eigenfunctions

\phi_n(X)=\frac{(\omega/\pi)^{1/4}}{\sqrt{2^n n!}}H_n(\sqrt\omega X)e^{-\omega X^2/2}

The normalized Hermite recurrence generates levels zero through nine. An eigenstate changes only its global phase; its density stays still while kinetic and potential energies remain nonzero.

Adjacent-level interference

\psi=\sqrt{1-w}\,\phi_n e^{-iE_n\tau}+\sqrt w\,e^{i\varphi}\phi_{n+1}e^{-iE_{n+1}\tau}

Changing relative phase moves interference in position space. Occupation weights remain constant during ideal evolution. Energy bars are not positions inside the trap.

Displaced ground state

\langle X\rangle=a\cos(\omega\tau),\quad\langle P\rangle=-a\omega\sin(\omega\tau),\quad\Delta X=\frac1{\sqrt{2\omega}}

The centroid follows the classical oscillator while the packet retains its width. This is an ensemble mean, not the path of a hidden point particle. The analytic displaced state is not truncated to the displayed levels.

Coherent-state occupation

P_n=e^{-\bar n}\frac{\bar n^n}{n!},\quad\bar n=\frac{\omega a^2}{2}

The first nine levels are shown separately and all higher probabilities are grouped. The model tracks the additional tail beyond level 32 without renormalizing the visible bars.

Separate preparations

\bar q=\frac{1}{N}\sum_j q_j,\qquad s_q^2=\frac{\sum_j(q_j-\bar q)^2}{N-1}

Each event starts from a new preparation. Position and momentum records are separate ensembles. One standard error estimates uncertainty of a sample mean; sample spread estimates preparation uncertainty.

Common difficulties

One dot or a cloud

Typical misconceptionA moving density is a known path of one particle.

Better mental modelThe cloud is a preparation distribution. Each recorded point comes from a new preparation, not continuous tracking.

Numerics versus inference

Typical misconceptionMore shots fix every error.

Better mental modelMore shots reduce sampling noise. They do not correct grid spacing, a periodic boundary, imperfect apparatus or a wrong physical model.

Run the experiment

01
Wait
Replay the ground state and inspect the complex amplitude.
What to observe: The global phase moves; the probability density does not.
02
Find nodes
Select Three nodes, then change the level and trap frequency.
What to observe: Nodes count the eigenstate index; the trap changes the scale.
03
Mix
Mix adjacent levels; vary their weight and phase.
What to observe: Density moves while the level occupations remain fixed.
04
Displace
Displace the ground state and sample nine position times.
What to observe: Measured centroids oscillate; the cloud retains nonzero width.