Orbitals
Sample the probability distribution of a single electron in hydrogen. The empty regions matter as much as the bright ones.
Each dot is a sample of position probability, not a separate electron. Cyan and amber distinguish the sign of a real wavefunction—not electric charge. The distribution is stationary; only the camera moves. Orbitals are auto-scaled for comparison.
OpenStax · The hydrogen atom ↗Interference
Change the slit separation and wavelength. Reveal which path was taken, and the interference term disappears.
The field is an illustrative scalar two-source wave. The detector curve uses the far-field two-slit formula with a finite slit-width envelope. Simulated hits sample that normalized curve. Which-path information sets visibility V to zero; it is a physical loss of coherence, not the act of a conscious observer.
OpenStax · Interference ↗Qubit
Prepare a pure state, apply a gate, then measure it in the computational basis. Measurement changes the state.
The Bloch sphere represents a two-level quantum state, not the physical path of a spinning particle. The arrow is the Bloch vector; z = +1 is |0⟩ and z = −1 is |1⟩. Global phase is omitted. H, X, Y, Z and S act as exact single-qubit gates in this model.
IBM Quantum · Quantum states ↗Entanglement
Rotate two spin measurement axes and sample a singlet pair. Each side is random. Their joint statistics are not.
The spheres depict analyzer directions, not pure local states: each member of the singlet is locally maximally mixed. Angles are spin axes, not photon polarizer angles. The dotted link denotes the joint state, not a signal. The ideal CHSH value 2√2 uses four fixed optimal settings; the two current sliders alone are not a Bell test.
IBM Quantum · CHSH inequality ↗Tunneling
Raise a barrier above the particle’s energy. Transmission becomes unlikely—not impossible.
This is an exact stationary plane-wave solution for a one-dimensional rectangular barrier. Cyan is Re(ψ); violet is Im(ψ). The amber outline indicates the barrier. The plotted vertical scales for potential and amplitude are illustrative and separate. Readouts include above-barrier reflection and the continuous E = V₀ limit. Units: ℏ = 1 and 2m = 1.
OpenStax · Quantum tunneling ↗Uncertainty
Compress a Gaussian wave packet in space and its momentum spread grows. Let it evolve, and it disperses.
An analytic free Gaussian packet starts with minimum uncertainty. Cyan and violet show real and imaginary amplitudes; amber shows position probability density. The view follows the packet’s center through a fixed ±12 spatial window. The initial mean momentum is 2. Time is controlled by the slider; camera motion is not time evolution. Units: ℏ = m = 1.
OpenStax · Uncertainty ↗Energy levels
Compare a particle in a box with a harmonic oscillator. Their boundary conditions permit different energy ladders.
The curves are normalized stationary eigenfunctions, offset by their energy; their wave amplitudes use an independent illustrative scale. Cyan and amber mark positive and negative sign. Box states start at n = 1, oscillator states at n = 0. The oscillator ground state has nonzero zero-point energy. Units: ℏ = m = ω = 1; box width L = 4.
OpenStax · Harmonic oscillator ↗Decoherence
Watch an initial |+⟩ state lose phase coherence to an environment. A pure state becomes a mixture.
This is a Markovian pure-dephasing model, not energy relaxation. The cyan arrow is the average Bloch vector. Violet samples illustrate an ensemble of randomized phases, one possible representation of the same density matrix. Purity tends to ½ while the populations remain 50/50. The time slider controls evolution.
IBM Quantum · Density matrices ↗