Layer 14 · Quantum matter

How do fields and quantum rules become stable matter?

Atoms are not tiny solar systems. Stable matter appears because quantum states can bind, energies become discrete, wavefunctions develop nodes and symmetries, and identical electrons obey the Pauli exclusion principle. This lab makes those constraints visible.

bound stateshydrogen spectrumorbital probabilityPauli principlemolecular-orbital overlap

01 · BINDAttractive interactions can support discrete quantum states instead of unrestricted motion.

02 · SHAPEQuantum numbers constrain spatial probability patterns and their nodes.

03 · FILLIdentical fermions cannot all occupy the same one-particle state.

04 · COMBINEAtomic states can overlap and form new collective molecular states.

EXPERIMENT 01

Discrete energy levels become spectral lines.

Hydrogen's Coulomb potential supports bound states with energies Eₙ = −13.6057 eV / n². Choose a transition and connect an invisible change in quantum state to a photon with a definite energy and wavelength.

Initial principal quantum number nᵢ

Final principal quantum number n𝒇

Initial energy—

Final energy—

Photon ΔE—

Wavelength—

Choose two different levels.

Eₙ = −13.6057 eV / n²
λ = hc / |ΔE|

EXPERIMENT 02

An orbital is a probability amplitude, not an orbit.

Select a hydrogenic orbital and inspect a two-dimensional probability-density slice. Nodes are places where the wavefunction vanishes; changing quantum numbers changes the allowed spatial pattern.

Orbital slice

ℓ0

angular nodes0

radial nodes0

The canvas shows a 2D slice of |ψ|² in dimensionless Bohr-radius units. Brightness is rescaled for visibility. It is not a trajectory of an electron and not the full 3D orbital.

nodes: angular = ℓ
radial = n − ℓ − 1

EXPERIMENT 03

Identical electrons cannot all pile into one state.

Choose an atom from hydrogen through argon and watch electrons fill one-particle orbitals. Pauli exclusion limits an orbital to two electrons with opposite spin; Hund's rule sets the lowest-energy arrangement within degenerate p orbitals in this simplified ground-state filling model.

Atomic number

Neon1s² 2s² 2p⁶

Pauli exclusion principle: no two identical fermions can occupy the same complete quantum state. The opposite arrows in one orbital box represent opposite spin projections, so their full quantum-number sets differ.

capacity of one spatial orbital = 2 electrons
s subshell: 1 orbital · p subshell: 3 orbitals

EXPERIMENT 04

When atomic states overlap, new states can form.

Bring two hydrogenic 1s centers together. Constructive and destructive linear combinations produce bonding-like and antibonding-like molecular orbitals with very different density between the nuclei.

Internuclear separation R / a₀

2.00 a₀

Combination

1s overlap S(R)—

density at midpoint—

The plotted line is a schematic internuclear-axis slice of linear combinations of two 1s-like basis functions. The overlap S(R)=e⁻ᴿ(1+R+R²/3) is the exact overlap integral for equal hydrogenic 1s Slater orbitals in atomic units; this panel does not solve the full molecular Hamiltonian or claim a bond energy.

Scientific boundary. Hydrogen energies and the transition calculation are exact for the nonrelativistic Coulomb model with the usual reduced-mass corrections neglected here. Orbital images are rescaled 2D slices. Electron filling through argon is a simplified ground-state Aufbau/Hund picture. The two-center panel is a linear-combination teaching model, not a full molecular calculation. Relativistic effects, electron correlation, spin-orbit coupling, nuclear structure and quantum electrodynamic corrections are beyond this layer.