Physic Labs

Quantum mechanics

The wavefunction, the Schrödinger equation, and the hydrogen atom

Directly observe quantum tunneling, the uncertainty principle through a position measurement, and the true shapes of s, p, d, f orbitals. Measure transmission probability, packet widths, and orbital energy, and compare them with ΔxΔp≥ℏ/2ΔxΔp ≥ ℏ/2 and En=−13.6,eV/n2E_n = −13.6,eV/n².

Undergraduate

Equipment

  • 1D wave packet solved numerically via the split-step Fourier method
  • Rectangular potential barrier with adjustable height
  • Monte Carlo sampler for the hydrogen atom's |ψₙₗₘ|²

Procedure

  1. Observe quantum tunneling

    Select 'Tunneling through a barrier', set the barrier height V₀ above the particle energy E. Watch the transmission probability T shown below — it's nonzero even though a classical object could never cross. Compare the observed quantities with T∝e−2κaT \propto e^{−2κa}.

  2. Measure position and see the uncertainty principle

    In 'Free wave packet', click 'Measure position'. The wavefunction collapses into a very narrow packet (definite position), then spreads out very quickly afterward because momentum is now highly uncertain. Compare the observed quantities with ΔxΔp≥ℏ/2ΔxΔp ≥ ℏ/2.

  3. Compare orbital shapes

    Switch to the Hydrogen Atom section, change n, l, m. Compare the 1s orbital (solid sphere) with 2p (two lobes) and 3d (four lobes), and read the number of nodal surfaces shown below. Compare the observed quantities with En=−13.6 eV/n2E_n = −13.6\,eV/n².

Simulation

Experiment history

Quantum mechanics emerged in the early twentieth century to explain observations that classical mechanics could not describe at atomic scales. Schrödinger developed a wave equation for quantum states, while Max Born connected the squared amplitude of the wavefunction with the probability of finding a particle. This interpretation distinguishes a probability cloud from a miniature planetary orbit: the theory predicts measurement distributions, not a known path traced by an electron. The equation iħ∂ψ/∂t = Ĥψ describes time evolution; its stationary form leads to definite-energy states such as those of hydrogen. A particle's wave in a potential can also have nonzero probability beyond a barrier, the effect called quantum tunneling. Position and momentum spreads are related by ΔxΔp ≥ ħ/2. These simulations illustrate consequences under specific assumptions; they do not replace a complete atomic model or an experimental measurement.

Related physicists

Erwin SchrödingerMax BornMax PlanckNiels BohrWerner Heisenberg

Related library topics

The Schrödinger equationThe Heisenberg uncertainty principleThe particle in a box and quantum tunnelingThe hydrogen atom: energy levels and orbitals