Physic Labs

Quantum mechanics

The Schrödinger equation and wave packets

Observe a wave packet evolving under the time-dependent Schrödinger equation and the model's relation E = k₀²/2. Measure packet energy and wave number in model units, then verify E=k02/2E = k₀²/2.

Undergraduate

⚠ This quantum simulation is illustrative and does not fully describe a real physical system.

Equipment

  • One-dimensional wave packet and potential profile V(x)
  • k₀, σ, V₀, and width a sliders; run and reset controls

Procedure

  1. Adjust the wave packet and potential barrier

    Reset the simulation, adjust k₀ to change energy E = k₀²/2 and σ to change the packet's initial width; watch propagation and reflection on the graph. Move V₀ across the energy E and change barrier width a, then press Reset to start with the new configuration. Compare the transmitted portion as V₀ or a changes; this is a qualitative model in units ħ = m = 1. Compare the displayed values with E=k02/2E = k₀²/2.

  2. Observe the packet spreading

    Reset with the barrier off, run the simulation, and increase σ to make the initial packet wider. Observe its position and width over time; vary k₀ to change E = k₀²/2 without confusing energy with packet width. Compare the displayed values with ΔxΔp≥ℏ/2ΔxΔp ≥ ℏ/2.

  3. Investigate reflection at the barrier

    Turn on the barrier, vary V₀ around the energy E, and adjust its width a; reset before each run. Observe the reflected and transmitted parts as V₀ or a changes, comparing the barrier with E = k₀²/2 in units ħ = m = 1. Compare the displayed values with E=k02/2E = k₀²/2.

Simulation

Experiment history

The equation bearing Erwin Schrödinger's name appeared in 1926, in the early years of quantum mechanics, when physicists needed a rule for how a quantum state changes in time. Schrödinger developed a wave description that complemented Heisenberg's matrix mechanics with a continuous mathematical framework. For a particle in a potential V(x), the wavefunction ψ is not a classical trajectory; |ψ|² gives the probability density for finding the particle at a position in a measurement. The time-dependent equation iħ∂ψ/∂t = [−ħ²/(2m)∂²/∂x² + V(x)]ψ governs state evolution. A packet combines many wave numbers and can spread, reflect, or cross a potential barrier; transmission probability does not mean that a particle follows a known classical path. This simulation uses units ħ = m = 1, so a free packet has energy E = k₀²/2. Its results connect packet shape with potential energy, while remaining an illustration rather than a complete three-dimensional model or a direct measurement of a real system.

Related physicists

Erwin Schrödinger

Related library topics

The Schrödinger equationThe wavefunction and its probabilistic interpretationThe particle in a box and quantum tunneling