Quantum Mechanics of a Particle in a Potential Well
Theory: 1D Particle in a Box (Infinite Potential Well)
In quantum mechanics, the behavior of particles confined to a one-dimensional box can be analyzed using the Schrödinger equation. This experiment provides an interactive visual simulation of a particle's wavefunction in a 1D infinite potential well using the time-dependent Schrödinger equation.
Introduction
The "particle in a box" model is one of the fundamental problems in quantum mechanics. It describes a particle that is free to move within a region of space (the "box") but is completely confined by infinite potential barriers at the boundaries. This simple yet powerful model helps illustrate key quantum mechanical concepts.
Key Concepts
1. Wavefunction Ψ(x,t)
The wavefunction describes the quantum state of the particle. In our simulation:
- The complete wavefunction is: Ψ(x,t) = A·sin(nπx/L)·cos(ωt)
- Spatial Component: sin(nπx/L) - describes the distribution within the box
- Time Component: cos(ωt) - describes how the wavefunction evolves over time
2. Quantum Number (n)
The quantum number n = 1, 2, 3, ... determines:
- The energy level of the particle
- The number of nodes (n-1 nodes for quantum number n)
- The oscillation frequency of the wavefunction
3. Energy Levels
The energy of each quantum state is given by:
Eₙ = n²h²/(8mL²)
Where:
- h is the Planck's constant
- Eₙ represents the energy corresponding to quantum number n
- n is the quantum number (1, 2, 3, ...)
- L is the length of the box
- m is the mass of the particle
4. Probability Density |Ψ|²
The probability of finding the particle at a given position is given by |Ψ(x,t)|². In the simulation:
- Enable the "Show |Ψ|² (Probability)" toggle to visualize this
- Notice how probability is maximum at antinodes and zero at nodes
5. Box Length (L)
Changing the box length affects:
- The wavelength of the wavefunction (λ = 2L/n)
- The energy levels (inversely proportional to L²)
- Smaller boxes lead to higher energy states
Simulation Features
Parameters Panel
Display Options
- Show |Ψ|²: Toggle probability density visualization
- Show Grid: Enable/disable background grid
- Show Axis Labels: Toggle axis labels on canvas
Live Measurements
The simulation displays real-time values for:
- Energy (Eₙ): Current energy level in eV
- Wavelength (λ): de Broglie wavelength in nm
- Nodes: Number of zero-crossing points
- Period: Oscillation period in frames
- Time (t): Current simulation time step
- State: Current quantum state description
Key Observations
- Higher quantum numbers (n) result in more oscillations within the box
- Energy increases as the square of the quantum number (Eₙ ∝ n²)
- The number of nodes equals (n - 1)
- Probability density |Ψ|² is always positive and shows where the particle is likely to be found
- The wavefunction must be zero at the boundaries (x = 0 and x = L)
Mathematical Foundation
Schrödinger Equation
The time-independent Schrödinger equation for a particle in a box:
- Inside the box: -ℏ²/(2m) · d²Ψ/dx² = EΨ, here V = 0
- At boundaries: Ψ(0) = Ψ(L) = 0
Wavefunction Solution
The normalized spatial wavefunction: Ψₙ(x) = √(2/L) · sin(nπx/L)