Particle in a box

An idealized quantum system confines a particle to a region with impenetrable boundaries. Its allowed wavefunctions and energies are discrete, illustrating quantum confinement.

Schrödinger equation: A differential equation governing how a quantum state changes in space and time. Its stationary form determines the particle’s allowed wavefunctions and energies.
Quantum mechanics: The physical theory describing matter and energy at atomic and subatomic scales. The box is a simple model for its rules of states and measurement.
Quantum confinement: The alteration of a system’s quantum states when its motion is restricted to a small region. The box is the simplest setting for seeing confinement produce discrete energies.
Finite square well: A potential well with finite barriers that permits bound states and wavefunction penetration outside the well. Unlike the ideal box, its walls allow tunneling into the surrounding region.
Infinite square well: A one-dimensional potential that is zero inside a region and infinite outside it. This standard version makes the box’s boundary conditions exact and solvable.
Potential energy: Energy associated with a system’s position or configuration in a force field. The box is defined by its potential inside and at its boundaries.
Quantum dot: A nanoscale semiconductor structure that confines electrons in all three spatial dimensions. Its size-dependent electronic states resemble a three-dimensional box spectrum.
Harmonic oscillator: A quantum system whose potential energy is proportional to the square of its displacement from equilibrium. Its equally spaced energy levels contrast with the box’s increasing level gaps.
Boundary condition: A restriction that a mathematical solution must satisfy at the edge of a domain. The wavefunction must vanish at the impenetrable walls.
Eigenvalue: A scalar associated with a linear operator acting on one of its eigenvectors. Allowed energies arise as eigenvalues of the Hamiltonian.

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