The Particle in a Box
The infinite square well — a particle trapped between two perfectly rigid walls — is the 'hydrogen atom' of quantum mechanics pedagogy. Despite its simplicity, it captures the essential features of quantum confinement: quantized energy levels, standing wave patterns, and zero-point energy. Every physics student meets this problem in their first quantum mechanics course.
Energy Quantization
Unlike a classical particle that can have any energy, the quantum particle can only occupy discrete energy levels E_n = n^2 pi^2 hbar^2 / (2mL^2). The energies scale as n^2 — the second level has 4 times the ground state energy, the third has 9 times, and so on. The energy level diagram on the right shows these quantized levels. This quantization arises from the boundary conditions: the wave function must be zero at both walls, which restricts it to standing sine waves with integer half-wavelengths fitting inside the box.
Wave Functions and Probability
The wave function psi_n(x) = sqrt(2/L) sin(n pi x / L) tells us the probability amplitude for finding the particle at position x. The probability density |psi|^2 (shown as the shaded region) gives the actual probability. For the ground state (n=1), the particle is most likely found at the center. For higher states, the probability develops nodes — points where the particle will never be found.
Zero-Point Energy
The lowest possible energy E_1 is not zero — the particle always retains a minimum kinetic energy called the zero-point energy. This is a direct consequence of Heisenberg's uncertainty principle: confining the particle's position increases the uncertainty in its momentum, and hence its kinetic energy. This effect is physically real and has measurable consequences, from the stability of atoms to the Casimir effect.
Superposition and Time Evolution
Toggle 'Superposition' to see what happens when the particle occupies a mixture of the n=1 and n=2 states simultaneously. The probability density now oscillates in time — the particle 'sloshes' back and forth in the box at a frequency proportional to the energy difference between the two levels. This demonstrates quantum dynamics and the principle of superposition.