Infinite Square Well Calculator
Calculates the quantized energy levels, de Broglie wavelengths, and momenta of a particle confined in a one-dimensional infinite square well. Ideal for introductory quantum mechanics courses and nanoscale device modeling.
About this calculator
The infinite square well (particle-in-a-box) is the foundational exactly solvable problem in quantum mechanics. A particle of mass m confined to a box of width L can only occupy discrete energy levels given by E_n = n²h²/(8mL²), where n = 1, 2, 3, … is the quantum number, h = 6.626×10⁻³⁴ J·s is Planck's constant, and the result is typically expressed in electron-volts. The allowed de Broglie wavelengths inside the well are λ_n = 2L/n, corresponding to standing waves that fit an integer number of half-wavelengths into the box. The momentum magnitude is p_n = nℏπ/L = nh/(2L). The wave function is ψ_n(x) = √(2/L) sin(nπx/L), representing a standing wave with n antinodes. These results explain why electrons in nanometer-scale quantum dots and wells absorb and emit light at discrete, size-tunable wavelengths.
How to use
Consider an electron (m = 9.109×10⁻³¹ kg) in a well of width L = 1 nm = 1×10⁻⁹ m, quantum number n = 2. Energy: E₂ = (2² × (6.626×10⁻³⁴)²) / (8 × 9.109×10⁻³¹ × (1×10⁻⁹)²) = (4 × 4.390×10⁻⁶⁷) / (7.272×10⁻⁴⁸) = 1.756×10⁻⁶⁶ / 7.272×10⁻⁴⁸ ≈ 2.415×10⁻¹⁹ J ÷ 1.602×10⁻¹⁹ ≈ 1.508 eV. Wavelength: λ₂ = 2 × 1 nm / 2 = 1.000 nm. Momentum: p₂ = (2 × 6.626×10⁻³⁴) / (2 × 1×10⁻⁹) = 6.626×10⁻²⁵ kg·m/s.
Frequently asked questions
Why are energy levels in an infinite square well quantized and not continuous?
Quantization arises from the boundary conditions: the wave function must be exactly zero at both walls (x = 0 and x = L) because the potential is infinite outside the box. This restricts the allowed standing-wave solutions to those where an integer number of half-wavelengths fits inside the well. Only specific wavelengths — and therefore specific momenta and energies — satisfy this condition. This is directly analogous to the discrete harmonics of a vibrating string fixed at both ends, but applied to matter waves.
How does well width affect the energy levels of a confined particle?
Because E_n = n²h²/(8mL²), the energy scales as 1/L². Halving the well width quadruples each energy level, which is why electrons in atomic-scale quantum dots have much higher energies than those in larger nanostructures. This strong size dependence is exploited in semiconductor quantum wells (used in laser diodes) and colloidal quantum dots (used in QLED displays) to tune the emission color simply by controlling the physical size of the confinement region. Making the box smaller pushes levels further apart and blue-shifts the emitted photons.
What is the ground state energy of an electron in an infinite square well and why is it not zero?
For an electron in a 1 nm well, the ground state (n = 1) energy is approximately 0.377 eV, not zero. This non-zero minimum energy is a direct consequence of the Heisenberg uncertainty principle: confining the particle to a finite region Δx = L gives it a minimum momentum uncertainty Δp ≥ ℏ/(2L), which translates to a minimum kinetic energy. A particle at rest inside the box would violate the uncertainty principle. This zero-point energy has real physical consequences, including contributing to the stability of atoms and the properties of superfluid helium at absolute zero.