Infinite Square Well Energy Calculator
Calculates the quantised energy levels of a particle confined in an infinite square potential well, given the quantum number, well width, and particle mass. Widely used in semiconductor physics and quantum device design.
About this calculator
A particle trapped in an infinitely deep potential well of width L can only occupy discrete energy states, because the wavefunction must vanish at the walls. The energy of the nth level is: E_n = n²h² / (8mL²), where n = 1, 2, 3, … is the quantum number, h = 6.626×10⁻³⁴ J·s is Planck's constant, m is the particle mass in kg, and L is the well width in metres. Energy scales as n², so the gap between successive levels grows with n. It also scales as 1/L², meaning a narrower well pushes energy levels higher — this is the quantum confinement effect exploited in quantum dot displays and semiconductor lasers. For an electron in a 1 nm well, the ground-state energy is roughly 0.38 eV, which falls in the visible photon range and explains why quantum dot colour depends on size.
How to use
Find the ground-state energy (n = 1) of an electron (m = 9.109×10⁻³¹ kg) in a well of width L = 1×10⁻⁹ m (1 nm). Step 1 — compute n²h²: 1² × (6.626×10⁻³⁴)² = 4.390×10⁻⁶⁷ J²·s². Step 2 — compute 8mL²: 8 × 9.109×10⁻³¹ × (1×10⁻⁹)² = 7.287×10⁻⁴⁸ kg·m². Step 3 — divide: E₁ = 4.390×10⁻⁶⁷ / 7.287×10⁻⁴⁸ = 6.025×10⁻²⁰ J. Step 4 — convert: 6.025×10⁻²⁰ / 1.602×10⁻¹⁹ ≈ 0.376 eV. The n = 2 level would be 4 × 0.376 ≈ 1.50 eV.
Frequently asked questions
Why do energy levels in an infinite square well increase with the square of the quantum number?
The allowed wavefunctions in the well are standing waves with wavelengths λ_n = 2L/n, so higher quantum numbers correspond to shorter wavelengths and higher spatial frequencies. Since kinetic energy is proportional to momentum squared, and momentum is inversely proportional to wavelength (p = h/λ), the energy goes as p² ∝ n²/L². This quadratic scaling means higher levels become increasingly spaced apart, unlike the harmonic oscillator where levels are equally spaced. The n² dependence is a direct consequence of confining a free particle between rigid walls.
How does well width affect the energy levels of a confined particle?
Energy levels scale as 1/L², so halving the well width increases all energies by a factor of four. This is the quantum confinement effect: a narrower box forces shorter-wavelength wavefunctions, increasing kinetic energy. In quantum dot nanocrystals, the dot diameter plays the role of L, and engineers tune the dot size to control the colour of emitted light. Larger dots emit red (lower energy), smaller dots emit blue (higher energy). This tunability without changing material chemistry makes quantum dots ideal for display technology and biological imaging.
What is the difference between an infinite and a finite potential well in quantum mechanics?
In an infinite well, the potential at the walls is assumed to be infinitely large, so the wavefunction is strictly zero outside and the particle is perfectly confined. Energy levels are E_n = n²h²/(8mL²) with no exceptions. In a finite well, the potential has a finite height V₀, and the wavefunction decays exponentially but is non-zero outside the well — the particle has a small probability of being found in the classically forbidden region. This leads to fewer allowed bound states, slightly lower energy levels than the infinite case, and the possibility of quantum tunnelling through the barrier. Real semiconductor heterostructures are modelled as finite wells.