Compute infinite square-well energies, wavelengths, normalized wavefunctions, and visualize |psi(x)|^2. Adjust length units, particle preset, mass, and quantum number to see how the ladder changes.
| n | Energy J | Energy eV | Wavelength | Nodes |
|---|
psi_n(x) = sqrt(2/L) sin(n pi x/L). The probability curve is rescaled so |psi|^2 peaks at 1.
v1.1 (May 18, 2026)
For an electron confined to a 1 nm infinite square well, the ground-state energy is:
E₁ ≈ 0.376 eV
The third level is E₃ = 9E₁ ≈ 3.38 eV, because energy scales with n². This is why nanoscale confinement can create energy gaps large enough to matter in semiconductor and quantum-dot systems.
This Schrödinger equation solver is a visual particle-in-a-box calculator for the 1D infinite square well. It calculates the allowed energy levels and wavefunctions for a particle trapped in an infinite square well, then plots the wavefunction and the probability density so you can see what the math means. Think of it as a guided look at how quantum confinement creates discrete, ladder-like energy states instead of a smooth, continuous spectrum.
The model assumes perfectly rigid walls at x=0 and x=L. Inside the box, the Schrodinger equation becomes a standing-wave problem, so only specific sine-wave patterns are allowed. Each pattern is labeled by the quantum number n. As n increases, the wavefunction gains more nodes and the energy grows with n squared. The calculator uses the standard formulas for a 1D infinite square well to show the energy E_n in joules and electron volts, the wavelength for that mode, and the normalized wavefunction psi_n(x) along with |psi_n(x)|^2.
To use the calculator, start by entering the box length L and the particle mass. You can choose a common particle or enter a custom value, then pick the quantum number n you want to examine. Press Solve to update the energy readout, the energy ladder, and the plots. The ladder compares relative energy levels up to the number you select, while the graph shows the wavefunction shape and the probability density, which indicates where the particle is most likely to be found.
This tool is useful for building intuition about quantum wells and related systems. The same ideas appear in semiconductor quantum dots, nanowires, and thin films where electrons are confined to tiny regions. Students often use the particle-in-a-box model to understand why smaller systems have higher energy gaps, why heavier particles have lower energies, and how standing waves translate into measurable probabilities. Since everything runs in your browser, it is a quick, private way to explore the Schrodinger equation without setup or downloads.
The infinite square well has V(x)=0 inside the box from x=0 to x=L, while V(x) is infinite outside the box. The particle can exist only inside the well.
The wavefunction must be zero at both walls, so psi(0)=0 and psi(L)=0. Those boundary conditions allow only standing sine waves that fit an integer number of half-wavelengths inside the box.
Normalization requires the total probability across the box to equal 1. That gives psi_n(x)=sqrt(2/L) sin(n pi x/L), where n is a positive integer.
Each allowed standing wave has wavelength lambda_n=2L/n. Substituting the corresponding momentum into the kinetic-energy expression gives E_n = n²h² / 8mL².
It solves the 1D infinite square well, also called the particle-in-a-box model.
No. It focuses on the analytic particle-in-a-box case rather than arbitrary potentials or time-dependent wavefunctions.
E_n = n²h² / 8mL².
Because E is proportional to 1/L², stronger confinement produces higher allowed energies.
The nth state has n − 1 internal nodes, excluding the walls.
In a 1D box, each n is unique. Higher n simply adds nodes and raises energy by n^2.
Cut L in half and the whole ladder rises by 4x. Stronger confinement raises the allowed energies.
|psi|^2 is largest where the sine peaks. For odd n, the center is a maximum; for even n, it is a node.
Finite wells allow tunneling and change the energies. This ideal model is the clean textbook limit.
The same math powers quantum dots, nanowires, and the particle-in-a-ring/2D well variants.
