No degeneracy here
In a 1D box, each n is unique. Higher n simply adds nodes and raises energy by n^2.
n^2 scaling
Schrodinger Equation Solver (1D Particle in a Box)
Compute infinite square-well energies, normalized wavefunctions, and visualize |psi(x)|^2. Adjust length, mass, and quantum number to see how the ladder changes.
Setup
Selected state (n = 1)
Energy E_n (J)
—
E_n = n^2 h^2 / (8 m L^2)
Energy E_n (eV)
—
Uses 1 eV = 1.602e-19 J
Wavelength lambda_n
—
lambda_n = 2L / n (one spatial period)
Nodes
—
Nodes = n − 1 (excluding walls)
Energy vs ground (n=1)
Energy ladder
Wavefunction & probability density
psi_n(x) = sqrt(2/L) sin(n pi x/L). The probability curve is rescaled so |psi|^2 peaks at 1.
How this solver works
- Model: Infinite square well of length L with impenetrable walls at x=0 and x=L. Solutions are standing sine waves that vanish at the walls.
- Energy spectrum: E_n = n^2 h^2/(8 m L^2). Halve the length and the energy jumps by 4x. Heavier masses sit lower in the ladder.
- Wavefunctions: psi_n(x) = sqrt(2/L) sin(n pi x/L). We plot both psi and |psi|^2 (rescaled to peak at 1) so you can see nodes and antinodes.
- Energy ladder view: Bars show relative energies up to the count you choose. The height is normalized to the highest level displayed.
- Privacy: Everything runs in your browser—no uploads, no saves.
Quick quantum facts
Shorter box = hotter states
Cut L in half and the whole ladder jumps up by 4x. Confinement is expensive.
E ∝ 1/L^2
Probability piles at antinodes
|psi|^2 is largest where the sine peaks. For odd n, the center is a maximum; for even n, it is a node.
Standing waves
Real boxes leak
Finite wells allow tunneling and change the energies. This ideal model is the clean textbook limit.
Idealization
Use it for intuition
The same math powers quantum dots, nanowires, and the particle-in-a-ring/2D well variants.
Nanotech link