Analyze wells, barriers, oscillators, and custom potentials numerically. Tune constants and domains for stable solutions. Plot wavefunctions, densities, and potentials for deeper insight today.
This page solves the one-dimensional time-independent Schrodinger equation on a finite grid using a numerical Hamiltonian matrix. Use consistent arbitrary units for mass, energy, distance, and the reduced Planck constant.
Supported models: infinite well, finite well, harmonic oscillator, barrier, step, double well, and custom polynomial potential.
| Scenario | Potential | Domain | Grid | Key Inputs | Expected Use |
|---|---|---|---|---|---|
| Ground-state oscillator | Harmonic oscillator | -5 to 5 | 61 | m=1, ħ=1, ω=1 | Checks evenly spaced low-lying energy levels. |
| Bound finite well | Finite square well | -6 to 6 | 61 | width=4, depth=20 | Shows localized states below the outside potential. |
| Tunneling barrier | Finite barrier | -6 to 6 | 61 | width=2, height=12 | Highlights penetration and shape changes near the barrier. |
| Asymmetric landscape | Polynomial potential | -4 to 4 | 71 | c2=1, c3=0.2, c4=0.05 | Tests custom confining or tilted energy landscapes. |
Continuous equation:
-(ħ² / 2m) (d²ψ / dx²) + V(x)ψ = Eψ
Finite-difference second derivative:
(d²ψ / dx²) ≈ [ψ(i+1) - 2ψ(i) + ψ(i-1)] / Δx²
Hamiltonian entries:
Diagonal: Hii = 2k + Vi
Off-diagonal: Hi,i±1 = -k
where k = ħ² / (2mΔx²)
After building the symmetric Hamiltonian matrix, the calculator computes approximate eigenvalues and eigenvectors numerically. The eigenvalues are energy levels, and the eigenvectors are normalized wavefunctions on the grid.
Accuracy improves when the domain is wide enough, the grid is dense enough, and the potential changes smoothly relative to the spacing. Very sharp barriers or overly small domains can distort the spectrum.
For most interactive use, a grid between 41 and 81 points is a practical balance between speed and stability in a single-file implementation.
It solves the one-dimensional time-independent Schrodinger equation numerically for several potential models and returns approximate bound-state energies, wavefunctions, and probability densities.
They are numerical approximations. Accuracy depends on domain size, grid density, and whether the chosen potential is represented well by the finite-difference mesh.
Use any consistent unit system. If distance, mass, energy, and ħ are not internally consistent, the computed energies and shapes will not represent a meaningful physical system.
A normalized wavefunction should integrate to one. The normalization check confirms that the probability density has been scaled correctly on the chosen numerical grid.
The grid controls how accurately derivatives and sharp potential features are represented. A finer grid usually improves convergence, though it also increases computational cost.
Nodes are interior sign changes of the wavefunction. Lower states generally have fewer nodes, while higher excited states have more oscillations across the domain.
Yes. Use the polynomial model to create symmetric, tilted, or anharmonic shapes by tuning coefficients around the chosen center coordinate.
Numerically, it is approximated with a very large wall height. That approach closely reproduces the expected confinement while keeping the Hamiltonian finite and computable.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.