Schrödinger Equation Solver

The time-independent Schrödinger equation is Ĥψ = Eψ, where Ĥ is the Hamiltonian operator (kinetic + potential energy), ψ is the wavefunction, and E is the energy eigenvalue. Solving it yields the allowed stationary energy states of a quantum system. In 1D it takes the form −(ℏ²/2m)d²ψ/dx² + V(x)ψ = Eψ.

The infinite square well confines a particle to a region of width L with infinitely high walls. Inside the box the potential is zero; outside it is infinite, so the particle cannot escape. The exact energy eigenvalues are Eₙ = n²π²ℏ²/(2mL²), giving discrete quantized energy levels proportional to n².

For a harmonic oscillator with angular frequency ω, the energy eigenvalues are Eₙ = (n − ½)ℏω using the convention n = 1, 2, 3… (or Eₙ = (n + ½)ℏω with n = 0, 1, 2…). The levels are equally spaced by ℏω, and the ground state has non-zero zero-point energy E₁ = ½ℏω.

Zero-point energy is the lowest possible energy a quantum system can have — even at absolute zero temperature. It arises from the Heisenberg uncertainty principle: a perfectly stationary particle would violate ΔxΔp ≥ ℏ/2. For the infinite square well, E₁ = π²ℏ²/(2mL²); for the harmonic oscillator, E₁ = ½ℏω.

The quantum number n labels the energy eigenstates of a bound system. n = 1 is the ground state (lowest energy), n = 2 is the first excited state, and so on. Each state has a distinct energy and wavefunction shape with (n − 1) nodes inside the well.

Energy eigenvalues are inversely proportional to particle mass. A heavier particle has lower, more closely spaced energy levels for the same potential geometry. This is why protons in a nucleus have much larger energy spacings than electrons in an atom of similar size.

A finite square well has walls of finite height V₀. Unlike the infinite well, the wavefunction penetrates into the classically forbidden region outside the well (exponential decay), and only a finite number of bound states exist — those with energy E < V₀. The allowed energies must be found by solving a transcendental equation, and they are always slightly lower than the corresponding infinite-well levels.

The de Broglie wavelength λ = h/p relates a particle's momentum p to a wave. For a particle with kinetic energy equal to the computed eigenvalue, λ = h/√(2mE). This calculator displays it as a useful cross-check: for the infinite square well, the ground-state de Broglie wavelength equals 2L (half a wavelength fits in the box).