Finite Square Well Bound States Calculator

The finite square well is a more realistic version of the particle-in-a-box model: a particle sits in a region of lower potential energy (depth V0, half-width a) bounded by walls of finite, not infinite, height. Unlike the infinite well, the particle's wavefunction can tunnel a short distance into the classically forbidden region outside the well before decaying away.

Matching the wavefunction and its derivative at the well's edges gives two transcendental equations, one for even-parity states and one for odd-parity states, that cannot be solved algebraically. This calculator finds every root numerically (via bisection) inside the range 0 < z < z0, where z0 = a√(2mV0)/ħ is the dimensionless well-strength parameter, then converts each root into an exact bound-state energy.

In one dimension, any attractive well, no matter how shallow or narrow, supports at least one bound state, the ground state, which is always even parity. This is a general feature of 1D quantum wells and differs from 3D wells, which need a minimum depth-times-width² product before they trap any bound state at all.

The well's potential is symmetric about its centre, so every eigenstate must be either symmetric (even) or antisymmetric (odd) under reflection. Ordering all the bound states by increasing energy, the ground state is always even, the first excited state is always odd, and they continue to alternate, mirroring the node-counting pattern of the infinite square well.

In the infinite well, the wavefunction is exactly zero at the walls and there are infinitely many bound states. In the finite well, the wavefunction penetrates into the classically forbidden region outside the walls (decaying exponentially), and there are only a finite number of bound states, states above the well's rim (E ≥ 0) are unbound and form a continuum instead.

The number of bound states is controlled entirely by the dimensionless parameter z0 = a√(2mV0)/ħ, which combines the well's width, depth, and the particle's mass. Roughly one new bound state appears every time z0 increases by π/2, so wider, deeper wells, or lighter particles, support more bound states.

As V0 becomes very large, the lower bound-state energies (measured from the well's bottom) approach the infinite-square-well formula E_n = n²π²ħ²/(8ma²), since the walls become effectively impenetrable and the wavefunction is squeezed almost entirely inside. This is a useful consistency check: a very deep, wide finite well should reproduce the particle-in-a-box result.

No, a bound state, by definition, always has energy below the well's rim (E < 0 in the convention used here, where the well bottom is at −V0 and the rim is at 0). States with E ≥ 0 are not bound, they form a continuum of scattering states that are not trapped by the well and are not covered by this calculator.

The finite square well is a standard first approximation for electrons trapped in a semiconductor quantum well, for nucleons bound loosely inside a nucleus (modeled as a finite well rather than the infinite well used for rough nuclear energy-scale estimates), and generally for any particle confined by a short-range attractive potential with a finite barrier height.

The matching equations, √(z0²−z²)cos z = z sin z for even states and √(z0²−z²)sin z = −z cos z for odd states, mix polynomial and trigonometric terms and have no closed-form algebraic solution. This is completely standard in quantum mechanics: bisection or Newton's method reliably finds every root to any desired precision, giving exact energies rather than an approximation.