5Experiment 5

The Finite Square Well: Bound States and Barrier Penetration

Observe wavefunction penetration into classically forbidden regions, count bound states as depth and width vary, and compare with the infinite-well ground state.

A. Bound states of a finite square well

Hamiltonian discretised on a 181-point grid with 5-point O(Δx⁴) finite differences; diagonalised exactly by the Jacobi method. Accuracy: ~0.5% vs. exact transcendental-equation result.

Particle:

Width L (nm)1.00 nm

Depth V₀ (eV)2.00 eV

Bound states: —

Equiv. infinite well E₁ (same L): 0.37607 eV
Finite well E₁ < infinite well E₁ (wavefunction penetrates beyond L).

Click a level to view its wavefunction.

ψₙ(x) |ψₙ(x)|² V(x)

Note: the wavefunction does not vanish at the well edges (dashed vertical lines). It penetrates exponentially into the classically forbidden region — the same effect that drives quantum tunneling.

Observations

Record your measured data in the tables below. Refer to the lab manual for the full procedure.

Table 5.1 — Effect of well depth (electron, $L = 1$ nm)

V₀ (eV)	Bound states	E₁ (eV)

Rows: 0.5, 2.0, 4.0 eV

Table 5.2 — Finite vs. infinite well ground state ( $L = 1$ nm, $V_{0} = 2$ eV)

Quantity	Value (eV)

Rows: Finite-well E₁; Infinite-well E₁ (same L)