A Particle in a Well of Finite Height

In an infinite well, the wavefunction is strictly zero outside the well. Real wells are finite: if the particle’s energy is below the wall height, the wavefunction decays outside the well but is not zero. This is evanescent penetration; if the forbidden region has finite width and another allowed region lies beyond it, the same mathematics permits tunnelling transmission.

1. Finite-square-well model

We use the common “finite square well” model:

• V(x) = 0 for 0 < x < L
• V(x) = V₀ for x ≤ 0 and x ≥ L

For a bound state, the particle has:

E < V₀

The finite-well schematic below shows the key bound-state signature: oscillatory ψ(x) inside the well with exponential tails outside.

2. Solutions in each region

Start from the time-independent Schrödinger equation:

-(ħ²/2m)d²ψ/dx² + Vψ = Eψ

Region II (inside the well, 0 < x < L)

Here V = 0, so:

d²ψ/dx² = -k²ψ where k² = 2mE/ħ²

General solution:

ψ_II(x) = A sin(kx) + B cos(kx)

Regions I and III (outside the well)

Here V = V₀ and for a bound state E < V₀, so:

d²ψ/dx² = κ²ψ where κ² = (2m(V₀-E))/ħ²

General solution:

ψ(x) = Ce^(κ x) + De^(-κ x)

Physical acceptability requires ψ → 0 far from the well:

• Region I ( x < 0): discard e^(-κ x) (it blows up as x → -∞), so ψ_I(x) = F e^(κ x)
• Region III ( x > L): discard e^(+κ x) (it blows up as x → + ∞), so ψ_III(x) = G e^(-κ x)

So the wavefunction is non-zero outside the well but decays exponentially.

3. Boundary conditions and quantisation

At a finite potential step, both ψ and its first derivative must be continuous:

• at x = 0: ψ_I = ψ_II and dψ_I/dx = dψ_II/dx
• at x = L: ψ_II = ψ_III and dψ_II/dx = dψ_III/dx

These conditions do not force ψ to be zero at the walls. Instead, they select only certain values of E that make the three-region solution “fit” smoothly, so the bound-state energies are still discrete.

4. Penetration depth and tunnelling

Outside the well, ψ ∝ e^(-κ x), so the probability density falls like:

|ψ|² ∝ e^(-2κ x)

A common way to describe how far the wavefunction penetrates is the penetration depth:

δ = 1/κ

Larger V₀ (or smaller E) makes κ larger, so the tail decays faster and the finite well approaches the infinite-well limit.

5. Common traps

1. Saying “the particle cannot be outside because E < V₀” (quantum mechanically it can, with small probability).
2. Using sinusoidal solutions outside the well for E < V₀ (they should be exponential).
3. Forgetting the continuity of both ψ and dψ/dx at boundaries.
4. Confusing ψ with |ψ|².

6. Quick check

A. For a bound state ( E < V₀), what is the qualitative form of the wavefunction outside the well?

B. If V₀ increases while E stays the same, what happens to the tunnelling tail?