Questions: The Finite Square Well

Question 1

In one dimension, every finite square well, no matter how shallow or narrow, has at least one bound state.

• T. True
• F. False

Answer: True

In the graphical solution, the even-state curve tan z always crosses √(z₀²/z² − 1) at least once for any z₀ > 0, so there is always an even ground state.

Question 2

Compared with an infinite well of the same width, the ground-state energy of a finite well (measured from the bottom of the well) is:

• A. Higher
• B. Lower
• C. Exactly the same
• D. Zero

Answer: (B)

The finite-well wavefunction extends into the walls, so it bends less sharply inside. Less curvature means less kinetic energy and a lower energy.

Question 3

A finite well has the dimensionless strength z₀ = (a/ħ)√(2mV₀) = 4. How many bound states does it have?

• A. 1
• B. 2
• C. 3
• D. 4

Answer: (C)

A new state appears each time z₀ passes a multiple of π/2. Since 4 lies between π (about 3.14) and 3π/2 (about 4.71), there are three states: two even and one odd.

Question 4

An electron is bound with energy E = −1.0 eV relative to the region outside the well. Estimate the distance over which its wavefunction decays by a factor e outside the well.

Model answer: About 0.2 nm

κ = √(2m|E|)/ħ = √(2 × 9.11 × 10⁻³¹ × 1.60 × 10⁻¹⁹)/(1.055 × 10⁻³⁴) ≈ 5.1 × 10⁹ m⁻¹. The decay length is 1/κ ≈ 0.2 nm.