Questions: Particle in a Box (Infinite Square Well)

Question 1

A physicist is designing quantum dots (semiconductor nanocrystals) for use in displays. She wants dots that emit lower-energy (longer wavelength, redder) light. Using the particle-in-a-box relationship E_n ∝ 1/L², which design change would achieve this?

• A. Make the dots smaller — tighter confinement lowers the energy levels
• B. Make the dots larger — a bigger box reduces the confinement energy, lowering all energy levels
• C. Use a lighter semiconductor material — lower mass increases the energy
• D. Increase the quantum number n — higher states always have lower energy gaps

Answer: (B)

Since E_n = n²π²ℏ²/(2mL²), energy scales as 1/L² — doubling the box size reduces energy levels by a factor of four. Larger quantum dots have lower energy transitions, emitting lower-energy (redder) photons. This is why the size-tunable color of quantum dots is one of their most commercially valuable properties: the same material emits blue when made very small and red when made larger. Option A reverses the relationship — smaller boxes give *higher* energy, bluer light.

Question 2

Why does a particle confined in a box have nonzero energy even in its ground state (n=1)?

• A. The particle gains potential energy by pressing against the walls of the box
• B. The wavefunction normalization constant contributes kinetic energy to the ground state
• C. The Pauli exclusion principle prevents two particles from sharing the zero-energy state
• D. Confinement to a region of width L imposes a position uncertainty, which requires a nonzero momentum spread and thus nonzero kinetic energy

Answer: (D)

This is the zero-point energy, and its origin is the Heisenberg uncertainty principle. Confining a particle to a box of width L means Δx ~ L. By ΔxΔp ≥ ℏ/2, this forces Δp ≥ ℏ/(2L) — the momentum cannot be precisely zero. Nonzero momentum spread means nonzero average kinetic energy. A particle at rest inside the box would have zero momentum (precisely), violating the uncertainty bound set by its confinement. The zero-point energy is not a measurement artifact — it is irreducible and fundamental.

Question 3

A particle in a larger box (greater L) has higher energy levels for each quantum number n compared to a particle in a smaller box.

• T. True
• F. False

Answer: False

The energy levels scale as E_n = n²π²ℏ²/(2mL²), which is *inversely* proportional to L². Larger box → lower energy levels for all n. This is counterintuitive if you think of confinement as 'squeezing' energy into the particle, but it follows directly from the standing-wave condition: a larger box accommodates longer wavelengths, which correspond to lower momenta and lower kinetic energies. The common misconception reverses this relationship.

Question 4

The zero-point energy of a particle in a box arises because the wavefunction must satisfy boundary conditions, and the lowest-energy solution that satisfies ψ(0) = ψ(L) = 0 has n=1, not n=0.

• T. True
• F. False

Answer: True

n=0 would give ψ(x) = 0 everywhere — a zero wavefunction, which means no particle at all. The minimum physically meaningful quantum number is n=1, corresponding to a half-wavelength fitting exactly inside the box. This gives E₁ = π²ℏ²/(2mL²) > 0. Both the boundary condition explanation (minimum standing wave) and the uncertainty principle explanation (nonzero momentum spread from confinement) are correct and complementary ways of understanding why zero-point energy is nonzero.

Question 5

Why can't a particle in a box have zero energy in its ground state? Connect your answer to the Heisenberg uncertainty principle.

Model answer: Zero energy would require zero momentum — the particle would be at rest. But the Heisenberg uncertainty principle states ΔxΔp ≥ ℏ/2. Confinement inside a box of width L bounds the position uncertainty to Δx ≤ L, which forces the momentum uncertainty to satisfy Δp ≥ ℏ/(2L). This is a nonzero lower bound on momentum spread, which means the particle cannot have precisely zero momentum. Since kinetic energy KE = p²/(2m), a nonzero momentum spread implies a nonzero average kinetic energy — this is the zero-point energy E₁ = π²ℏ²/(2mL²).

The zero-point energy is one of the clearest demonstrations that quantum mechanics fundamentally limits how 'quiet' a confined system can be. It explains why atoms don't collapse (electrons can't fall into the nucleus without infinite confinement energy), why liquid helium remains liquid at absolute zero (zero-point motion prevents it from freezing at normal pressure), and why quantum dot size controls emission color. The boundary condition and uncertainty principle descriptions are two faces of the same physical constraint.