Infinite Potential Well: Ground State Energy When Width Doubles

The ground-state energy in a 1D infinite potential well scales inversely with the square of the well width, so doubling the width from L to 2L reduces the energy to one-fourth of the original.

Energy Formula Basics

The ground-state energy E₁ for width L follows E₁ = h²/(8mL²) where h is Planck’s constant and m is particle mass. For the new width 2L, E₂ = h²/(8m(2L)²) = h²/(32mL²). Thus, E₂ = E₁/4 since the L² dependence makes energy proportional to 1/L².

Option Analysis

• a. E₂ = 1.41 × E₁: Incorrect; 1.41 (≈√2) applies if energy halves, requiring width √2 L, not 2L.
• b. E₂ = 4 × E₁: Wrong; this occurs if width halves to L/2, increasing energy fourfold.
• c. E₂ = E₁ / 2: False; linear scaling would imply 1/L dependence, but quantum energy scales as 1/L².
• d. E₂ = E₁: No; wider well allows lower momentum states, reducing energy.

Correct answer: None listed match E₂ = E₁/4 exactly, but closest conceptual error analysis points to misunderstanding the quadratic scaling; standard problems confirm 1/4 ratio.

Quantum Mechanics Context

In quantum mechanics, the 1D infinite potential well models particle confinement, with 1D infinite potential well width doubled ground state energy dropping to 1/4 due to E ∝ 1/L². This key concept appears in exams testing energy quantization.

Derivation Steps

Start with Schrödinger equation inside well (V=0): -ℏ²/2m d²ψ/dx² = Eψ. Solutions are ψₙ(x) = √(2/L) sin(nπx/L), energies Eₙ = n²π²ℏ²/(2mL²). Ground state (n=1): E₁ ∝ 1/L². Doubling L to 2L gives E₂ = E₁/4.

Why Energy Decreases

Larger L means longer de Broglie wavelengths fit, lowering minimum kinetic energy (zero-point energy). All levels scale identically by 1/4, unlike harmonic oscillator.

Exam Relevance

For CSIR NET Life Sciences/Physics crossovers or quantum chemistry, recognize this scaling avoids traps like linear (option c) or inverse assumptions. Visualize: wavefunction spreads, probability density flattens.