Particle in a box

Particle in a box

1. What the Model Is

Particle in a 1D infinite potential well (box) = particle mass $m$ confined to $0<x<L$ where $V=0$ inside, $V=∞$ outside (walls). Classic solvable quantum system showing quantized energies and wavefunctions due to boundary conditions; approximates quantum well, dot.

2. TISE Inside Box

Inside, $V=0$:

where

General:

Outside, $φ=0$ (infinite $V$).

Boundary: $φ(0)=0 ⇒ B=0$; $φ(L)=0 ⇒ sin kL =0 ⇒ kL=nπ$, $n=1,2,3,...$ (n=0 trivial, negative same).

Thus quantized:

Energy grows $∝n²/L²$ — confinement raises energy.

3. Normalized Wavefunctions

for $0<x<L$, zero outside. Probability

has $n$ half-waves, $n-1$ nodes inside (excluding walls). Ground state $n=1$ has no interior node; $n=2$ has one at $L/2$.

Full $ψ_n(x,t)=φ_n(x) e^{-iE_n t/ħ}$.

4. 3D Box (mention)

For $L_x×L_y×L_z$:

degeneracy when dimensions equal.

5. Worked Examples

(a) Electron in L=0.1 nm (atom size): E1=h²/(8mL²)=6.02e-18 J≈37.6 eV ; E2=4E1≈150 eV ✓

(b) L=1 nm: E1≈0.376 eV (nanoscale, relevant for quantum dots vs atom's 37 eV)

(c) Compare classical: particle can have any E, quantum only discrete; as L→∞, spacing E1∝1/L²→0 ⇒ continuum.

(d) Probability to find particle between L/3 and 2L/3 for n=1: P=∫_{L/3}^{2L/3}2/L sin²(πx/L)dx≈0.61 (more central)

6. Exam Points

• Derive $φ_n=√(2/L)sin(nπx/L)$, $E_n=n²h²/8mL²$ with $φ=0$ at walls — 7-10 marks, very frequent (the most common quantum derivation).
• Sketch $φ_1, φ_2$ and $|φ|²$ — 5 marks, frequent.
• Compute $E_n$ for electron in nm box — 5 marks, frequent.
• Explain quantization due to boundary — 3 marks.