Playground
Standing wave modes inside infinite square well. Shows psi_n(x) and |psi_n|^2 for selected n, with energy ladder.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Energy of level noutput Quantized energy of the nth stationary state | J | M*L^2*T^-2 | 0 – 1e-15 | |
| Quantum number Positive integer mode index | dimensionless | 1 | 1 – 50 | |
| Reduced Planck constant h / (2*pi) | J*s | M*L^2*T^-1 | 1.054571817e-34 – 1.054571817e-34 | |
| Particle mass Mass of the confined particle | kg | M | 1e-31 – 1e-24 | |
| Box length Width of the 1D infinite well | m | L | 1e-12 – 0.000001 |
Deep dive
Derivation
Inside 0 < x < L, V = 0 so -hbar^2/(2m)*psi'' = E*psi. General solution psi = A*sin(k*x) + B*cos(k*x) with k = sqrt(2mE)/hbar. Boundary conditions psi(0) = psi(L) = 0 force B = 0 and sin(k*L) = 0 → k*L = n*pi. Therefore k_n = n*pi/L and E_n = hbar^2*k_n^2/(2m) = n^2*pi^2*hbar^2/(2m*L^2).
Experimental verification
Quantum dots, semiconductor heterostructures, and quantum wires all exhibit discrete energy spectra following the box-confinement scaling. Observed in photoluminescence and tunneling spectroscopy.
Common misconceptions
- The ground state is NOT zero energy — zero-point energy is a physical consequence of confinement
- Wavefunctions must vanish AT the walls, not just be small
- Level spacing grows with n — it is not uniform like a harmonic oscillator
Real-world applications
- Quantum dots and nanocrystals (tunable color by size)
- Semiconductor quantum well lasers
- Conjugated molecule absorption bands (free-electron model)
- Nanowire transistors
Worked examples
Electron in 1 nm box, ground state
Given:
- m:
- 9.1093837015e-31
- L:
- 1e-9
- n:
- 1
Find: E_1
Solution
E_1 ≈ 6.02e-20 J ≈ 0.376 eV
Transition n=2 → n=1 in the same box
Given:
- m:
- 9.1093837015e-31
- L:
- 1e-9
Find: emitted photon wavelength
Solution
lambda ≈ 1100 nm (near-infrared)
Scenarios
What if…
- scenario:
- What if L doubles?
- answer:
- E_n drops by factor 4 — confinement energy ∝ 1/L^2.
- scenario:
- What if the walls were finite height?
- answer:
- Wavefunctions tunnel into the barrier; fewer bound states; energies slightly lower.
- scenario:
- What if the particle were a proton instead of an electron?
- answer:
- Masses ratio 1836 — energies drop by 1836 for same L.
Limiting cases
- condition:
- n = 1
- result:
- E_1 = pi^2*hbar^2/(2mL^2) — zero-point energy
- explanation:
- Ground state is NOT zero; uncertainty principle forbids it.
- condition:
- L → ∞
- result:
- E_n → 0, continuous spectrum
- explanation:
- Free particle limit — quantization disappears.
- condition:
- Large n
- result:
- Level spacing Delta_E = (2n+1)*E_1 → quasi-continuum
- explanation:
- Correspondence principle — classical limit recovered.
Context
Erwin Schrödinger (canonical example) · 1926
The simplest exactly-solvable bound-state problem in the new wave mechanics — a pedagogical cornerstone ever since.
Hook
What happens when you trap an electron in a nanometer-wide well?
A particle confined to a rigid 1D box has quantized energies that depend on box length L. What are the allowed energies?
Dimensions:
- lhs:
- E_n → [M*L^2*T^-2]
- rhs:
- [M*L^2*T^-1]^2 / ([M]*[L]^2) → [M*L^2*T^-2]
- check:
- Both sides are energy. ✓
Validity: Valid for idealized infinite square well. Finite wells allow tunneling and have fewer bound states. 3D boxes give E_nlm = (n^2 + l^2 + m^2) * E_1.