Module 03 · Quantum mechanics
Quantum Mechanics
The Schrödinger equation turns a potential into allowed energies and wave functions. Choose the potential — a box, a finite well, a barrier, a parabola — and you are designing quantum dots, flash memory, tunnelling microscopes and infrared gas sensors.
Interpretation
What ψ means for an engineer
- |ψ(x)|² dx is the probability of finding the particle between x and x+dx; ∫|ψ|²dx = 1 (normalisation).
- ψ must be finite, single-valued and continuous; dψ/dx continuous where the potential is finite.
- Measured averages are expectation values: ⟨x⟩ = ∫ψ* x ψ dx, ⟨p⟩ = ∫ψ* (−iħ d/dx) ψ dx.
- The time-dependent equation iħ∂Ψ/∂t = ĤΨ evolves the state; for a fixed potential, stationary states obey Ĥψ = Eψ.
Operator dictionary
Observables as operators
| Observable | Operator |
|---|---|
| Position | x̂ = x |
| Momentum | p̂ = −iħ ∂/∂x |
| Kinetic energy | −(ħ²/2m) ∂²/∂x² |
| Hamiltonian | Ĥ = −(ħ²/2m)∂²/∂x² + U(x) |
| Total energy | Ê = iħ ∂/∂t |
Physics → industry. Walls at x = 0 and L force ψ = √(2/L) sin(nπx/L) and Eₙ = n²h²/8mL². Energy is quantised, the ground state has non-zero (zero-point) energy, and level spacing scales as 1/L². Display makers tune quantum-dot size to hit exact red and green emission lines.
Physics → industry. With finite walls U the wave function penetrates the barrier as e−Cx, so every level sits below its infinite-well value and only a finite number of states are bound. Quantum-well lasers and HEMT transistors are engineered on this model.
Physics → industry. A particle with E < U still crosses a barrier with probability T ≈ e−2CL, C = √(2m(U−E))/ħ. Exponential sensitivity to thickness L is why flash memory programs by tunnelling, why STMs image single atoms, and why gate oxides leak as they thin.
Physics → industry. For U = ½kx² the levels are equally spaced, Eₙ = (n+½)ħω, with zero-point energy ½ħω. Molecular bonds behave this way, so each gas absorbs at its own ħω — the fingerprint used by NDIR and open-path laser gas detectors.