Quantum Mechanics
Core concepts
Understand wave–particle duality, the de Broglie relation λ = h/p, the photoelectric effect, the Schrödinger equation, wavefunctions and normalization, expectation values, the infinite square well, the harmonic oscillator, and the hydrogen atom. Know operators for position, momentum, and energy, and the meaning of commutation relations. Familiarity with the uncertainty principle ΔxΔp ≥ ħ/2 and with quantization of angular momentum is essential.
Traps
- Probability density is |ψ|², not ψ; normalization requires ∫|ψ|²dx = 1.
- Boundary conditions quantize energy: an infinite well of width L gives E_n = n²π²ħ²/(2mL²), so E scales as 1/L² and gaps grow with n.
- Commutators matter: [x, p] = iħ; non-commuting observables cannot be simultaneously sharp.
- Photoelectric effect depends on frequency, not intensity, for whether emission occurs; intensity controls rate.
- Tunneling probability decays exponentially with barrier width and height.
Worked example
An electron is confined in an infinite well of width L = 1 nm. Ground-state energy E₁ = π²ħ²/(2mL²). With ħ ≈ 1.055×10⁻³⁴ J·s and m ≈ 9.11×10⁻³¹ kg, E₁ ≈ 6.0×10⁻²⁰ J ≈ 0.37 eV. Trap: forgetting to square L or using h instead of ħ - both change the answer by factors of 4 or 2π².
Rule of thumb
When a problem mentions confinement size, immediately estimate the energy scale via E ~ ħ²/(2mL²).