Heisenberg uncertainty principle

Heisenberg uncertainty principle

1. Statement

Heisenberg's uncertainty principle (1927): it is impossible to simultaneously measure position $x$ and momentum $p_x$ (conjugate pair) with arbitrarily high precision:

where $Δx,Δp$ are rms uncertainties (standard deviations) over many identical measurements. Similarly:

(energy-time),

(angular).

$ħ = h/2π =1.054×10^{-34}$ J·s.

2. Why It Happens (wave picture)

Particle is wave packet of size $Δx$; to localize to $Δx$, need superposition of many $k$ values with spread $Δk ∼1/Δx$; since $p=ħk$, $Δp =ħΔk ⇒ ΔxΔp∼ħ$. Not due to instrument imperfection — fundamental due to wave nature.

• Measuring $x$ (slit) diffracts momentum; illuminating to see position disturbs momentum (Heisenberg microscope).

3. Illustrations

(a) Electron through slit width a ⇒ Δx≈a ⇒ Δp_x≳ħ/(2a) ⇒ angular spread Δθ≈Δp_x/p≈ħ/(2a p)= λ/(4π a) comparable to diffraction

(b) Atomic electron: confine to Δx≈0.5 Å ⇒ Δp∼ħ/Δx≈2×10^{-24} kg·m/s ⇒ kinetic ∼(Δp)²/2m≈ 3 eV, order of binding — explains why electron not collapse to nucleus (localizing further raises kinetic).

(c) Lifetime of excited atom Δt∼10^{-8} s ⇒ ΔE∼ħ/(2Δt)∼3×10^{-8} eV ⇒ natural linewidth Δλ∼...

4. Consequences

• No classical orbit: electron cannot have simultaneous exact $x$ and $p$ as Bohr orbit suggests; only probabilistic $ψ$.
• Zero-point energy: confining particle to box $L$ gives $Δp∼ħ/L$ ⇒ ground energy $∼ħ²/(8mL²)$ not zero (next topic particle in box).
• Tunneling allowed because energy-time uncertainty permits barrier violation short time.

5. Exam Points

• State $ΔxΔp≥ħ/2$ (or $h$) — 3-5 marks, very frequent.
• Explain with wave packet / slit diffraction — 5 marks, frequent.
• Estimate ground state energy or $Δp$ for given $Δx$ — 5-7 marks, very frequent (common numerical: slit, nucleus, box).
• Energy-time form and lifetime broadening — 3-5 marks, frequent.