Tunneling

Tunneling

1. What Quantum Tunneling Is

Tunneling (quantum leakage) = transmission through a classically forbidden barrier ($E<V_0$) with exponentially small but non-zero probability, due to evanescent wave penetrating barrier. Particle does not go over but through — like light evanescent in fibre cladding.

2. Transmission for Thick Barrier ($E<V_0$)

From previous barrier with $κ=√[2m(V_0-E)]/ħ$, width $a$:

Often approximated as

The huge sensitivity to $a$ and $√(V_0-E)$ means 1 Å change alters $T$ by order.

For arbitrary-shaped barrier $V(x)$ between $x_1,x_2$ where $E<V$, WKB approximation:

3. Why It Happens (Heisenberg view)

Energy-time uncertainty $ΔE·Δt∼ħ/2$ allows borrowing $ΔE=V_0-E$ for time $Δt∼ħ/(2ΔE)$ to cross; or wave's exponential tail already exists in barrier — matter wave not zero.

4. Applications (what examiners love)

Application	Barrier	$T$ relevance
Alpha decay	Nucleus Coulomb wall (~MeV)	$α$ tunnels out, Geiger-Nuttall law $log T ∝1/√E$
STM (Scanning Tunneling Microscope)	Vacuum gap tip-sample (~4 eV, ~1 nm)	Current $I∝e^{-2κd}$ ⇒ $0.1$ Å height sensitivity
Tunnel diode / Flash memory	Thin $SiO_2$ (~3 eV, 1–10 nm)	Electrons tunnel through gate oxide
Fusion in stars	Coulomb repulsion	Protons tunnel at keV despite MeV barrier
Chemical reactions	Potential saddle	H tunnels at low $T$

5. Worked Estimates

(a) Electron E=1 eV, V0=5 eV, a=0.5 nm: V0-E=4 eV ⇒ κ=√(2·9.1e-31·4·1.6e-19)/1.05e-34≈1.02e10 m⁻¹ ⇒ κa≈5.1 ⇒ e^{-10.2}=3.7e-5 ⇒ T≈ prefactor ≈0.2·3.7e-5≈7e-6 ✓

(b) STM gap 0.5→0.6 nm increase Δa=0.1 nm: T ratio e^{-2κΔa}=e^{-2·1e10·1e-10}=e^{-2}=0.135 ⇒ current drops 7× per Å — why STM atomic resolution.

(c) Alpha decay: width ~10 fm, height 20 MeV, α E=5 MeV ⇒ κ≈√(2·6.6e-27·15MeV)/ħ≈~? Gives T~10^{-25} but attempt frequency 10^{21} ⇒ lifetime ~days.

6. Exam Points

• Explain tunneling as $E<V_0$ transmission with $T∝e^{-2κa}$, $κ=√[2m(V0-E)]/ħ$ — 5-7 marks, very frequent (the most common tunneling question).
• Derive or state $T≈e^{-2κa}$ for thick barrier — 5 marks, frequent.
• Mention WKB integral — 3 marks.
• Applications: α-decay, STM, tunnel diode — 5 marks, very frequent (at least one example required).