Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

← Part III/Scattering & Tunneling

6. Scattering & Tunneling

Reading time: ~30 minutes | Pages: 8

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Quantum tunneling is one of the most remarkable quantum phenomena—particles can penetrate barriers that would be impenetrable in classical physics. This purely quantum effect powers nuclear fusion in stars, enables modern electronics, and revolutionized microscopy.

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Video Lecture

Quantum Tunneling - Understanding the Phenomenon

Comprehensive explanation of quantum tunneling, barrier penetration, and scattering in quantum mechanics

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Discovery of Quantum Tunneling

1928

Gamow's Theory of Alpha Decay

— George Gamow

Explained alpha particle emission from radioactive nuclei as quantum tunneling through the Coulomb barrier

Significance: First application of quantum tunneling, explaining why alpha decay rates vary by many orders of magnitude

1957

Esaki Diode (Tunnel Diode)

— Leo Esaki

Discovered electron tunneling in heavily doped p-n junctions, creating negative differential resistance

Significance: First practical electronic device based on quantum tunneling; Nobel Prize 1973

1981

Scanning Tunneling Microscope (STM)

— Gerd Binnig & Heinrich Rohrer

Invented STM using quantum tunneling current between sharp tip and surface to image individual atoms

Significance: Revolutionized surface science and nanotechnology; Nobel Prize 1986

2019

Measurement of Tunneling Time

— Riken Team (Japan)

Used attosecond spectroscopy to show tunneling through barriers appears to take zero time

Significance: Resolved 90-year debate about whether tunneling is instantaneous or takes finite time

💡 Understanding the historical development helps contextualize why certain concepts emerged and how they fit into the broader quantum revolution.

General Scattering Setup

Consider a wave incident on a potential barrier from the left. The general solution in three regions:

$$\psi(x) = \begin{cases} Ae^{ikx} + Be^{-ikx} & x < 0 \quad \text{(incident + reflected)} \\ Ce^{iqx} + De^{-iqx} & 0 < x < a \quad \text{(inside barrier)} \\ Fe^{ikx} & x > a \quad \text{(transmitted)} \end{cases}$$

Match at boundaries x = 0 and x = a using continuity of ψ and ψ′

Boundary conditions: ψ and dψ/dx must be continuous (unless potential has infinite discontinuity)

💡 Physical Interpretation

The coefficients have physical meaning:

  • A: Incident wave amplitude (normalized to 1)
  • B: Reflected wave amplitude
  • F: Transmitted wave amplitude
  • Reflection coefficient: R = |B/A|²
  • Transmission coefficient: T = |F/A|² × (k_transmitted/k_incident)

Conservation of probability: R + T = 1

Square Barrier

Rectangular barrier potential:

$$V(x) = \begin{cases} V_0 & 0 < x < a \\ 0 & \text{otherwise} \end{cases}$$

Case 1: E > V₀ (classical allowed—particle has enough energy to surmount barrier)

$$T = \frac{1}{1 + \frac{V_0^2\sin^2(qa)}{4E(E-V_0)}}$$

where $q = \sqrt{2m(E-V_0)}/\hbar$ is the wave number inside the barrier

Resonances: When qa = nπ, sin²(qa) = 0, giving perfect transmission T = 1!

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Video Lecture

Quantum Tunneling Animation

Animated visualization showing wave function behavior during barrier penetration and tunneling

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

📝 Worked Example: Transmission Through Square Barrier

Problem: An electron (E = 5 eV) encounters a square barrier of height V₀ = 10 eV and width a = 0.2 nm. Calculate the transmission probability.

Step 1: Identify the regime

Since E < V₀, this is quantum tunneling (classically forbidden).

Step 2: Calculate decay constant κ

$$\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar} = \frac{\sqrt{2(9.11\times10^{-31})(10-5)(1.6\times10^{-19})}}{1.055\times10^{-34}}$$
$$\kappa \approx 1.15 \times 10^{10} \text{ m}^{-1}$$

Step 3: Calculate transmission

$$T \approx e^{-2\kappa a} = e^{-2(1.15\times10^{10})(0.2\times10^{-9})} = e^{-4.6} \approx 0.01$$

💡 Result: About 1% transmission probability—the electron has a 1 in 100 chance of tunneling through this classically forbidden barrier!

Quantum Tunneling (E < V₀)

When E < V₀ (energy below barrier height), classical physics predicts zero transmission. But quantum mechanically:

$$T \approx e^{-2\kappa a}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$

Valid for thick barriers (κa >> 1)

Key insight: Wave function decays exponentially inside barrier (evanescent wave) but doesn't vanish—if barrier is finite, it "leaks through"

Inside the barrier: ψ(x) ∝ e^(-κx) (exponential decay, not oscillation)

The thicker the barrier or higher it is above E, the smaller the transmission

🤔 Self-Check Question

Question: Why does increasing the barrier width by just 10% dramatically reduce the tunneling probability?

Show Answer

Answer: Because transmission probability depends exponentially on barrier width: T ∝ e^(-2κa).

If you increase a by 10%, the exponent changes from -2κa to -2κ(1.1a) = -2.2κa. For example, if 2κa = 10, then:

Original: T ∝ e^(-10) ≈ 4.5 × 10^(-5)
New: T ∝ e^(-11) ≈ 1.7 × 10^(-5)

A 10% increase in width reduces transmission by factor of e ≈ 2.7 (almost 3×). This exponential sensitivity makes tunneling extremely dependent on barrier properties!

Tunneling Time

How long does tunneling take? This has been a controversial question for 90 years!

Phase time approach:

$$\tau_{\text{phase}} = \frac{d\phi}{dE}$$

where φ is the phase shift acquired tunneling through the barrier

Larmor time approach: Measure spin precession during tunneling

Recent 2019 experiments using attosecond spectroscopy suggest tunneling appears to be instantaneous within measurement precision—the particle doesn't "spend time" inside the barrier in the classical sense

▶️

Video Lecture

Alpha Decay and the Gamow Factor

How quantum tunneling explains radioactive alpha decay and the enormous range of nuclear lifetimes

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Resonant Tunneling

For certain "magic" energies, transmission through a barrier can be perfect (T = 1) even when E < V₀!

Resonance condition (for rectangular barrier with E > V₀):

$$qa = n\pi, \quad n = 1, 2, 3, \ldots$$

where q = √[2m(E-V₀)]/ℏ is the wave number inside the barrier

Physical interpretation: The barrier width is an integer multiple of half-wavelengths—constructive interference between multiple reflections leads to perfect transmission!

This is analogous to Fabry-Pérot resonances in optics

📝 Worked Example: Resonant Energy

Problem: A square barrier has V₀ = 20 eV and width a = 0.5 nm. Find the lowest energy E > V₀ at which resonant transmission occurs for an electron.

Step 1: Apply resonance condition

For n = 1 (fundamental resonance): qa = π

Step 2: Solve for E

$$\frac{\sqrt{2m(E-V_0)}}{\hbar} \cdot a = \pi$$
$$E - V_0 = \frac{\hbar^2\pi^2}{2ma^2} = \frac{(1.055\times10^{-34})^2\pi^2}{2(9.11\times10^{-31})(0.5\times10^{-9})^2} \times \frac{1}{1.6\times10^{-19}}$$
$$E - V_0 \approx 1.5 \text{ eV} \quad \Rightarrow \quad E \approx 21.5 \text{ eV}$$

💡 Result: At E ≈ 21.5 eV, electrons will have 100% transmission through the 20 eV barrier due to resonance!

Double Barrier & Quantum Wells

Two barriers create a quantum well between them—a particle can be trapped in quasi-bound states:

  • Sharp resonances at quasi-bound state energies (like infinite well, but finite lifetime)
  • Fabry-Pérot analogy: Multiple reflections between barriers create interference patterns
  • Resonant tunneling diodes (RTDs): Show negative differential resistance at resonance energies
  • Quantum cascade lasers: Use cascaded quantum wells for terahertz emission

Key applications: High-frequency oscillators, THz sources, quantum computing architectures

WKB Approximation for Tunneling

For arbitrary barrier shape V(x) (not just rectangular), the WKB approximation gives:

$$T \approx \exp\left(-2\int_{x_1}^{x_2}\frac{\sqrt{2m(V(x)-E)}}{\hbar}dx\right)$$

where x₁, x₂ are classical turning points where V(x) = E

Physical meaning: The integral measures the "action" accumulated while classically forbidden—thicker or higher barriers accumulate more phase and have lower transmission

🤔 Self-Check Question

Question: For a triangular barrier (V(x) = V₀ - αx for 0 < x < V₀/α), would you expect higher or lower tunneling probability compared to a rectangular barrier of the same height and base?

Show Answer

Answer: Higher tunneling probability for the triangular barrier.

The WKB integral ∫√[V(x)-E] dx is smaller for the triangular barrier because V(x) decreases linearly, making V(x)-E smaller on average compared to the constant V₀-E of the rectangular barrier.

Intuitively: The particle has to tunnel through "less material" on average. This is why field emission (triangular Coulomb barrier) is much easier than tunneling through uniform barriers of the same maximum height.

Physical Applications

Nuclear Physics

  • Alpha decay: Nucleus tunnels through Coulomb barrier (Gamow theory)
  • Nuclear fusion: Protons tunnel through Coulomb repulsion (powers stars!)
  • Fission: Heavy nuclei tunnel through fission barrier

Electronics & Materials

  • Flash memory: Electrons tunnel to/from floating gate
  • Tunnel diodes: Esaki diode, RTDs for oscillators
  • MOSFET leakage: Gate oxide tunneling (scaling limit)

Instrumentation

  • STM: Images individual atoms via tunneling current
  • Josephson junctions: Cooper pair tunneling (superconducting qubits)
  • Field emission: Electron emission via triangular barrier

Chemistry & Biology

  • Enzyme catalysis: Proton/electron tunneling in active sites
  • DNA mutations: Proton tunneling between base pairs
  • Hydrogen bonding: Proton tunneling in ice and water

💡 Application: Nuclear Fusion in the Sun

The Sun's core temperature is about 15 million K (kT ≈ 1.3 keV). But the Coulomb barrier between two protons is about 500 keV at contact distance!

Classical physics: Protons can't get close enough to fuse—the Sun shouldn't shine.

Quantum reality: Protons tunnel through the Coulomb barrier. Even though T ~ e^(-30) is incredibly small for individual collisions, the enormous number of protons in the Sun (10^56) makes fusion happen at the observed rate.

Without quantum tunneling, stars wouldn't burn and life wouldn't exist!

Gamow Factor

For Coulomb barrier V(r) = Z₁Z₂e²/r (alpha decay, fusion):

$$T \sim e^{-G}, \quad G = \frac{2\pi Z_1 Z_2 e^2}{\hbar v} = \frac{2\pi Z_1 Z_2 \alpha c}{v}$$

where v is the velocity of the particle and α ≈ 1/137 is the fine structure constant

The Gamow factor explains the enormous variation in nuclear decay lifetimes (femtoseconds to billions of years) based on small changes in nuclear charge and energy

Example: U-238 (α decay): G ≈ 60, lifetime ~ 4.5 billion years

Po-212 (α decay): G ≈ 30, lifetime ~ 0.3 microseconds

Factor of 2 change in Gamow factor → 10^15 change in lifetime!

📝 Chapter Summary

Key Equations

Tunneling (thick barrier): T ≈ e^(-2κa)
Decay constant: κ = √[2m(V₀-E)]/ℏ
WKB tunneling: T ≈ exp(-2∫√[2m(V-E)]dx/ℏ)
Gamow factor: G = 2πZ₁Z₂e²/ℏv
Resonance: qa = nπ for perfect transmission

Key Concepts

  • Particles can tunnel through classically forbidden barriers
  • Wave function decays exponentially inside barrier
  • Transmission exponentially sensitive to barrier properties
  • Resonant transmission at special energies (qa = nπ)
  • WKB approximation for arbitrary barrier shapes
  • Tunneling enables nuclear fusion, alpha decay, STM
  • Gamow factor explains nuclear lifetime variation

Quantum tunneling is a purely quantum phenomenon with no classical analog. It powers the Sun, enables modern nanotechnology, and sets fundamental limits on electronic devices. Understanding barrier penetration is essential for nuclear physics, solid-state devices, and quantum computing.

🔗 Related Topics:WKB Approximation • Delta Function Potential • Scattering Theory

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