Quantum tunneling is a process in which a particle has a chance to pass through an energy barrier even when it does not have enough energy to cross it classically. It matters because microscopic particles such as electrons behave like waves, not just tiny balls. When a particle wave reaches a barrier, part of the wave can reflect and part can continue through.
This effect is small in everyday objects but essential in atoms, electronics, nuclear physics, and stars.
In quantum mechanics, the wave function describes the probability of finding a particle at each position. Inside a barrier where the particle's energy is less than the potential energy, the wave function does not become zero, but it decays exponentially. If the barrier is thin enough, a smaller wave can emerge on the far side, giving a nonzero transmission probability.
Tunneling is used in scanning tunneling microscopes, helps explain alpha decay, and allows fusion in the Sun to occur at temperatures lower than classical physics would predict.
Key Facts
- Tunneling can occur when particle energy E is less than barrier height U0, so E < U0.
- The wave function gives probability density by P(x) = |ψ(x)|^2.
- Inside a constant barrier with E < U0, the wave function decays approximately as ψ(x) ∝ e^(-κx).
- The decay constant is κ = sqrt(2m(U0 - E))/ℏ.
- For a rectangular barrier of width L, the tunneling probability is roughly T ∝ e^(-2κL).
- Tunneling becomes less likely for larger particle mass, taller barriers, wider barriers, or lower particle energy.
Vocabulary
- Quantum tunneling
- Quantum tunneling is the nonzero chance that a particle passes through a potential energy barrier that it could not cross according to classical mechanics.
- Wave function
- A wave function is a mathematical function that contains information about the probability of finding a quantum particle in different places.
- Potential energy barrier
- A potential energy barrier is a region where a particle would need extra energy to pass through classically.
- Transmission probability
- Transmission probability is the probability that a particle will appear on the far side of a barrier after interacting with it.
- Reflection probability
- Reflection probability is the probability that a particle will bounce back from a barrier instead of passing through it.
Common Mistakes to Avoid
- Thinking tunneling means the particle breaks energy conservation. This is wrong because total energy is conserved, while the particle's wave function has a nonzero probability amplitude inside and beyond the barrier.
- Drawing the wave function as zero inside the barrier when E < U0. This is wrong because the wave function decays exponentially in the barrier rather than stopping abruptly.
- Assuming tunneling is equally likely for all barriers. This is wrong because the probability drops rapidly as barrier width, barrier height, or particle mass increases.
- Treating the transmitted wave as the same size as the incoming wave. This is wrong because only part of the probability amplitude is transmitted, so the emerging wave is usually much smaller.
Practice Questions
- 1 An electron has energy E = 4.0 eV and approaches a rectangular barrier of height U0 = 6.0 eV. Is tunneling possible in quantum mechanics, and is the electron classically allowed to cross the barrier?
- 2 For a barrier with κ = 5.0 nm^-1 and width L = 0.40 nm, estimate the relative tunneling factor using T ≈ e^(-2κL).
- 3 A tunneling microscope measures current between a sharp tip and a surface. Explain why increasing the gap between the tip and the surface causes the current to decrease sharply.