Potential Barrier V(x)
Wave Function ψ(x)
Tunneling Probability
Wave Packet Animation
Classical vs Quantum
Classical
Particle reflects (E < V₀)
Quantum
Probability of tunneling: T ≈ 0.01%
System Parameters
Energy Parameters
Particle Properties
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Quantum Tunneling Equations
What is Quantum Tunneling?
Quantum tunneling is a quantum mechanical phenomenon where a particle can pass through a potential barrier even when its energy is less than the barrier height. This is impossible in classical physics, where a ball would always bounce back from a wall it cannot overcome. In quantum mechanics, the wave function of the particle extends into and through the barrier, giving a non-zero probability of finding the particle on the other side.
How Does It Work?
According to quantum mechanics, particles exhibit wave-like behavior described by a wave function ψ(x). When this wave encounters a potential barrier, it doesn't simply reflect - part of it penetrates into the barrier and decays exponentially. If the barrier is thin enough, some of the wave emerges on the other side, meaning there's a probability of finding the particle there. The transmission probability T depends exponentially on the barrier width and the square root of the barrier height: T ≈ e^(-2κa), where κ = √[2m(V₀-E)]/ħ.
Key Factors Affecting Tunneling
Particle Energy (E): Higher energy particles tunnel more easily as they have smaller decay constants.
Barrier Height (V₀): Taller barriers reduce tunneling probability exponentially.
Barrier Width (a): Thinner barriers allow much more tunneling - the dependence is exponential.
Particle Mass (m): Lighter particles (like electrons) tunnel much more easily than heavy particles.
Applications of Quantum Tunneling
Scanning Tunneling Microscope (STM): Uses tunneling current between a sharp tip and surface to create atomic-resolution images. This earned the 1986 Nobel Prize in Physics.
Flash Memory: Stores data using tunneling to inject and remove charge from floating gates.
Nuclear Fusion in Stars: Protons tunnel through Coulomb barrier to fuse, powering the Sun and stars.
Alpha Decay: Alpha particles escape atomic nuclei by tunneling through the nuclear potential barrier.
Tunnel Diodes: Electronic devices that use tunneling for ultra-fast switching and negative resistance.
Classical Limit
For macroscopic objects, quantum tunneling is negligible because the large mass makes the transmission probability vanishingly small. This is why we don't see people walking through walls! The transition from quantum to classical behavior occurs when the action scale is much larger than Planck's constant.