Quantum Tunneling in Quantum Mechanics

What is Quantum Tunneling?

Quantum tunneling is a fundamental phenomenon in quantum mechanics where particles can pass through potential barriers, even when classical physics would predict that they do not have enough energy to overcome the barrier. In classical mechanics, a particle would need energy greater than the height of the barrier to pass through it. However, in quantum mechanics, particles exhibit wave-like behavior, and there is a probability that a particle can "tunnel" through a barrier even if its energy is less than the barrier's height.

A Concrete Example of Quantum Tunneling

One of the most well-known examples of quantum tunneling is radioactive decay, specifically alpha decay. In this case, an alpha particle is trapped within the nucleus by a potential barrier, but due to quantum tunneling, there is a finite probability that it will escape from the nucleus, resulting in radioactive decay. Quantum tunneling also plays a crucial role in modern technologies like scanning tunneling microscopes (STM) and the operation of semiconductors.

Schrödinger Equation and Quantum Tunneling

Quantum tunneling can be mathematically described using the Schrödinger equation. The time-independent Schrödinger equation is expressed as:

Where:

• $\hbar$ is the reduced Planck constant,
• $m$ is the mass of the particle,
• $V(x)$ is the potential energy,
• $E$ is the energy of the particle,
• $\psi(x)$ is the wave function at position $x$.

To understand tunneling, let’s consider a simple case of a particle encountering a one-dimensional potential barrier defined as:

​This represents a barrier of height $V_0$ between $x = 0$ and $x = L$.

Mathematical Interpretation of Tunneling

When the particle's energy $E$ is less than the barrier height $V_0$, classical mechanics predicts that the particle cannot cross the barrier. However, in quantum mechanics, the wave function $\psi(x)$ does not become zero within the barrier but rather decays exponentially. If the barrier is not too wide, the wave function may still exist on the other side, meaning the particle has a probability of "tunneling" through the barrier.

Inside the barrier, the wave function is given by:

Where $\kappa$ is the decay constant, defined as:

​​

The transmission probability $T$, or the likelihood that the particle tunnels through the barrier, is proportional to an exponential decay:

Thus, the taller or wider the barrier, the smaller the probability that the particle will successfully tunnel through it.

Applications of Quantum Tunneling

(1) Scanning Tunneling Microscope (STM)

The STM uses quantum tunneling to visualize surfaces at the atomic level. By bringing a sharp tip very close to a surface, a tunneling current is generated as electrons tunnel between the tip and the surface. By measuring the tunneling current, the atomic structure of the surface can be mapped with extraordinary precision.

(2) Semiconductors and Tunneling

Quantum tunneling is crucial in the functioning of semiconductor devices like MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistors), especially as devices become smaller. Tunneling current in such nanoscale devices significantly impacts their performance and efficiency.

(3) Nuclear Fusion and Quantum Tunneling

In nuclear fusion, tunneling is essential for overcoming the Coulomb barrier between protons. Even though protons repel each other due to their positive charge, quantum tunneling allows them to get close enough for fusion to occur, despite not having sufficient thermal energy to overcome the barrier classically.

 

Conclusion

Quantum tunneling is a remarkable phenomenon unique to quantum mechanics, allowing particles to perform feats that seem impossible from the perspective of classical physics. It plays a crucial role in many natural phenomena and modern technologies. A deep understanding of tunneling requires mastering the mathematical framework of the Schrödinger equation, which governs the probabilistic nature of quantum mechanics.