Quatum Tunneling
Quantum tunneling is one of the most important concepts in quantum mechanics. According to classical physics, a particle cannot cross a barrier if its energy is less than the barrier height. However, quantum mechanics predicts that particles can sometimes penetrate and pass through such barriers. This strange phenomenon is called quantum tunneling.
Classical View of a Barrier
Imagine a ball rolling toward a hill.
- If the ball has enough energy, it climbs over the hill.
- If the ball does not have enough energy, it rolls back.
For example, suppose a ball has kinetic energy of $100\ \text{J}$ and encounters a hill requiring $200\ \text{J}$ to cross. According to classical mechanics, the ball can never appear on the other side of the hill.
The probability of crossing is:
$$
P = 0
$$
This means crossing the barrier is impossible in classical physics.
Quantum Mechanical View
Quantum mechanics treats particles such as electrons not only as particles but also as waves. Every particle is described by a wave function:
$$
\psi(x)
$$
This wave function spreads over space. When a quantum particle encounters a barrier, a small part of its wave can penetrate inside the barrier and emerge on the other side.
Therefore, even when the particle energy is less than the barrier height,
$$
E < U_0
$$
there is still a finite probability that the particle can cross the barrier.
This phenomenon is called quantum tunneling.
Quantum tunneling simulation
This simulation visually demonstrates how a quantum particle approaches a potential barrier, partially reflects, and partially tunnels through to the other side. Observe the behavior of the wave function in different regions and understand how tunneling probability changes with barrier width, barrier height, and particle energy.
Potential Energy Barrier
A finite potential barrier is represented as
$$
U(x)=
\begin{cases}
0 & x<0 \\
U_0 & 0<x<L \\
0 & x>L
\end{cases}
$$
where:
$$U_0 = \text{barrier height}$$
L = barrier width
The barrier divides space into three regions.
Region I – Before the Barrier
In this region,
$$
x < 0
$$
the particle moves freely toward the barrier.
The wave function is
$$
\psi_1 = Ae^{ikx} + Be^{-ikx}
$$
where:
- $Ae^{ikx}$ represents the incident wave
- $Be^{-ikx}$ represents the reflected wave
The wave number is
$$
k = \frac{\sqrt{2mE}}{\hbar}
$$
Region II – Inside the Barrier
Inside the barrier,
$$
0 < x < L
$$
and the potential energy is greater than the particle energy.
$$
U_0 > E
$$
The Schrödinger equation gives the solution
$$
\psi_2 = Ce^{\beta x} + De^{-\beta x}
$$
where
$$
\beta = \frac{\sqrt{2m(U_0-E)}}{\hbar}
$$
Unlike ordinary waves, this solution is exponential. The wave function decreases rapidly inside the barrier.
This means the probability of finding the particle decreases as it travels through the barrier.
Region III – After the Barrier
After the barrier,
$$
x > L
$$
a small transmitted wave emerges:
$$
\psi_3 = Fe^{ikx}
$$
This wave represents particles that successfully tunneled through the barrier.
Transmission Probability
The probability that a particle tunnels through the barrier is called the transmission coefficient or tunneling probability.
For a high and wide barrier, the approximate expression is
$$
T \approx e^{-2\beta L}
$$
This equation shows that tunneling depends strongly on the barrier width and barrier height.
Important Observations
Effect of Barrier Width
If the barrier width $L$ increases,
$$
T \downarrow
$$
The tunneling probability decreases exponentially.
Effect of Barrier Height
$$\text{If the barrier height } U_0 \text{ increases, the value of } \beta \text{ increases}$$ and tunneling becomes smaller. Higher barriers are more difficult to penetrate.
Effect of Particle Mass
Lighter particles tunnel more easily.
Electrons tunnel significantly because their mass is extremely small.
Large objects such as footballs or cricket balls have practically zero tunneling probability.
Physical Meaning of the Wave Function
|
Region |
Nature of Wave |
Physical Meaning |
|---|---|---|
|
Region I |
Oscillatory |
Incident and reflected particles |
|
Region II |
Exponential decay |
Wave attenuates inside barrier |
|
Region III |
Oscillatory |
Transmitted particle |
Applications of Quantum Tunneling
1. Alpha Decay
Alpha particles escape atomic nuclei through quantum tunneling.
This explanation was developed by George Gamow.
2. Nuclear Fusion in the Sun
Hydrogen nuclei tunnel through electrostatic repulsion and fuse together.
Without tunneling, nuclear fusion inside the Sun would not occur efficiently.
3. Scanning Tunneling Microscope (STM)
The STM works using electron tunneling between a sharp tip and a conducting surface.
It allows scientists to observe individual atoms.
4. Tunnel Diode
Tunnel diodes use tunneling for very fast electronic switching.
They are important semiconductor devices.
Why Quantum Tunneling is Important
Quantum tunneling demonstrates that microscopic particles do not obey the rules of classical mechanics completely. Their wave nature allows them to exist in classically forbidden regions.
This phenomenon plays a major role in:
- semiconductor physics
- nanotechnology
- nuclear physics
- modern electronics
- quantum devices
Conclusion
Quantum tunneling is the phenomenon in which particles penetrate through a potential barrier even when their energy is smaller than the barrier height.
The effect arises because particles behave like waves in quantum mechanics. Although the wave function decreases inside the barrier, it never becomes exactly zero, allowing a small probability for the particle to appear on the other side.
Quantum tunneling is one of the clearest demonstrations of the strange and powerful nature of quantum mechanics.