Quantum mechanical tunnelling
Introduction
Quantum mechanical tunneling is a fundamental concept in quantum mechanics, describing the phenomenon where particles traverse potential energy barriers that they classically should not be able to surmount. This process is a direct consequence of the wave-particle duality inherent in quantum systems and is crucial for understanding various physical phenomena and technological applications, from nuclear fusion in stars to the operation of modern electronic devices like tunnel diodes and transistors.
Historical Background
The concept of tunneling emerged in the early 20th century as physicists grappled with the implications of quantum mechanics. The term "tunneling" was first used in the context of alpha decay, where it was observed that alpha particles could escape the nucleus despite the presence of a substantial energy barrier. This observation could not be explained by classical physics, which predicted that the particles did not possess enough energy to overcome the barrier. The development of quantum mechanics, particularly the Schrödinger equation, provided the framework to understand this phenomenon.
Theoretical Framework
Schrödinger Equation and Wave Functions
The Schrödinger Equation is central to quantum mechanics, describing how the quantum state of a physical system changes over time. In the context of tunneling, the Schrödinger equation is used to calculate the probability amplitude of a particle's wave function, which can extend across a potential barrier. The wave function's square modulus gives the probability density, indicating the likelihood of finding the particle in a particular region.
Potential Barriers and Quantum Tunneling
In a classical context, a particle with energy less than the height of a potential barrier cannot pass through it. However, in quantum mechanics, the wave function of a particle does not abruptly drop to zero at the barrier but instead decays exponentially within it. This allows for a non-zero probability that the particle will be found on the other side of the barrier, a phenomenon known as tunneling.
Mathematical Description
Consider a one-dimensional potential barrier of height \( V_0 \) and width \( a \). For a particle with energy \( E < V_0 \), the wave function in the barrier region is described by:
\[ \psi(x) = A e^{-\kappa x} + B e^{\kappa x} \]
where \( \kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}} \), \( m \) is the particle's mass, and \( \hbar \) is the reduced Planck's constant. The probability of tunneling is given by the transmission coefficient \( T \), which can be approximated for a rectangular barrier as:
\[ T \approx e^{-2\kappa a} \]
This expression shows that the tunneling probability decreases exponentially with the barrier width and height.
Applications of Quantum Tunneling
Nuclear Fusion
Quantum tunneling plays a critical role in nuclear fusion, the process that powers stars, including our Sun. In stellar cores, temperatures and pressures are extremely high, yet not sufficient for nuclei to overcome the Coulomb barrier classically. Tunneling allows protons to fuse, releasing energy that sustains the star.
Semiconductor Devices
In modern electronics, tunneling is exploited in devices such as tunnel diodes and MOSFETs. Tunnel diodes use the tunneling effect to achieve fast switching speeds, while tunneling is a limiting factor in the miniaturization of transistors, as it can lead to leakage currents in very small devices.
Scanning Tunneling Microscopy
Scanning Tunneling Microscopy (STM) is a powerful technique for imaging surfaces at the atomic level. STM operates by measuring the tunneling current between a sharp metallic tip and the surface being studied, providing detailed topographical maps of surfaces.
Quantum Tunneling in Chemistry
In chemical reactions, tunneling can influence reaction rates, especially at low temperatures. For example, hydrogen tunneling is significant in reactions involving hydrogen transfer, where the light mass of hydrogen allows for a higher tunneling probability.
Quantum Tunneling in Biological Systems
There is growing evidence that tunneling may play a role in biological processes. For instance, enzyme-catalyzed reactions often involve hydrogen tunneling, which can enhance reaction rates beyond classical predictions. Additionally, tunneling has been proposed as a mechanism in the olfactory system, where it might contribute to the detection of odorant molecules.
Quantum Tunneling and Quantum Computing
Quantum tunneling is also a key consideration in the development of quantum computing. Quantum bits, or qubits, can exist in superpositions of states, and tunneling phenomena can be harnessed to perform computations that are infeasible for classical computers.