Quantum mechanical tunneling is one consequence of quantum theory. What is quantum mechanical tunneling?

Quantum mechanics deals with the behavior of particles at an atomic and subatomic level. The foundation of quantum mechanics is the wave-particle duality, which describes particles as showing both wave-like and particle-like properties. This behavior is in stark contrast to classical mechanics, where particles have well-defined positions and energies.

In quantum mechanics, particles are described by wave functions, which contain all the information about a particle's state. The Schrödinger equation is a partial differential equation that describes how the wave function of a particle evolves over time. The time-independent Schrödinger equation for one-dimensional systems can be written as: \[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\] It relates the kinetic energy term, potential energy term, and the total energy of a system, denoted by E, as eigenvalues of the wave-function eigenstates. Here, \(\hbar\) represents the reduced Planck constant, \(m\) denotes the mass of the particle, \(\psi(x)\) is its wave function, and \(V(x)\) is the potential energy at position \(x\).

In the context of tunneling, we typically consider three regions: the region before the potential barrier (Region I), the barrier region (Region II), and the region after the barrier (Region III). The potential-barrier region usually takes the form of a step potential well or a rectangular potential barrier. In Region I and Region III, the wave functions are oscillatory because the total energy of the particle (E) is higher than the potential energy (V(x)). In Region II, the wavefunctions become exponentially decaying or growing, correlating to the probability density of the particle being inside the barrier at position \(x\).

To find the probability of a particle tunneling through a barrier, we need to calculate the ratio of the transmitted wave function's intensity to the incident wave function's intensity. In other words, by solving the Schrödinger equation for each region and applying boundary conditions at the interfaces, we obtain coefficients of the transmitted and incident wave functions, which help us determine the tunneling probability. The tunneling probability depends on the height and width of the potential barrier, the mass and energy of the particle, and the initial position of the incident wave function. For a rectangular potential barrier, the tunneling probability can be calculated using the formula: \[P_{tunnel} = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0 - E)}}\] where \(P_{tunnel}\) is the tunneling probability, \(V_0\) represents the height of the potential barrier, \(E\) is the energy of the incident particle, \(L\) is the width of the barrier, and \(\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}\).

Quantum mechanical tunneling is a direct result of the wave-particle duality observed in quantum mechanics. It can be explained as the phenomenon where a particle with insufficient energy to surmount a potential barrier classically can still "tunnel" through it with non-zero probability. It's widely observed in various applications, such as radioactive decay, scanning tunneling microscopy, and quantum computing. The tunneling probability can be computed by solving the time-independent Schrödinger equation and analyzing the wave functions in different regions.