Quantum Tunneling through Potential Barriers

Page Claimed By: Shreenithi Katta Spring 22

Quantum tunneling is a phenomenon of quantum mechanics in which a wave function can travel through a potential barrier and be observed on the other side. It utilizes the Heisenberg uncertainty principle and the Schrödinger equation to explain the classically forbidden phenomenon. Using these principles, the probability that a particle can tunnel through a potential barrier can be calculated.

Main Idea

A potential barrier can be defined as a bounded region of significantly high potential than the space it is surrounded by. For the purpose of quantum tunneling, the potential barrier is finite with a specified height, width, and some unknown potential defined in terms of the particle that is incident upon it ([math]\displaystyle{ V_o \gt E }[/math]).

Classical Theory

According to classical mechanics, a particle of energy [math]\displaystyle{ E }[/math] does not have sufficient energy to pass through the barrier of energy [math]\displaystyle{ V_o \gt E }[/math] - the particle is classically forbidden to enter the region. However, due to the Wave-Particle Duality, quantum mechanics predicts that the wavefunction of the particle has a chance of entering the classically forbidden region.

Quantum Theory

Wave Mechanics

Wave mechanics tells us that when a wave travels through a medium are partially absorbed, transmitted, and reflected. With a finite potential barrier of certain dimensions, we can create three regions the wave function travels through a) the initial region before the potential barrier b) the finite potential barrier c) the region after the potential barrier. Using the wave properties, we know a portion of the incident wave function is reflected, absorbed, and transmitted into region B where the same happens on the border of regions B and C.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle can be used to explain quantum tunneling a bit more intuitively. The second uncertainty principle outlined by Heisenberg is concerned with energy and time. Given a short amount of time to examine a particle, the uncertainty in the energy it contains vastly increases. Applying that to this phenomenon is simple. If the uncertainty in the energy of the particle is great in a short amount of time, there is a possibility that the particle contains enough energy to enter the classically forbidden region. Whether it is reflected halfway through or transmits to the other side, the uncertainty in the energy essentially allows for a classical particle to exist in a potential barrier.

The Schrödinger Equation

The Schrödinger equation tells us that a wave function must be continuous at each boundary it encounters, and the derivative of the wave function must also be continuous except when the boundary height is infinite. Applying this to a finite potential barrier, we can find the probability a particle can tunnel through a potential barrier. Notably, the wave functions and calculations most resemble the solution for a single particle in a semi-infinite well when [math]\displaystyle{ V_o \gt E }[/math] in regions II and III.

Mathematical Model

Beyond the conceptual understanding of how quantum tunneling works, calculations can be made to determine the probability a wave function can tunnel through a specific finite potential barrier. This is known as transmission probability and to understand this equation, we must look at the wave equations associated with the potential barrier.

Wave Equations

As mentioned before, the finite potential barrier creates three different regions with their own unique Schrödinger equations. Let's assume that the potential barrier is from [math]\displaystyle{ 0 \le x \le L }[/math]. Given that, we can define the potential as follows: [math]\displaystyle{ V(x) = \begin{cases} 0 & x \lt 0 \\ V_o & 0\leq x\leq L \\ 0 & x\gt L \end{cases} }[/math]

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