Tunneling Through a Potential Energy Barrier

A potential energy function of this shape is called a square barrier, and U is called the barrier height.

According to the uncertainty principle, the particle could be within the barrier as long as the time interval during which it is in the barrier is short. If the barrier is relatively narrow, this short time interval can allow the particle to pass through the barrier.

The movement of the particle to the far side of the barrier is called tunneling or barrier penetration.

Tunneling Through a Potential Energy Barrier

The probability of tunneling can be described with a transmission coefficient T and a reflection coefficient R. The transmission coefficient represents the probability that the particle penetrates to the other side of the barrier, and the reflection coefficient is the probability that the particle is reflected by the barrier.

This quantum model of barrier penetration shows that T can be nonzero. That the phenomenon of tunneling is observed experimentally provides further confidence in the principles of quantum physics.

The Simple Harmonic Oscillator

The potential energy of the system is

where the angular frequency of vibration is

In the classical model, any value of E is allowed, including E=0, which is the total energy when the particle is at rest at x=0.

The Simple Harmonic Oscillator

Let’s investigate how the simple harmonic oscillator is treated from a quantum point of view. The Schrödinger equation for this problem is

The Simple Harmonic Oscillator

It turns out that the solution we have guessed corresponds to the ground state of the system, which has an energy

The energy levels of a harmonic oscillator are quantized as we would expect because the oscillating particle is bound to stay near x=0. The energy of a state having an arbitrary quantum number n is