A square potential barrier is a concept in quantum mechanics and solid-state physics. It refers to a scenario where a particle encounters a potential energy barrier with a square-shaped profile. This barrier may arise in various physical systems, including electron motion in a semiconductor device, nuclear physics, and other quantum systems.

We consider one dimensional motion of a particle having fixed energy E along X axis such that its potential energy.

Where V is the potential energy at position x, Vo is the height of the barrier, and a is the width of the barrier. This type of potential barrier is often assumed to be infinitely high and infinitely thick.

When a particle with a certain energy approaches the barrier, there are two main possibilities:

Particle with energy less than Vo: If the particle's energy is less than the height of the barrier (E < Vo), it does not have enough energy to surmount the barrier. Therefore, the particle will experience total internal reflection and will not be able to pass through the barrier.
Particle with energy greater than Vo: If the particle's energy is greater than the height of the barrier (E > Vo), it has enough energy to overcome the barrier. In this case, there is a non-zero probability that the particle can tunnel through the barrier and appear on the other side.

Tunneling is a quantum mechanical phenomenon where particles can penetrate energy barriers that would be classically impossible to surmount. The probability of tunneling decreases as the barrier width (L) and height (V0) increase. The phenomenon of quantum tunneling is crucial in understanding various physical processes, such as nuclear fusion in stars and electron transport in semiconductor devices.

In the region I: X < -a

d²u(X)/ dX² + 2mEu(X)/ ℏ² = 0

or, d²u(X)/ dX² + Ko²u(X) = 0

The solution is

In the region II: X in between -a and a

d²u(X)/ dX² + 2m(E- Vo)u(X)/ℏ² = 0

Case I: E> Vo

Put 2m (E- Vo) /ℏ² = k²

or, d²u(X)/ dX² + k² u(X) = 0

The solution is

In the region III: X>a, V= 0

d²u(X)/ dX² + 2mE u(X)/ ℏ² = 0

The solution is

Here also two states of particle for the value of energy E are possible. One state represents the motion of particle travelling from left to right. In this case Ae^iKox represents the incident wave. Be^-iKox represents the reflected wave from the barrier and A'e^iKox represents the transmitted wave. In this case B' = 0.

Since there is no any barrier in the region X>a. Another state represents the motion of the particle travelling from right to left. In this case, B'e^-iKox represents the incident wave and A'e^iKox represents the transmitted wave. In this case A = 0.

So the energy E is doubly degenerated. We consider the case of particle travelling from left to right.

Boundary condition

Solving equation 3 and 4

Putting the expressions for C and D in 1 and 2 and then solving 1 and 2 we obtain

The width of the barrier = 2a = l (say)

Since these relations are valid for any value of Ko, the spectrum of energy E is continuous, it cab have any value in continuous range.

Reflection coefficient

Transmission coefficient

kl = nπ

l = nπ / k = nπ / 2π/𝜆 = n𝜆 /2

Whenever width of the barrier is an integral multiple of 1/2 of wavelength. T_E becomes equal to 1 and total transmission takes place. The barrier becomes transparent for E>Vo so resonance scattering is said to take place.

In summary, the square potential barrier is an essential concept in quantum mechanics that demonstrates the wave-like nature of particles and the phenomenon of quantum tunneling. It plays a significant role in understanding and describing various physical systems.

This note is a part of the Physics Repository.

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