In quantum mechanics, a potential step refers to a sudden change in the potential energy of a particle when it encounters a step-like barrier or potential profile. This potential step can be either an increase or a decrease in the potential energy.

This phenomenon of partial transmission and reflection at a potential step is a manifestation of wave-particle duality in quantum mechanics. Particles, such as electrons or photons, can exhibit wave-like behaviour and undergo interference effects, leading to the observed transmission and reflection probabilities.

The potential step concept is relevant in various areas of physics, including semiconductor devices, quantum tunneling, and wave optics. In semiconductor physics, for example, potential steps play a crucial role in understanding the behaviour of electrons at junctions between different materials.

ELI5: Imagine you are playing with a ball on a playground. There's a little hill in the middle of the playground. On one side of the hill, the ground is higher, and on the other side, it's lower.

Now, let's pretend you're the ball, and you want to roll over the hill. If you don't have enough energy, you won't make it over the hill, and you'll roll back to where you started.

But if you have a lot of energy, you'll be able to roll over the hill and continue rolling on the other side.

In the same way, particles like electrons can behave like this too. They can come across a "hill" made of energy, called a potential step. If they don't have enough "energy," they can't pass the hill, just like the ball couldn't roll over the hill. But if they have enough "energy," they can "roll" through the potential step and continue on the other side.

So, the potential step is like a little energy hill that particles need to overcome to move from one place to another. If they have enough energy, they can make it, but if they don't, they'll be stuck on one side.

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

V(x) = 0 in region X< 0

V(x) = Vo in region X⩾ 0

Hamiltonian operator in the region X< 0

H = - ℏ²/ 2m d²/dX²

and in the region X⩾ 0

H = - ℏ²/ 2m d²/dX² + Vo

This is the case of stationary state. We write energy eigen value equation

Hu(x) = Eu(x)

In the region X < 0

It's general solution

Probability density is given by

Probability current density due to incident wave

In the region x⩾ 0

Hu(x) = Eu(x)

Case I : E > Vo

Put 2m/ ℏ² ( E - Vo) = k²

d² u(X)/ dX² + K² u(X) = 0

Its general solution is

Probability current density

If we consider the case of particles travelling from left to right, no wave is travelling in the direction from right to left in the region X>0. So D = 0.

Note: If we consider the case of particle travelling from right to left De^-ikx represents the wave reflected from the potential step and Be^-ikox represents the transmitted wave. In this case A =0. So, here energy eigen value is doubly degenerated.

Two states: One state of particles travel from left to right and another state of particles travel from right to left.

We continue the case of particle travelling from left to right

Using the boundary condition as potential has finite jump at X = 0, the wavefunction and it's first derivative both are continuous at X =0

We define transmission coefficient. T_E and reflection coefficient R_E as

T_E = Probability current density due to transmitted wave/ Probability current density due to incident wave

Overall, the potential step is an essential concept in quantum mechanics that illustrates the wave nature of particles and the probabilistic nature of their interactions with potential energy barriers.

This note is a part of the Physics Repository.

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