# Smooth Step

Quantum · ψ under V(x)

`V(x) = 1 + erf(x/2)`

[Open in the app](https://www.wavelace.com/app#p=114) · [This page](https://www.wavelace.com/presets/smooth-step)

### What you see

The formula is the potential. `erf` runs from `−1` far to the left to `+1` far to the right, so `1 + erf(x/2)` climbs from 0 to 2. The ground is flat on both sides and the whole rise happens in the middle.

The 2 inside sets how long that rise takes. At `x = 2` the potential has reached `1 + erf(1) = 1.84`, and at `x = −2` it is only `0.16`, so the step is essentially complete across four units of x. Its half height falls at `x = 0`.

### The physics

`Momentum k₀` 2 and `Packet width` 0.7 give an energy of `k₀²/2 + 1/(4σ²) = 2 + 0.51 = 2.51`. The step tops out at 2. The packet is therefore travelling *above* the step, and a classical particle would cross every time, merely slowing down as it climbed.

A quantum one does not. Part of it turns back from a rise it has more than enough energy to climb, which is called quantum reflection, and it depends on how abruptly the potential changes rather than on how high it goes. A step that turns on gently over several wavelengths is almost invisible to the packet. A step that turns on within one is a wall it can partly bounce off.

### What T and R read

By `t ≈ 5` at `Speed` 1 the readouts hold near `T = 0.76`. About a quarter of the packet comes back from a step it should have cleared. Read the numbers before `t ≈ 7`, after which the wall at `x = −8` returns the reflected part and mixes the two.

### Try

- Make the step abrupt with `2 · (x > 0)`. Same height, no gradient at all, and T drops to about 0.48. The edge alone accounts for the difference.
- Make it gentler with `1 + erf(x/4)`. T rises to about 0.83, closer to the classical answer of everything through.
- Make it steeper with `1 + erf(x/0.5)`. T falls to about 0.54, nearly the sharp step again.
- Raise `Momentum k₀` to 3. T reaches about 0.95: a faster packet has a shorter wavelength, so the same rise looks gentler to it.

### Read more

- [Solution of Schrödinger equation for a step potential](https://en.wikipedia.org/wiki/Solution_of_Schr%C3%B6dinger_equation_for_a_step_potential)
- [Error function](https://en.wikipedia.org/wiki/Error_function)
- [Adiabatic theorem](https://en.wikipedia.org/wiki/Adiabatic_theorem)
- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)