# Reflectionless Well

Quantum · ψ under V(x)

`V(x) = −((λ · (λ +3))/(2 · cosh(3x)²))`

[Open in the app](https://www.wavelace.com/app#p=145) · [This page](https://www.wavelace.com/presets/reflectionless-well)

### What you see

The formula is the potential. `cosh(3x)` is 1 at the origin and grows fast either side, so `1/cosh(3x)²` is a narrow bump, down to a quarter of its height by `x ≈ 0.44`. The factor in front, `−λ(λ + 3)/2`, turns it over and sets how deep the hole goes: 9 at `λ = 3`, 16.9 at the opening value 4.5, 54 at `λ = 9`.

The packet comes in from `Packet centre` `−8` with `Momentum k₀` 1.5, at an energy of 1.22. A drop in the ground reflects a wave much as a rise does, and at the opening `λ` about a fifth of the packet comes back. Below zero the factor changes sign, so `λ` between `−3` and 0 draws a small bump instead of a hole, and the stops under `−3` repeat those above 0.

### Why multiples of three reflect nothing

This shape is the Pöschl–Teller well, and at this width its order is `λ/3`. At whole order it does what no other well here does: it reflects nothing at all, at any energy. The plane wave answer is `R = sin²(πλ/3)/(sinh²(πk/3) + sin²(πλ/3))`, whose numerator is zero whenever `λ` is a multiple of 3, whatever `k` does. It is largest halfway between, which is why the preset opens at 4.5: the first run shows reflection, and one drag removes it.

Read `R` once the packet is past the well, by `t ≈ 14` at `Speed` 1. It settles near 0.19 at `λ = 1.5`, 4.5 and 7.5, and near 0.003 at 3 and 6, a sixtyfold difference one and a half units of the slider apart. Four reflectionless stops sit on the slider, 0, 3, 6 and 9, and the last holds in a well 54 deep, where `R` reads about 0.009.

### History

Gertrud Pöschl and Edward Teller wrote this family down in 1933, as one of the few potentials the Schrödinger equation can be solved for exactly. The reflectionless members turned up again much later in another subject, as the potentials of the multiple soliton solutions of the Korteweg–de Vries equation. A hole that lets every wave past and a wave that survives a collision unchanged are two views of the same mathematics.

### Try

- Drag `λ` to 3. `R` falls to about 0.002: the well is there, 9 deep, and the packet passes as though it were not.
- Drag `λ` to 6. Still about 0.003, in a well three times deeper than at `λ = 3`.
- Drag `λ` to 1.5, a well 3.4 deep against the 9 at `λ = 3`. `R` climbs back to about 0.19, so depth is not what decides it.
- Leave `λ` at 4.5 and raise `Momentum k₀` to 2.5. The faster packet is clear by `t ≈ 10`, and `R` reads about 0.03: away from a multiple of 3, reflection is something that happens to slow packets.

### Read more

- [Pöschl–Teller potential](https://en.wikipedia.org/wiki/P%C3%B6schl%E2%80%93Teller_potential)
- [Korteweg–De Vries equation](https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equation)
- [Soliton](https://en.wikipedia.org/wiki/Soliton)
- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)