# Kronig–Penney

Quantum · ψ under V(x)

`V(x) = Σ_(k=0)⁶ 4 · exp(−30(x − 2k)²)`

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### What you see

The formula is the potential, and it is a sum. `Σ 4 · exp(−30(x − 2k)²)` over `k = 0…6` puts seven gaussian barriers on the axis, centred at `x = 0, 2, 4` and so on to `12`. Each stands 4 tall and falls to `1/e` of that within `1/√30 ≈ 0.18` of its centre.

Halfway between two of them the potential has effectively gone: at `x = 1` it is `4 · exp(−30) ≈ 4 · 10⁻⁸`. The packet therefore crosses flat ground seven times and meets seven walls, all identical and all equally spaced. That regularity is the whole point.

### The physics

`Momentum k₀` 1.9 and `Packet width` 1.6 give an energy of `k₀²/2 + 1/(4σ²) = 1.81 + 0.10 = 1.90`, well under the barrier tops. Every crossing is tunnelling, so a single barrier would already turn much of the packet back.

Seven do something a single one cannot. Each barrier reflects a little, and those reflections interfere with each other. At some momenta they cancel and the lattice is nearly clear; at others they reinforce and it is nearly closed. The closed intervals are *band gaps*, and a solid conducts or insulates for this reason. A weak lattice of spacing `a` opens its first gap near `k = π/a = 1.57`; here the barriers also lift the average potential by `0.65`, which pushes the measured gap a little higher.

### What T and R read

By `t ≈ 12` at `Speed` 1 the numbers settle near `T = 0.29` and `R = 0.71`. This momentum sits inside a gap, so most of the packet comes back.

### Try

- Raise `Momentum k₀` to 2.6. T climbs to about 0.83: the same lattice, now transparent, because this momentum lies in a pass band.
- Lower `Momentum k₀` to 1.3. T is about 0.50, between the two, on the shoulder of the gap.
- Keep one barrier: `4 · exp(−30x²)`. It passes about 0.64. Seven acting independently would pass `0.64⁷ ≈ 0.04`, and the real seven pass 0.29, so they are not independent at all.
- Widen the spacing to 3 with `sum(4 · exp(−30(x − 3k)²), k, 0, 4)`. T rises to about 0.55: a different spacing puts this momentum outside the gap.

### Read more

- [Particle in a one-dimensional lattice](https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice)
- [Electronic band structure](https://en.wikipedia.org/wiki/Electronic_band_structure)
- [Bloch's theorem](https://en.wikipedia.org/wiki/Bloch%27s_theorem)
- [Quantum tunnelling](https://en.wikipedia.org/wiki/Quantum_tunnelling)