# Potential Step

Quantum · ψ under V(x)

`V(x) = 2(x > 0)`

[Open in the app](https://www.wavelace.com/app#p=70) · [This page](https://www.wavelace.com/presets/potential-step)

### What you see

The formula is the potential: `x > 0` is 1 to the right of the origin and 0 to the left, so `V` is a cliff of height 2. There is flat ground on the near side and a raised plateau on the far side. The dashed line across it is the packet's energy, `k₀²/2 + 1/4σ² = 2 + 0.51 = 2.51`, just above the top of the step.

The packet starts at `Packet centre` `−4` with `Momentum k₀` 2, and reaches the step about two seconds in at `Speed` 1. A ball with more energy than the step would climb it, slow down and carry on. This packet does something else: it divides.

### The physics

What matters is not the packet's average energy but each momentum in it separately. A component of momentum `k` has energy `k²/2` and can only pass if `k > 2`, exactly the packet's central momentum. Half of what it is built from is under the threshold and is turned back whole. The rest crosses and slows to `k′ = √(k² − 4)`, so a component at 2.5 leaves the step at 1.5. The transmitted hump therefore crawls away while the reflected one races back at the full speed 2. Even the components that clear the step are not all transmitted. A step reflects a wave whenever it changes its wavelength abruptly, in the ratio `4kk′/(k + k′)²`, which vanishes as `k′` goes to zero. Adding that up over the packet's momenta gives 0.44.

### Where T settles

`T` overshoots to about 0.63 near `t = 3`, while the packet is still lying across the origin. It then falls back as the reflected part pulls away. By `t = 6` it reads 0.44, the value the momentum sum gives.

### Try

- Set `Momentum k₀` to 3: the energy is well clear of the step now and `T` reaches about 0.9.
- Halve the step, `1 · (x > 0)`: about 0.84 goes on.
- Turn the cliff into a drop, `−2 · (x > 0)`. Nothing is in the way and the particle speeds up, yet about a tenth of it still comes back. Reflection needs a change, not an obstacle.
- Press `Side` in the deck to read the split against the silhouette of the step.

### 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)
- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)
- [Matter wave](https://en.wikipedia.org/wiki/Matter_wave)
- [Probability current](https://en.wikipedia.org/wiki/Probability_current)