Coherent tunnelling in a double well

A state localized in one well tunnels coherently to the other and returns, even though its energy lies below the central barrier.

QPE preset #12

This page describes the phenomenon using the current QPE preset and numerical modules as its primary source.

Physical idea

Left-right transfer is not a sequence of random hops. In an isolated system the population oscillates coherently because the double-well eigenstates have slightly different energies and accumulate relative phase.

Key relation. A localized state is a coherent combination of symmetric and antisymmetric double-well eigenstates; their energy splitting sets the tunnelling oscillation period.

Guided protocol from the preset

What to observe. The initial state is built from the two lowest eigenstates of the double well and is localized mostly on the left. No packet is launched at a barrier.

Press Play. Probability slowly leaves the left well, appears in the right, reaches almost a maximum and then returns. In the Double well tab P_left and P_right exchange almost periodically.

Why. The two lowest eigenstates form an approximately symmetric/antisymmetric pair. Their energy splitting DeltaE rotates the relative phase: a left-localized combination becomes right-localized after about pi*hbar/DeltaE and returns after 2*pi*hbar/DeltaE.

The surprising part. The mean pair energy remains below the central barrier: this is not classical over-the-barrier motion, but coherent tunnelling.

Try this. Select the central barrier in Potential Scene, increase Width or V slightly and Apply changes. Recompute the states: the splitting shrinks and transfer becomes slower.

What to observe and how to study it

Run the guided preset unchanged and first identify the observable that changes.
Change one parameter at a time, then Reset and repeat.
Always compare rendering with the Analysis readouts: the image is a view, not a complete measurement.

History and experimental status

The double well is a universal model for molecular inversion, qubits, junctions and atomic traps. QPE shows the coherent single-particle version without a bath or many-body interactions.

Representative physical scale

Representative platform	ultracold Rb-87 atom / trapped matter wave
Mapping basis	Rb-87 kinetic-unit mapping
1 QPE length unit	1 µm
1 QPE time unit	1.37 ms
Energy scale E₀=ħ/T₀	0.481 peV
Equivalent frequency E₀/h	0.116 kHz
Note	Representative cold-atom mapping. Trap frequencies, optical geometries and internal-state protocols set the actual laboratory scale; this mapping is intentionally order-of-magnitude.

The scale is an order-of-magnitude bridge from QPE dimensionless units to one plausible platform; it does not turn the preset into a quantitative simulation of that specific apparatus.

Essential references

E. Schrödinger — Quantisierung als Eigenwertproblem (1926) (Annalen der Physik)

Continue studying

Open Model & Constraints to see exactly what QPE computes and what is left outside the model. For current live values use Help → Current Experiment → Current Parameters.