1D quantum thermometer Zero-point delocalization and the classical limit

Quantum spread / classical spread

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How much more spread out the quantum particle is than a classical one at the same temperature. When k_BT is well above the vibrational quantum ħω the two agree. Below it, the classical particle settles into the bottom of the well while the quantum one keeps its zero-point spread, so the ratio grows. Heavier isotopes stay closer to classical.

Preset

Well and isotope

Vibration and temperature

At this temperature

Averages over the live simulations — they settle within a few seconds and restart whenever a control changes.

The well is set by the vibrational wavenumber of H. D and T feel the same force constant, so they vibrate at ω̃/√2 and ω̃/√3. Quantum effects dominate when k_BT is below about a third of ħω; well above ħω the particle is nearly classical. For the quartic well, ω̃ sets the energy and length scales in the same way.

Both particles are jostled by the same kind of random thermal kicks. The classical one is a single point; where it is found does not depend on its mass, only how fast it moves. The quantum one is a ring of 32 beads joined by springs — the same method used in real quantum simulations. Both results carry a little statistical noise.