Quantum mechanics

Uncertainty in position and momentum

Explore the Δx–Δp trade-off for Gaussian packets and recognize ΔxΔp ≥ ħ/2 as a state property, not an instrument error. Measure position and momentum widths, then verify $ΔxΔp ≥ ℏ/2$ .

Undergraduate

⚠ The plots illustrate quantum states rather than direct measurements; do not infer medical or real-device outcomes.

Equipment

Wavefunction and probability-density plots in position space and momentum spectrum
σₓ, p₀, packet separation d sliders and state selector

Procedure

Compare the two Fourier-related spaces
In Gaussian packet mode, increase σₓ and observe the position distribution broaden while the momentum spectrum narrows; check the uncertainty product ΔxΔp in the readout. Change p₀ to shift the spectrum's center without confusing that shift with its width. Select Two superposed Gaussian packets and vary d to inspect interference in the density; each distribution still has finite width. Compare the displayed values with $ΔxΔp ≥ ℏ/2$ .
Vary the Gaussian packet width
In Gaussian packet mode, increase and then reduce σₓ while watching the position distribution and momentum spectrum. Read Δx, Δp, and their product; check ΔxΔp ≥ ħ/2 as the widths of the two plots change in opposite directions. Compare the displayed values with $ΔxΔp ≥ ℏ/2$ .
Separate a momentum shift from uncertainty
Keep σₓ fixed, vary p₀, and observe the momentum-spectrum center shift while its width stays nearly unchanged. Select the two-packet superposition, increase d, and inspect the probability-density fringes; compare each packet with ΔxΔp ≥ ħ/2. Compare the displayed values with $ΔxΔp ≥ ℏ/2$ .

Simulation

Experiment history

Werner Heisenberg's uncertainty principle took shape in 1927, as quantum mechanics forced physicists to reconsider what quantities such as position and momentum mean. It does not claim that imperfect instruments merely leave us uninformed; rather, a quantum state cannot have arbitrarily small spreads in both members of a conjugate pair at once. For a Gaussian packet, the bound ΔxΔp ≥ ħ/2 is reached by a minimum-uncertainty state. In position and momentum representations, the distributions are related by a Fourier transform: narrowing a packet in x broadens its momentum spectrum, while shifting the spectrum's center with p₀ is not the same as changing its width. A superposition can produce interference fringes in probability density even though each component packet remains finite. The simulation plots mathematical distributions of a state, not a direct measurement; the inequality describes a property of the prepared state, not a limitation of the display or a sensor.

Related physicists

Werner Heisenberg

Equipment

Procedure

Simulation

Experiment history

Related physicists

Related library topics