Track a Gaussian phase without changing its position density

Define the state, source and evidence

A normalized Gaussian can carry more information than its visible density. Take psi(x)=(2πsigma²)^−1/4 exp[−x²/(4sigma²)+i beta x²/(4sigma²)], centered at zero. The real parameter beta is a quadratic phase chirp. Its magnitude squared is the same Gaussian position density for every beta, with intrinsic position standard deviation sigma.

Derive the quantitative relation

Use the Fourier convention phi(k)=(1/sqrt(2π)) integral psi(x)exp(−ikx)dx. Completing the square in the Gaussian integral gives phi(k) proportional to (1−i beta)^−1/2 exp[−sigma²k²/(1−i beta)]. The magnitude squared is Gaussian with variance (1+beta²)/(4sigma²). Since p=hbar k, sigmaP=hbar sqrt(1+beta²)/(2sigma).

Check normalization, limits and inference

Directly apply p-hat=−i hbar d/dx to the state. The real symmetrized position-momentum covariance is hbar beta/2. Therefore sigmaX sigmaP=hbar sqrt(1+beta²)/2, while sigmaX²sigmaP²−covariance²=hbar²/4. A nonzero chirp raises the simple product; it is incorrect to call every Gaussian phase a minimum-product state.

Compare a changed premise

These are intrinsic state spreads, not a detector error bar or uncertainty in an estimated population mean. For 100 independent position preparations, the mean’s standard error is sigma/10 under the stated identical-distribution assumption; each preparation still has intrinsic spread sigma. Translating the center or adding a linear phase shifts a mean, whereas this quadratic phase changes correlation and momentum spread.

Worked original case

At sigma=0.2 nm and beta=0, sigmaP=hbar/(0.4 nm), and the product is hbar/2. At beta=2 the position density is unchanged, sigmaP grows by sqrt(5), covariance is hbar, and the product becomes sqrt(5)hbar/2. The covariance-corrected determinant remains hbar²/4.

Investigation

Compare beta=0,+2,−2 at the same sigma. Retain the identical position curves, equal momentum spreads for opposite chirps and opposite covariance signs. Then halve sigma and predict the momentum change before revealing it. Explain why repeated sampling does not shrink the intrinsic state width.

Changed-case practice

Try the changed case before feedback

For beta=1, find sigmaX sigmaP divided by hbar.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions, numerical packets and diagrams are original. These models do not establish actual beam, radiation, vacuum, optical or electrical laboratory performance. Public feedback is practice, not authenticated independent mastery. Preserve first reasoning, criterion critique, revision and a genuinely new assigned packet in the course evidence workspace.

The Fourier convention and wave-packet interpretation are also checked against MIT Quantum Physics I lecture 7. The chirped Gaussian integral and covariance calculation above are the original worked extension, not a copied source exercise.