PHY.9:4 - Solution

Choose the wanted distinction → select a responsive physical effect → construct the preparation and comparison → derive the readout and its disturbance → determine what can be resolved → revise the arrangement where needed.

PHY.9:4.1 - Specify the physical distinction and conditions

Name two physical possibilities that the work needs to distinguish, or a quantity range within which an estimate is needed. State where and when the property is sought. A pressure before connecting a probe and the pressure maintained during readout can be different targets.

Choose the needed consequence. Detecting a change, estimating a value and deciding which side of a limit applies place different demands on the same arrangement. C.16 provides the characteristic and scale; C.16.IR determines what a given indication can resolve.

Include the conditions the work permits you to change. A prepared reference, imposed force, interrogation time or detector orientation is a potential experimental choice only when it can actually be controlled.

PHY.9:4.2 - Choose an effect that carries the distinction into a response

Find a physical interaction through which the sought difference changes a state, transition, force, flow or other observable response. Derive that dependence far enough to see whether the proposed readout retains it. PHY.4 can constrain an unknown law; PHY.6 or PHY.7 can construct the relevant physical evolution.

Compare a small number of plausible arrangements. For each, ask what differs in the predicted readout between the selected physical possibilities under the same preparation. Reject an insensitive arrangement for this question even if it uses a well-established effect. The falling-body case in :5.1 shows why the occurrence of a quantity in one equation is insufficient.

Identify what supplies a reference. A known imposed quantity, a second preparation or a stable response can make the comparison interpretable. State the physical relation that allows the reference to be transported into this measurement. Numerical labels on two devices do not establish that relation.

Retain an unknown coupling as unknown. An available bound or a calibration can be enough; a proposed response without grounds for its coupling remains conditional. Select new characterization only when it can change the intended use.

PHY.9:4.3 - Construct the preparation and comparison

Specify how the subject, probe and reference are prepared before coupling. Then choose what is held fixed, what is varied and what is read. Include the order when the first interaction changes the next state.

Use a differential, reversed, ratio or null comparison when its physical invariance removes a consequential influence. Derive the cancellation from the arrangement. For example, reversing a known applied force while preserving a common force offset changes the wanted response but leaves the offset shared. A reversal that also changes the offset does not support that subtraction.

For a quantum probe, choose a prepared state, the parameter-dependent interaction and the measurement together. The physical interaction may be described by a channel taking rho to rho_theta; measurement operators E_k give outcome probabilities P(k|theta)=Tr(rho_theta E_k). Here rho is the prepared state, theta the sought parameter, and the positive operators E_k sum to the identity. If those probabilities do not change with theta, that preparation and readout supply no information about it.

A reference can also select which part of a response is read. A phase-sensitive measurement needs a phase reference; a differential displacement needs a reference position. Preserve the conditions that make this reference stable during the comparison.

PHY.9:4.4 - Include the effect of measuring on the subject

Construct the coupled subject-probe account for the duration that matters. Determine whether connecting the probe changes the sought quantity, changes only another quantity, or leaves its relevant value effectively unchanged. Use the physical coupling rather than a universal claim that measurement must disturb the target.

If the wanted value precedes coupling, derive how it can be recovered from the disturbed response, or change the arrangement to reduce the consequential disturbance. Keep uncertainty in that recovery. When correction is poorly determined, a different interaction or a conditional range can be more useful than more decimal places.

Compare the retained effect with the allowed error or decision margin. Loading, finite response time and backaction matter through their consequences for this question. PHY.5 helps justify an approximation. A usable result does not require modeling every interaction.

A nondestructive or non-demolition measurement protects a specified property under particular conditions; it does not establish that every later use of the subject is unaffected. Preserve any disturbance that changes the receiving experiment or work.

PHY.9:4.5 - Determine the distinctions the readout supports

Compose the physical response with the actual recording procedure through C.16.MR. Use MMP.7 when the conclusion requires a probability law for those records. Retain common influences, saturation, timing and finite resolution when they affect the conclusion.

Determine whether different target values remain compatible with the same possible records. If they do, identify which interaction, preparation or readout could separate the relevant alternatives. A local slope measures a local response; it does not resolve a periodic ambiguity or a shared unknown parameter.

Use the uncertainty or error model appropriate to the inference. Strongly nonlinear conversion can change both an estimate and its uncertainty; do not assume that converting an average is equivalent to averaging the converted quantity. Apply the necessary mathematical propagation or inference method, rather than treating a sensitivity coefficient as a complete answer.

When several parameters or performance goals matter, compare achievable combinations under the available resources. Different measurements can favor different parameters. Use C.16 to characterize those combinations and C.11 when choosing between available arrangements.

PHY.9:4.6 - Obtain a first result and revise the physical arrangement

Work one admissible preparation through to a predicted readout and an interpreted result. This can reveal an unresolved physical relation before an instrument is built. It also gives collaborators a concrete construction to implement or compare.

Change a condition that could defeat the intended use. Check the shared premise of a subtraction, the range of an inverse, the time available for a response or the disturbance caused by the probe. Return to the smallest implicated part of the arrangement.

Use available observations where they can change confidence or selection. C.11.DUA helps choose between another measurement, a conditional result and proceeding with uncertainty. A statistical comparison or a calibration certificate does not substitute for a missing physical response.

Return the proposed arrangement, its preparation and readout, the result it supports and the conditions that would change that use. The next work may be implementation, calibration, inference or a different measuring interaction. PHY.10 constructs a physical test when the purpose is to separate rival accounts.