LM.1 · Counting statistics, uncertainty and dimensional models

A detector with ten percent efficiency does not register exactly ten photons out of every hundred. The count fluctuates even with stable efficiency.

Prerequisites: 47.

• Use binomial and Poisson count means and fluctuations
• Propagate small uncertainties with stated correlation assumptions
• Solve dimensional exponent constraints and check a model’s units

For N independent incident particles each detected with probability p, count X is binomial: mean Np and variance Np(1−p). The standard deviation is sqrt[Np(1−p)], not the variance itself. With large N and small p, a Poisson approximation 泊松近似 has mean λ=Np and variance λ, so deviation sqrt(λ). For N=200 and p=0.05, the exact mean is 10 and deviation sqrt(9.5)≈3.08, while Poisson gives sqrt(10)≈3.16. Neither model promises a fixed count. Correlated detections or dead time can invalidate independence.

For a smooth measured function y(x1,…), linearise changes using its partial derivatives. Independent small standard uncertainties combine in quadrature: σ_y²≈Σ(∂y/∂xi)²σ_i². Correlated inputs require covariance cross terms. For y=x^a, fractional standard uncertainty 标准不确定度 is approximately |a|σ_x/|x|. Thus kinetic energy K=½mv² with negligible mass uncertainty has fractional uncertainty twice that of speed. With independent mass uncertainty, combine (σ_m/m)²+(2σ_v/v)². These approximations need small errors and a suitable local linear model.

Random scatter measures precision; a common calibration bias affects accuracy and does not disappear by averaging repeats. For independent repeated readings, the standard uncertainty of the mean falls as 1/sqrt(n), but shared systematic error does not. The uncertainty of a physical spread and uncertainty of its estimated mean are different quantities. State whether a quoted percentage is a standard uncertainty, confidence interval or worst-case bound. Summing absolute contributions is a conservative bound, not the independent-standard-error quadrature rule.

Dimensional analysis equates powers of mass, length and time, rather than numerical sizes. For a Planck-length form G^a $\hbar$^b c^d, dimensions are [G]=L³/(MT²), [$\hbar$]=ML²/T and [c]=L/T. Requiring length gives −a+b=0, 3a+2b+d=1 and −2a−b−d=0. Hence a=b=1/2 and d=−3/2, so length is sqrt(G$\hbar$/c³). Dimensional analysis cannot determine an arbitrary dimensionless coefficient or prove that the chosen constants are physically sufficient. Reject a formula with wrong units before inserting numbers.

For independent small standard uncertainties in $K=mv^2/2$, differentiate first.

With relative uncertainties 3% in mass and 2% in speed, A worst-case bound would use a sum, not this quadrature. Correlations require covariance terms.

Do not confuse mean count with guaranteed count, variance with standard deviation, or a random standard error with a calibration bias. Dimensional consistency is necessary but not sufficient.

standard uncertainty: Uncertainty expressed as a standard deviation under a stated measurement model.

Poisson approximation: Rare independent-event count model with variance equal to its mean.