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Fix a hidden variable in the deterministic local hidden-variable model, so that every $A_j,B_j$ is an integer. Let $[x{]}_d$ be the representative of $x$ modulo $d$ in $\{0,\dots ,d-1\}$. The terms in the chained modular Bell inequality alternate between the two parties. Before reduction their sum telescopes:After reduction, the sum $T$ is a nonnegative integer congruent to $-1$ modulo $d$. For $d\ge 2$, the smallest possible such integer is $d-1$. Thus $T\ge d-1$ for each hidden variable separately. Averaging over the setting-independent distribution givesHere each expectation is the expectation value of the reduced random variable, not the residue of its expectation. Each term can be measured using one setting at each site. The proof uses their common deterministic assignments rather than any joint quantum measurement of incompatible local settings. Stochastic local hidden-variable theories satisfy the same bound: include their local random seeds in $\lambda$ and average the resulting deterministic assignments.

The printed final coefficient in the definition of the average is typographically incomplete. The expectation used here is the usual $\mathbb{E}X={\sum }_{x=0}^{d-1}xP(X=x)$, with final coefficient $d-1$.

Distinct eigenvalues of the self-adjoint angular momentum operator imply $⟨{\psi }_1,{\psi }_2⟩=0$. The given combination is not normalized: its squared norm is $({\cos }^2\omega t+3{\sin }^2\omega t)/4$, which is always positive. Therefore its properly normalized expectation value isIt vanishes when ${\tan }^2\omega t=1/3$. The first positive solution has $\omega t=\pi /6$, soNormalization changes the value of the expectation value away from zero, but not its zero times.

The expectation value is the mean outcome over many identically prepared measurements of the observable. For real $\lambda$, positivity of the norm in the Hilbert space givesThis real quadratic has nonpositive quadratic discriminant, soApply this to the centered observables $Q-⟨Q⟩$ and $P-⟨P⟩$ to obtain the Robertson uncertainty principle

For the quantum harmonic oscillator, $[x,p]=i\hslash$ andThe arithmetic-geometric mean inequality and $\Delta x\Delta p\ge \hslash /2$ now give

A stationary state has the formFor a time-independent quantum observable $Q$, the global phases cancel in the expectation value:It is therefore independent of time.

For the infinite square well,The possible energy measurement results arewith respective Born rule probabilities $|c_1{|}^2$ and $|c_2{|}^2$. Normalization gives $|c_1{|}^2+|c_2{|}^2=1$.

At time $t$,The required matrix elements areConsequentlywhere the angular frequency isFinally, $2|c_1{c}_2|\le |c_1{|}^2+|c_2{|}^2=1$, so