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For coprime integers $p,q$ with $q>0$,This follows directly from the real and p-adic absolute values, or from the height-Mahler measure formula for the primitive polynomial $qX-p$.

An Abelian K-matrix Chern--Simons theory has $|\det K|$ topologically distinct ground states on a torus.

The K-matrix is the integral bilinear form multiplying the Abelian Chern--Simons gauge fields. Integer unimodular basis changes send $K$ to $WK{W}^T$ without changing the phase.

For $J^{\mu }=Q{n}^{\mu }+Q^{\mu }$ with spatial $Q^{\mu }$, current conservation gives $n^{\mu }{\nabla }_{\mu }Q=KQ-D_{\mu }{Q}^{\mu }-Q^{\mu }{D}_{\mu }\log \alpha$ under the stated extrinsic-curvature convention.

With future unit normal $n^{\mu }$, the convention $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$ measures how the hypersurface is embedded in spacetime.

The shift vector describes how spatial coordinates move tangentially between neighboring hypersurfaces.

The lapse function measures proper-time separation between neighboring spatial hypersurfaces along their unit normal.

Writing $v=e^S$ in $v^{\prime \prime }=q(x)v$ produces the Riccati equationExpanding $S^{\prime }$ in inverse powers of $x$ determines the exponential phase and algebraic prefactor recursively.

To classify infinity, set $z=1/x$ and classify $z=0$. For a second-order equation in normal form, a transformed coefficient with a pole of order greater than two makes infinity an irregular singular point.

Forthe substitutiongives the normal form

For an oscillatory interval terminated by a hard boundary and a simple turning point, matching the boundary condition to the decaying solution gives an action integral equal to an integer multiple of $\pi$ with the turning-point phase correction.

For a one-dimensional energy $E$ and potential $V(x)$, a classical turning point satisfies $V(x)=E$. The classically allowed region has $E>V(x)$ and supports an oscillatory leading-order WKB solution.

Near a simple turning point, rescaling the independent variable reduces a second-order equation to the Airy equation. The decaying Airy solution on the forbidden side matches an oscillatory solution with phase shift $\pi /4$ on the allowed side.

A Gaussian wave packet is a wavefunction whose spatial amplitude is a Gaussian function. Free quantum evolution preserves its Gaussian form while changing its complex width and spreading its probability density.

For a nonrelativistic particle of mass $m$, the probability current is

A probability density is a nonnegative function whose integral over an event gives its probability.

A wavefunction is normalizable when the integral of its squared modulus is finite, so multiplication by a constant can make its total probability one.

A probability amplitude is a complex number that describes the contribution of a pure state to a specified measurement outcome. For a normalized quantum state $|\psi ⟩$ and a rank-one outcome associated with a normalized vector $|j⟩$, it is the inner product $a_j=⟨j|\psi ⟩$. The Born rule gives the corresponding probability $p_j=|a_j{|}^2=a_j^{\ast }{a}_j$, using the complex conjugate. Thus an amplitude can be negative or complex, while a probability is real and nonnegative. A wavefunction is the analogous position amplitude: its squared modulus gives a probability density, which must be integrated over a region to obtain a probability. ^[ref].

The extra information is the quantum phase. If $a=r{e}^{i\theta }$, its modulus $r$ fixes $|a{|}^2=r^2$, but its angle $\theta$ controls how it adds to another contribution. In the complex plane, picture amplitudes as arrows: add the arrows, then square the resulting length. For coherent contributions $A$ and $B$ to the same outcome,The last term is interference. Adding probabilities first loses it. Contributions with different relative quantum phases can reinforce or cancel even when their individual squared moduli are fixed.

For the qubit quantum superposition $|{\psi }_{\varphi }⟩=(\sqrt{3}/2)|0⟩+(e^{i\varphi }/2)|1⟩$, project onto the plus vector of the Hadamard basis. By linearity, the two contributions combine asHere $|A{|}^2=3/8$ and $|B{|}^2=1/8$, while the interference term is $(\sqrt{3}/4)\cos \varphi$. At $\varphi =0$, the arrows align and the plus probability is $1/2+\sqrt{3}/4\simeq 0.9330127$. At $\varphi =\pi$, they oppose and that probability is $1/2-\sqrt{3}/4\simeq 0.0669873$; cancellation is incomplete because the lengths differ. At $\varphi =\pi /2$, the arrows are perpendicular and the cross term vanishes. The minus amplitude uses $A-B$, reversing the cross term. These are alternate projective measurements on identically prepared copies, rather than a sequence of measurements on one copy.

A global phase multiplies every contribution by the same factor $e^{i\gamma }$, givingIt therefore changes no outcome probability. Changing a relative quantum phase can change the angle between contributions and hence their interference. Keeping amplitudes until the last step makes quantum circuits and matter-wave interference calculable; discarding their angles prematurely discards physically testable information. The Born rule example compares two measurement bases explicitly.

A plane wave has constant amplitude and a phase linear in space and time. It is a momentum eigenstate and is not spatially normalizable.

Expanding $(1+t{)}^{-1}$ at the endpoint and applying Watson's lemma gives