For $p(x)=\sqrt{2m[E-V(x)]}$ in a classically allowed region, a local validity condition is $|\hslash p^{\prime }/p^2|\ll 1$. The momentum changes little during a local oscillation. It fails at an ordinary turning point where $p=0$, requiring an Airy turning-point connection formula.

For an opaque barrier between simple turning points, the leading transmission probability is $T\simeq e^{-2S_B/\hslash }$, where $S_B=\int \sqrt{2m[V(x)-E]}\mathrm{d}x$ over the forbidden interval. The exponent for probability is twice that for amplitude. Matching determines prefactors and the domain where this approximation applies.

At fixed incident energy and fixed potential-energy profile, the forbidden-region WKB action is $\int \sqrt{2m(V-E)}dx$. Its leading transmission exponent grows as the square root of mass. Equal electrostatic voltages need not define equal potential-energy barriers for particles with different charges. The leading exponential comparison assumes a sufficiently opaque barrier; prefactors and the WKB turning-point conditions remain relevant.

Write $(3/(4\pi n){)}^{1/3}=r_s{a}_B^{\ast }$, where $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$. Here $r_s$ is dimensionless, unlike a Wigner-Seitz radius measured as a length. Electron-gas energy formulas quoted in effective atomic units commonly use this convention.

The wheel modulus is the product $W$ of a chosen finite set of small distinct prime numbers in wheel factorization. Except for those primes themselves, every prime candidate must lie in a residue class coprime to $W$. For $W=2\cdot 3\cdot 5=30$, the allowed residues are $1,7,11,13,17,19,23,29$, repeating every thirty integers. The proportion retained is $\phi (W)/W$. Passing the wheel only avoids these small divisors; it does not certify primality, since composite values such as $49$ still pass.

The Zeeman effect need not be diagonal between different $J$ multiplets. The familiar Landé g-factor gives the interaction projected into one isolated multiplet. Inter-multiplet matrix elements cause higher-order weak-field shifts and become important in the Paschen-Back effect.

A position or set of positions where a wavefunction vanishes. Its Born rule position probability density is also zero there. For a one-dimensional bound state, counting interior nodes distinguishes successive energy eigenfunctions; hard-wall endpoint zeros are usually counted separately.

The wavefunction is the probability amplitude of a quantum state in the generalized position basis. For a scalar particle on a line, the position operator multiplies by $x$ and the momentum operator acts as $-i\hslash d/dx$ on their appropriate domains. This makes the canonical commutation relation $[x,p]=i\hslash$ explicit.

A top-hat wavefunction is constant on an interval and zero elsewhere. It has a uniform position distribution and a sinc function momentum amplitude. Its discontinuous edges place it outside the finite-kinetic-energy domain of the free Hamiltonian. Smooth-edged localized packets can approximate its position profile while having different momentum tails.

The Fourier transform in the title represents a one-dimensional state in the wavevector basis. Its probability density is $|\varphi (k){|}^2$, normalized by $\int |\varphi (k){|}^{2}dk=1$ when $\int |\psi (x){|}^{2}dx=1$. For physical momentum $p=\hslash k$, the density per unit momentum is $|\varphi (p/\hslash ){|}^2/\hslash$. A confined stationary energy eigenstate need not be a momentum eigenstate.