For an infinite square well of width $a$, the normalized trial $\tilde{\psi }=\sqrt{30/a^5}x(a-x)$ obeys both Dirichlet boundary conditions and has no interior nodes of a wavefunction. Its kinetic energy isThe ratio to the exact ground-state energy is $10/{\pi }^2=1.01321\dots$. Its expansion in the normalized sine eigenfunctions has coefficients $4\sqrt{60}/({\pi }^3{n}^3)$ for odd $n$ and zero for even $n$. This supplies a concrete check of the quantum variational principle and shows that a small energy error can coexist with excited-state admixtures.

For a well on $0<x<L(t)$ with real normalized sine modes, the displayed coupling holds for $n\ne p$ and the diagonal coupling vanishes. Differentiate both the normalization and the sine at fixed position, then integrate on the instantaneous interval. The moving boundary contributes no endpoint term because the modes vanish there. The resulting real antisymmetric matrix preserves total probability in the instantaneous eigenstate expansion.

A centred width-$L$ infinite square well has ground state wavefunction $\sqrt{2/L}\cos (\pi x/L)$ inside and zero outside. Its momentum representation in quantum mechanics follows from the Fourier transform of a truncated cosine. The probability density in the title has removable singularities at $k=\pm \pi /L$, with limit $L/(4\pi )$, and an oscillatory $k^{-4}$ tail envelope. The distribution is even, with $⟨k⟩=0$ and $⟨k^2⟩={\pi }^2/L^2$.

Every stationary state in $0<x<w$ has mean position $w/2$. Splitting a uniform field coupling into $-qE(t)(x-w/2)-qE(t)w/2$ separates its transition-driving odd operator from an identity term. The latter gives the common phase $qw\int E(t)dt/(2\hslash )$ and cannot change populations. Removing it by an origin shift is equivalent to a time-dependent scalar phase; relative transition amplitudes are unchanged.

For $a>0$, this integral is an entire function of $x$: the exponential series converges uniformly on bounded subsets after multiplication by the integrable bound $e^{|x|}{u}^{a-1}$. ThusFor positive real $x$, a change of variables gives $F_a(x)=x^{-a}\gamma (a,x)$, where here $\gamma (a,x)={\int }_0^x{e}^{-t}{t}^{a-1}dt$ denotes the lower incomplete gamma function. The singular-looking factor $x^{-a}$ is removed by the integral over $[0,1]$. Applying integration by parts to the derivative of $u^a{e}^{-xu}$ gives $x{F}_a^{\prime }+a{F}_a=e^{-x}$; differentiating and eliminating $e^{-x}$ proves $x{F}_a^{\prime \prime }+(a+1+x)F_a^{\prime }+a{F}_a=0$.

In a massless Fermi interaction approximation, a charged-current decay channel contributes a common mass-to-the-fifth width multiplied by its mixing and final-state multiplicity factors. A final quark-antiquark pair has three color choices, while a lepton pair has one. An initial quark does not acquire a factor of three in its lifetime merely from its possible initial colors.

Selection associated with the indirect reproductive consequences of actions affecting genetically associated recipients. It can favour cooperation without removing all evolutionary conflicts among relatives.

Reduced reproductive or survival performance associated with mating among relatives or self-fertilization in the relevant comparison. Measuring seed set alone can miss later offspring costs.

A sequence of bound-to-bound optical transitions of a hydrogenic donor state or hydrogenic acceptor state, converging on a photoionization threshold $R^{\ast }$. For an initially occupied $1s$ state, ordinary electric dipole transitions reach $np$ states; above the threshold, the final carrier can enter a band continuum.

An interaction between a nucleus's magnetic or electric multipole moments and its electronic surroundings. Magnetic contact and dipolar terms can couple nuclear spin to electron spin or orbital magnetic moments. The resulting small energy shifts contribute to hyperfine structure; the contact term can also mediate conduction-electron-mediated exchange between nuclear moments in a metal.

For electron spin $s=1/2$, first add orbital and spin angular momenta to obtain $j=L\pm 1/2$. For nuclear spin $I=1$, add again to obtain $F=|j-1|,|j-1|+1,\dots ,j+1$. The projection has $2F+1$ possible values. A hydrogenic bound state still requires $L\le n-1$.

A transition whose frequency is stationary to first order in a specified perturbation at an operating point. For magnetic sensitivity, the derivative shown vanishes, but second and higher derivatives can remain nonzero. A clock transition need not have zero individual energy slopes: equal slopes suffice. Optical light shifts and other perturbations require their own sensitivity analysis.

Replace instantaneous acceleration and braking by legs of constant proper-acceleration magnitude $a$, retaining the same cruise rapidity $\eta$ and fixed endpoint separation. Each leg takes proper time $c\eta /a$ and displaces $(c^2/a)(\cosh \eta -1)$ in the endpoint rest frame. Subtracting the proper cruise time displaced by both legs gives the displayed correction. A cruise segment exists only if the total endpoint separation exceeds twice this acceleration distance.

In units $c=1$, constant signed proper acceleration $a$ givesPosition data determine $t_0,x_0$ but leave the initial rapidity ${\eta }_0$ free. Proper-acceleration magnitude alone also leaves its orientation unspecified. The familiar centred hyperbola $x^2-t^2=a^{-2}$ follows for the additional initial data $t_0=0$, $x_0=1/a$, ${\eta }_0=0$ and acceleration toward increasing $x$.