Physics

Added mass is the effective inertia contributed by fluid accelerated together with an immersed body.

A sphere of volume $V$ translating through unbounded inviscid incompressible fluid has fluid kinetic energyIts added mass is therefore one half of the displaced fluid mass.

A sphere of density ${\rho }_s$ falling in inviscid fluid of density $\rho$ has effective inertia $({\rho }_s+\rho /2)V$ and net weight $({\rho }_s-\rho )Vg$. Hence

The Compton wavelength of a particle of mass $m$ is ${\lambda }_C=h/(mc)$.

For a photon scattered through angle $\theta$ by an initially stationary electron,

After reducing conserved cyclic momenta, an equilibrium at a strict local minimum of the resulting effective potential is stable in the reduced one-degree-of-freedom dynamics.

Motion-dependent induced current can produce a magnetic force opposing motion and dissipating energy resistively.

An ideal superconducting loop supports persistent current that preserves linked magnetic flux when resistance vanishes.

A uniformly charged spherical shell produces zero field inside and the field of a point charge outside.

For two particles of masses $m_1,m_2$ interacting through a central Coulomb potential, separation of the centre of mass leaves the relative-coordinate Schrödinger equation with

For reduced mass $\mu$, the Coulomb length scale is

A no-radial-node Coulomb eigenstate has principal quantum number $n=l+1$ and radial factorIts energy is $-e^2/[8\pi {ϵ}_{0}a(l+1{)}^2]$.

For $R(r)\propto r^l{e}^{-r/[a(l+1)]}$, the radial probability density is proportional to $r^{2l+2}{e}^{-2r/[a(l+1)]}$ and

Neglecting spin, bound hydrogen eigenstates are labelled $|n,\ell ,m⟩$, where $\ell =0,\dots ,n-1$ and $m=-\ell ,\dots ,\ell$. Their unperturbed energy depends only on $n$.

For the $z$ component of position,requires $\Delta m=0$ and $\Delta \ell =\pm 1$. The first condition follows from $[L_z,z]=0$; the second follows from the vector-operator angular-momentum relation together with odd parity.

The even, spherically symmetric hydrogen ground state has no permanent electric dipole, so a uniform electric field gives no first-order energy shift. Odd-parity excited states mix in at second order, yielding a negative shift quadratic in the field.