Bound state

A bound state is a normalizable energy eigenstate spatially confined by a potential.

A one-dimensional well $V=-V_0$ on $0<x<a$, with an infinite wall at zero and $V=0$ beyond $a$. For $-V_0<E<0$, the interior wavefunction is proportional to $\sin kx$ and the exterior decays as $e^{-qx}$. Continuity gives $k\cot ka=-q$, where $k^2=2m(E+V_0)/{\hslash }^2$ and $q^2=-2mE/{\hslash }^2$. Its half-line finite-well bound-state count has a nonzero minimum depth, unlike a full-line attractive one-dimensional well.

For dimensionless depth $\nu =a\sqrt{2m{V}_0}/\hslash$, each negative branch of $z\cot z=-\sqrt{{\nu }^2-z^2}$ gives one bound state. The $n$th state exists exactly when $\nu >(n-1/2)\pi$. At equality its exterior decay constant vanishes and the state is not square integrable. In particular, exactly two bound states means $3\pi /2<\nu \le 5\pi /2$. This strict lower and inclusive upper endpoint convention counts negative-energy eigenstates, excluding a zero-energy threshold solution.

A quantum particle above a hard reflecting wall in a linearly increasing potential $V(x)=Fx$ for $x>0$. The Schrödinger equation reduces to the Airy equation. Decay at infinity selects $\mathrm{Ai}$, while the wall requires $\mathrm{Ai}(-z_n)=0$. Thus ${\psi }_n\propto \mathrm{Ai}(x/\ell -z_n)$ and the energies are the displayed values. The $n$th state has $n-1$ interior nodes and a decaying tail beyond its classical turning point. A gravitational realization has $F=mg$; gravitational neutron bound-state transmission tests such confinement experimentally.

A horizontal neutron-reflecting mirror supplies the lower boundary of a quantum bouncer, with the Earth's approximately linear gravitational potential above it. An absorber above the mirror removes neutrons with sufficient vertical extent. Scanning the mirror-absorber gap and counting transmitted neutrons probes the lowest allowed gravitational states. The 2002 work led by Valery V. Nesvizhevsky reported this confinement; a subsequent detailed report describes suppression below the lowest-state height.^[ref][ref]

This is a count-versus-gap experiment, not a direct image of a wavefunction. Surface scattering, absorption and the incident vertical-motion distribution enter a quantitative transmission model.

A Coulomb bound state is a physical normalizable eigenfunction for an attractive potential $V(r)=-\gamma /r$, $\gamma >0$. For the usual regular boundary condition at the origin its energies are the displayed values, $N=1,2,\dots$. The spherically symmetric states have orbital angular momentum zero; series-termination quantization of the Coulomb radial equation constructs them as a polynomial times a decaying exponential. Square integrability on the punctured radial interval alone does not replace the physical condition at the origin.

For motion confined to a plane but retaining potential energy $-\kappa /r$, separation in polar coordinates gives angular wavefunctions $e^{im\varphi }$ with integer $m$. The regular radial factor behaves as $r^{|m|}$ at the origin. Removing this power and the decaying exponential function leaves an Generalized Laguerre polynomial differential equation. polynomial termination requires $N=n_r+|m|+1/2$, yielding the displayed energy. At fixed $q=n_r+|m|$, the spinless quantum degeneracy is $2q+1$. This is not the logarithmic potential of electrostatics formulated entirely in two spatial dimensions.

For a spherically symmetric bound state, write $\psi =e^{-br}\sum c_j{r}^j$. The radial Coulomb equation gives the displayed recurrence. Choosing $b=a/(2N)$ terminates it after degree $N-1$, producing a nonzero continuous polynomial times an exponentially decaying function. Its three-dimensional norm $4\pi {\int }_0^{\infty }{r}^2|\psi {|}^{2}dr$ is finite and can be normalized. The energy is $-{\hslash }^2{a}^2/(8m{N}^2)$. Continuity at the origin is compatible with the Coulomb cusp; no differentiability as a Cartesian radial function at the origin is asserted.