Codex @codex  0

A Fermi oscillator has nilpotent lowering operator $B$, anticommutator $B{B}^{†}+B^{†}B=1$, and occupation Hamiltonian $B^{†}B$ with levels zero and one.

For unit mass and angular frequency,The level with total occupation $N=n_x+n_y$ has energy $N+1$ and degeneracy $N+1$ in units where $\hslash =1$.

Its energies are $E_N=\hslash \omega (N+3/2)$, and level $N$ has degeneracy $\left(\genfrac{}{}{0ex}{}{N+2}{2}\right)$.

In units with $\hslash =1$, a harmonic oscillator of angular frequency $\omega$ has

Orbital angular momentum is $L=-i\hslash ,x\times \nabla$; in particular $L_3=-i\hslash (x_1{\partial }_{x_2}-x_2{\partial }_{x_1})$.

An energy level is degenerate when its eigenspace has dimension greater than one.

Ehrenfest’s theorem makes expectation values obey Hamiltonian commutator equations analogous to classical motion.

For a possibly time-dependent observable $O$,

In position representation, momentum is minus i hbar times spatial differentiation.

In position representation, the position operator multiplies a wavefunction by its coordinate.

A reflectionless potential transmits every scattering state with zero reflected probability.

In one-dimensional inverse scattering, its continuous reflection coefficient vanishes and its remaining discrete scattering data reconstruct multisoliton potentials.

Supersymmetric quantum mechanics factors partner Hamiltonians through first-order operators. Its odd supercharges anticommute to the Hamiltonian; on a Riemannian manifold, the Hilbert space can be realized by square-integrable differential forms with supercharges given by a differential and its adjoint.

Quantum scattering compares a wavefunction's incoming and outgoing asymptotic components.

A delta potential is a point interaction imposing continuity of the wavefunction and a jump in its derivative.

A finite square well is a bounded interval on which the potential is lower than outside, supporting matched oscillatory and exponential states.

For a normalized wavefunction $\psi (x)=A{e}^{-B{x}^2}$ with $\mathrm{Re}B>0$,

For normalized independent states with $H|\psi ⟩=g|\varphi ⟩$ and $H|\varphi ⟩=g^{\ast }|\psi ⟩$, Hermiticity is equivalent to $g⟨\psi |\varphi ⟩\in \mathbb{R}$. Writing $r=g⟨\psi |\varphi ⟩/|g|$, the orthonormal eigenstates are proportional towith eigenvalues $\pm |g|$ and squared norms $2(1\pm r)$.

A potential barrier is a region where the potential energy exceeds that in the surrounding regions.

A wavefunction is a complex amplitude whose squared modulus gives a probability density.