quantum-mechanics.bigb

= Quantum mechanics
{wiki}

Quantum mechanics models states by wavefunctions and observables by operators.

= Parity
{title2=$\mathcal P$}
{parent=Quantum mechanics}
{wiki=Parity_(physics)}

Parity is the eigenvalue $+1$ or $-1$ of a state under <spatial reflection>.

= Spatial reflection
{title2=$\mathbf x\mapsto-\mathbf x$}
{parent=Parity}
{wiki=Parity_(physics)}

Spatial reflection reverses every spatial coordinate.

= Photon
{parent=Quantum mechanics}
{wiki}

A photon is a quantum of the electromagnetic field with energy $E=h\nu=\hbar\omega$.

= Ground state
{parent=Quantum mechanics}
{wiki}

A ground state is an energy eigenstate with the smallest possible energy. For noninteracting particles, the many-body ground state is obtained by filling one-particle levels subject to the particles' exchange statistics.

= Energy eigenvalue
{parent=Quantum mechanics}

An energy eigenvalue $E$ is an <eigenvalue> of the Hamiltonian: $H|\psi\rangle=E|\psi\rangle$ for some nonzero state $|\psi\rangle$.

= Energy eigenstate
{parent=Energy eigenvalue}

An energy eigenstate is a nonzero <quantum state> represented by an eigenvector of the <Hamiltonian operator>. Its time evolution changes only its overall phase.

= Energy eigenspace
{parent=Energy eigenvalue}

The energy eigenspace for $E$ is $\ker(H-EI)$.

= Energy eigenspaces
{synonym}

= Excited state
{parent=Energy eigenvalue}
{wiki}

An excited state is an <energy eigenvalue>[energy eigenstate] whose energy exceeds the <ground state> energy.

= First excited state
{parent=Excited state}

The first excited state has the smallest energy strictly above the ground-state energy.

= Eigenstate
{parent=Energy eigenvalue}

An eigenstate of an observable is a <quantum state> represented by an <eigenvector> of that observable.

= Momentum eigenstate
{parent=Eigenstate}
{wiki=Momentum_operator#Eigenfunctions}

A momentum eigenstate is an eigenstate of the <momentum operator>. In one dimension it is a <plane wave> $e^{ikx}$ with momentum eigenvalue $\hbar k$.

= Quantum state
{parent=Quantum mechanics}
{wiki}

A pure quantum state is a ray in a complex Hilbert space and may be represented by a normalized state vector.

= Qubit state
{parent=Quantum state}

A pure qubit state is a ray in a two-dimensional complex Hilbert space.

= Qubit states
{synonym}

= Quantum number
{parent=Quantum state}
{wiki}

A quantum number is a discrete label, usually an observable's eigenvalue, used to distinguish quantum states.

= Quantum phase
{parent=Quantum state}
{wiki=Introduction_to_quantum_mechanics\#Mathematical_formulation}

Multiplying a quantum state by one global complex phase does not change any measurement probability; relative phases between components can affect interference.

= Observable
{parent=Quantum state}
{wiki=Observable}

A quantum observable is represented by a self-adjoint operator; its possible measured values are spectral values of that operator.

= Expectation value
{parent=Observable}
{wiki=Expected_value\#Quantum_mechanics}

In a normalized state $|\psi\rangle$, the expectation of an observable $A$ is $\langle\psi|A|\psi\rangle$.

= Hamiltonian operator
{parent=Observable}
{wiki=Hamiltonian_(quantum_mechanics)}

The Hamiltonian operator represents the total energy and generates time evolution through the Schrödinger equation.

= Quantum symmetry of a Hamiltonian
{parent=Hamiltonian operator}

A unitary operator $U$ is a symmetry of a time-independent Hamiltonian $H$ when
$$
U^\dagger H U=H,
$$
equivalently $[U,H]=0$. A differentiable one-parameter symmetry $U(s)=e^{-isQ/\hbar}$ has a self-adjoint generator $Q$ satisfying $[Q,H]=0$, so $Q$ is conserved.

= Conserved generator of a continuous quantum symmetry
{parent=Quantum symmetry of a Hamiltonian}

If $U(s)=e^{-isQ/\hbar}$ and $U(s)^\dagger H U(s)=H$, differentiation at $s=0$ gives
$$
[Q,H]=0.
$$
In the <Heisenberg picture>,
$$
\frac{dQ}{dt}=\frac i\hbar[H,Q]=0.
$$

= Unitary time evolution
{parent=Hamiltonian operator}
{wiki=Time_evolution}

A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hbar}$.

= Heisenberg picture
{c}
{parent=Unitary time evolution}
{wiki}

In the Heisenberg picture, states are fixed and an observable evolves as
$$
A_H(t)=U(t)^\dagger A_S U(t).
$$
For a time-independent Schrödinger-picture observable,
$$
\frac{dA_H}{dt}=\frac i\hbar[H,A_H].
$$

= Time-reversal symmetry in quantum mechanics
{parent=Unitary time evolution}
{wiki=Time_reversal_symmetry}

A time-reversal operator $T$ intertwines forward and backward evolution:
$$
U(t)T=TU(-t).
$$
It must be antiunitary in ordinary quantum mechanics. Its conjugate linearity sends $i$ to $-i$, allowing it to commute with a time-reversal-invariant Hamiltonian rather than reversing the sign of its energy.

= Unitary time reversal reverses the energy spectrum
{parent=Time-reversal symmetry in quantum mechanics}

If a unitary $T$ satisfied $U(t)T=TU(-t)$, differentiation at zero would give
$$
HT=-TH.
$$
It would map every energy $E$ to $-E$. A Hamiltonian unbounded above would consequently be unbounded below and have no stable ground state.

= Antiunitary time reversal preserves the energy spectrum
{parent=Time-reversal symmetry in quantum mechanics}

For antiunitary $T$, conjugate linearity gives $T(i\psi)=-iT\psi$. Differentiating
$$
U(t)T=TU(-t)
$$
therefore yields $HT=TH$. Time reversal maps an energy eigenspace to itself and creates no negative-energy instability.

= Bound state
{parent=Quantum state}
{wiki}

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

= Stationary state
{parent=Quantum state}
{wiki}

A stationary state is an energy eigenstate whose time dependence is only an overall phase.

= Planck constant
{title2=$h$}
{parent=Quantum mechanics}
{c}
{wiki}

The reduced Planck constant is $\hbar=h/(2\pi)$ and sets the scale of quantum commutators and phases.

= Wavefunction
{parent=Quantum mechanics}
{wiki=Wave_function}

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

= Plane wave
{parent=Wavefunction}
{wiki}

A plane wave has constant amplitude and a phase linear in space and time. It is a <momentum eigenstate> and is not spatially normalizable.

= Probability amplitude
{parent=Wavefunction}
{wiki}

A probability amplitude is a complex coefficient whose squared modulus gives the probability of the associated measurement outcome.

= Born rule
{parent=Probability amplitude}
{c}
{wiki}

The Born rule assigns probability $|\langle a|\psi\rangle|^2$ to outcome $a$ when a normalized state $|\psi\rangle$ is measured in an orthonormal eigenbasis.

= Wigner theorem
{c}
{parent=Born rule}
{wiki=Wigner%27s_theorem}

Every bijection of pure-state rays that preserves transition probabilities is induced by either a unitary or an antiunitary operator on the Hilbert space.

= Projective unitary representation
{parent=Wigner theorem}
{wiki=Projective_representation}

Because state vectors differing by a <quantum phase> represent the same ray, a unitary realization of a group action need only satisfy
$$
U(g_1)U(g_2)=e^{i\phi(g_1,g_2)}U(g_1g_2).
$$

= Energy measurement
{parent=Born rule}

An ideal energy measurement returns an eigenvalue of the Hamiltonian with probability equal to the squared norm of the state's projection onto its eigenspace. After obtaining a nondegenerate energy, the state is the corresponding energy eigenstate.

= Normalizable wavefunction
{parent=Wavefunction}
{wiki=Wave_function\#Normalization_condition}

A wavefunction is normalizable when the integral of its squared modulus is finite, so multiplication by a constant can make its total probability one.

= Normalizable
{synonym}

= Probability density
{parent=Wavefunction}
{wiki=Probability_density_function}

A probability density is a nonnegative <function> whose <integral> over an event gives its probability.

= Probability current
{parent=Wavefunction}
{wiki=Probability_current}

For a nonrelativistic particle of mass $m$, the probability current is
$$
J=-\frac{i\hbar}{2m}
\left(\psi^*\nabla\psi-\psi\nabla\psi^*\right).
$$

= Probability continuity equation
{parent=Probability current}

For a <wavefunction> satisfying the <Schrodinger equation> with a real potential,
$$
\frac{\partial|\psi|^2}{\partial t}+\nabla\cdot J=0.
$$
This <continuity equation> expresses local conservation of probability.

= Conservation of quantum probability
{parent=Probability continuity equation}

If the <probability current> of a <normalizable wavefunction> vanishes at spatial infinity, integrating the <probability continuity equation> over all space shows that the total probability $\int |\psi|^2$ is constant in time.

= Gaussian wave packet
{parent=Wavefunction}
{c}
{wiki}

A Gaussian wave packet is a <wavefunction> whose spatial amplitude is a <Gaussian function>. Free quantum evolution preserves its Gaussian form while changing its complex width and spreading its <probability density>.

= Potential barrier
{parent=Quantum mechanics}
{wiki=Potential_barrier}

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

= Quantum tunnelling
{parent=Potential barrier}
{wiki=Quantum_tunnelling}

Quantum tunnelling is transmission through a <potential barrier> even when the particle energy is below the barrier height.

= Finite square barrier transmission at half barrier height
{parent=Quantum tunnelling}

For a square <potential barrier> of width $2a$ and height $U_0=2E$, put
$$
k=\frac{\sqrt{2mE}}{\hbar}.
$$
The oscillatory wavenumber outside and decay constant inside are both $k$. Matching the <wavefunction> and its <derivative> at both faces gives transmission amplitude
$$
t=\frac{e^{-2ika}}{\cosh(2ka)}
$$
and transmission probability $\operatorname{sech}^2(2ka)$.

= Hermitian operator on a nonorthogonal two-state basis
{parent=Quantum mechanics}

For normalized independent states with $H|\psi\rangle=g|\phi\rangle$ and $H|\phi\rangle=g^*|\psi\rangle$, Hermiticity is equivalent to $g\langle\psi|\phi\rangle\in\mathbb R$. Writing $r=g\langle\psi|\phi\rangle/|g|$, the orthonormal eigenstates are proportional to
$$
(g^*/|g|)|\psi\rangle\pm|\phi\rangle
$$
with eigenvalues $\pm|g|$ and squared norms $2(1\pm r)$.

= Commuting time-dependent two-state Hamiltonian
{parent=Hermitian operator on a nonorthogonal two-state basis}

If a time-dependent perturbation remains diagonal in a fixed eigenbasis, Hamiltonians at different times commute. Each component therefore acquires the exact phase $\exp[-i\int E_\pm(t)dt/\hbar]$, with no time ordering required.

= Second moments of a complex Gaussian wave packet
{parent=Quantum mechanics}

For a normalized wavefunction $\psi(x)=Ae^{-Bx^2}$ with $\operatorname{Re}B>0$,
$$
\langle x^2\rangle=\frac1{4\operatorname{Re}B},
\qquad
\langle p^2\rangle=\frac{\hbar^2|B|^2}{\operatorname{Re}B}.
$$

= Finite square well
{parent=Quantum mechanics}
{wiki}

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

= Delta potential
{parent=Quantum mechanics}
{wiki}

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

= Bound state in a delta potential
{parent=Delta potential}
{wiki}

An attractive one-dimensional delta potential has one even exponentially decaying bound state.

= Multiple delta potential
{parent=Delta potential}

For delta interactions at points $x_j$, a Green-function solution reduces the Schrodinger equation to a finite linear system for the values $\psi(x_j)$.

= Scattering by a delta potential
{parent=Delta potential}

For $V(x)=V_0\delta(x)$, $E=\hbar^2k^2/(2m)$, and $\gamma=mV_0/(\hbar^2k)$, continuity and the derivative jump give
$$
t=\frac1{1+i\gamma},
\qquad
r=\frac{-i\gamma}{1+i\gamma}.
$$
They obey $|r|^2+|t|^2=1$ and $r^*t+t^*r=0$.

= Quantum scattering
{parent=Quantum mechanics}
{wiki}

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

= Reflected wave
{parent=Quantum scattering}

A reflected wave is an outgoing component travelling back toward the side from which the incident wave arrived.

= Transmission probability
{parent=Quantum scattering}
{wiki=Transmission_coefficient}

The transmission probability is the ratio of transmitted to incident <probability current>. When the asymptotic wavenumbers agree, it is the squared modulus of the transmission amplitude.

= One-dimensional S-matrix
{parent=Quantum scattering}
{c}

For a reflection-invariant localized potential, order incoming amplitudes from the left and right and outgoing amplitudes toward the left and right. Then
$$
S=\begin{pmatrix}r&t\\t&r\end{pmatrix},
$$
where $r$ and $t$ are the reflection and transmission amplitudes. Probability-current conservation makes $S$ unitary for a real potential.

= One-dimensional transfer matrix from scattering amplitudes
{parent=One-dimensional S-matrix}
{c}

If the left and right travelling-wave coefficients are related by
$$
B_L=rA_L+tB_R,
\qquad
A_R=tA_L+rB_R,
$$
then
$$
\begin{pmatrix}A_R\\B_R\end{pmatrix}
=\frac1t
\begin{pmatrix}t^2-r^2&r\\-r&1\end{pmatrix}
\begin{pmatrix}A_L\\B_L\end{pmatrix}.
$$

= Partial-wave S-matrix
{parent=Quantum scattering}
{c}
{wiki=S-matrix#Partial_wave}

For a central potential, each angular-momentum channel has a scalar scattering coefficient $S_l(k)=e^{2i\delta_l(k)}$.

= Unitarity and reflection identities for a partial-wave S-matrix
{parent=Partial-wave S-matrix}

For a real central potential and real $k$, radial-current conservation gives $S_l(k)^*S_l(k)=1$. Invariance of the radial equation under $k\mapsto-k$ gives $S_l(k)S_l(-k)=1$, so a continuous phase convention has $\delta_l(-k)=-\delta_l(k)$.

= Scattering length from a partial-wave S-matrix
{parent=Partial-wave S-matrix}

The $s$-wave scattering length is
$$
a_s=-\lim_{k\to0}\frac{\tan\delta_0(k)}k,
$$
and the low-energy total cross-section is $4\pi a_s^2$.

= Bound-state poles of a partial-wave S-matrix
{parent=Partial-wave S-matrix}

A pole at $k=i\kappa$ with $\kappa>0$ represents a normalizable bound state of energy $-\hbar^2\kappa^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.

= Scattering resonance
{parent=Partial-wave S-matrix}
{wiki=Resonance_(particle_physics)}

A scattering resonance is a metastable state represented by a pole of the analytically continued scattering matrix. Near an isolated narrow resonance, a cross-section has a Breit-Wigner peak whose energy width is the inverse lifetime up to $\hbar$.

= Lippmann-Schwinger equation
{parent=Quantum scattering}
{c}
{wiki}

The Lippmann-Schwinger equation rewrites the Schrodinger differential equation as an integral equation using a free Green function and a prescribed incoming state.

= Scattering amplitude
{parent=Quantum scattering}
{wiki}

A scattering amplitude is the coefficient of an outgoing asymptotic wave relative to the specified incoming wave.

= Momentum transfer
{title2=$Q$}
{parent=Scattering amplitude}
{wiki}

The momentum transfer in a scattering process is the difference between the incoming and outgoing <wavevector>[wavevectors], commonly $Q=k-k'$ up to a sign convention and a factor of $\hbar$.

= Elastic scattering
{parent=Scattering amplitude}
{wiki}

Elastic scattering preserves the total kinetic energy. For one particle scattered by a fixed target, the incoming and outgoing <wavevector>[wavevectors] therefore have equal magnitudes.

= Diffraction
{parent=Scattering amplitude}
{wiki}

Diffraction is interference produced when waves scatter from spatial structure. A periodic structure concentrates the scattered intensity at reciprocal-lattice conditions.

= Crystal scattering
{parent=Scattering amplitude}
{wiki=Diffraction}

In the <Born approximation>, scattering from identical atomic potentials translated to lattice sites factors into one atomic form factor times a finite sum of phases over the crystal sites.

= Bravais lattice
{parent=Crystal scattering}
{c}
{wiki}

A three-dimensional Bravais lattice is
$$
\Lambda=\{n_1a_1+n_2a_2+n_3a_3:n_i\in\mathbb Z\},
$$
where the primitive vectors $a_1,a_2,a_3$ are linearly independent.

= Lattice point
{parent=Bravais lattice}
{wiki=Lattice_(group)#Lattice_points}

A lattice point is a vector $l=n_1a_1+n_2a_2+n_3a_3$ whose coefficients in a chosen primitive <basis> are <integer>[integers].

= Reciprocal lattice
{title2=$\Lambda^*$}
{parent=Bravais lattice}
{wiki}

The reciprocal lattice is
$$
\Lambda^*=\{q:q\cdot l\in2\pi\mathbb Z\text{ for every }l\in\Lambda\}.
$$
If $\Omega=a_1\cdot(a_2\times a_3)$, its primitive vectors are
$$
b_1=2\pi\frac{a_2\times a_3}{\Omega},
\quad
b_2=2\pi\frac{a_3\times a_1}{\Omega},
\quad
b_3=2\pi\frac{a_1\times a_2}{\Omega},
$$
and satisfy $a_i\cdot b_j=2\pi\delta_{ij}$.

= Wigner-Seitz cell
{c}
{parent=Reciprocal lattice}
{wiki=Wigner%E2%80%93Seitz_cell}

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

= Reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice
{parent=Wigner-Seitz cell}

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt3/2)$, reciprocal primitive vectors are
$$
\left(2\pi,\frac{2\pi}{\sqrt3}\right),
\qquad
\left(0,\frac{4\pi}{\sqrt3}\right).
$$
The reciprocal Wigner--Seitz cell is a regular hexagon of area $8\pi^2/\sqrt3$ and circumradius $4\pi/3$.

= Body-centered cubic lattice
{parent=Bravais lattice}
{c}
{wiki=Body-centered_cubic}

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

= Reciprocal lattice of the 2023 Cambridge body-centered cubic basis
{parent=Body-centered cubic lattice}

For
$$
a_1=\frac a2(1,1,1),
\quad a_2=\frac a2(1,-1,1),
\quad a_3=a(0,0,1),
$$
a reciprocal basis is
$$
b_1=\frac{2\pi}{a}(1,1,0),
\quad b_2=\frac{2\pi}{a}(1,-1,0),
\quad b_3=\frac{2\pi}{a}(-1,0,1).
$$
Thus $\Lambda^*=(2\pi/a)\{(h,k,l)\in\mathbb Z^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt2\pi/a$.

= Crystal lattice structure factor
{parent=Crystal scattering}

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ is
$$
\Delta(Q)=\sum_{l\in S}e^{iQ\cdot l}.
$$
It multiplies the single-atom <scattering amplitude>.

= Reciprocal-lattice peaks of a finite crystal
{parent=Crystal lattice structure factor}

For $S=\{\sum_il_ia_i:-L_i/2\leq l_i\leq L_i/2\}$ and $\alpha_i=Q\cdot a_i$,
$$
\Delta(Q)=\prod_{i=1}^3
\frac{\sin((L_i+1)\alpha_i/2)}{\sin(\alpha_i/2)}.
$$
For large $L_i$, each factor is sharply peaked when $\alpha_i\in2\pi\mathbb Z$, exactly the condition $Q\in\Lambda^*$.

= Elastic Bragg scattering condition
{parent=Crystal scattering}
{wiki=Bragg%27s_law}

For elastic scattering with $k-k'=q\in\Lambda^*$ and $|k|=|k'|=k$,
$$
2k\cdot q=|q|^2,
\qquad
|q|=2k\sin\frac\theta2.
$$
The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

= Bound-state pole of the scattering amplitude
{parent=Scattering amplitude}

After analytic continuation in momentum, a pole at $k=i\kappa$ with $\kappa>0$ gives negative energy and spatial decay, and therefore represents a bound state.

= Supersymmetric quantum mechanics
{parent=Quantum mechanics}
{wiki}

Supersymmetric quantum mechanics factors partner Hamiltonians through first-order operators.

= Partner Hamiltonians
{parent=Supersymmetric quantum mechanics}
{wiki}

Partner Hamiltonians reverse the order of two first-order factors and share corresponding positive-energy states.

= Pöschl-Teller potential
{parent=Supersymmetric quantum mechanics}
{c}
{wiki=P%C3%B6schl%E2%80%93Teller_potential}

A Pöschl-Teller potential is a solvable one-dimensional potential built from $\operatorname{sech}^2 x$. The attractive member
$$
V(x)=-n(n+1)\alpha^2\operatorname{sech}^2(\alpha x)
$$
is reflectionless when $n$ is a positive integer.

= Supersymmetric factorization of the one-soliton potential
{parent=Pöschl-Teller potential}

Let
$$
A=\frac{d}{dx}+\chi\tanh(\chi x),
\qquad
A^\dagger=-\frac{d}{dx}+\chi\tanh(\chi x).
$$
Then
$$
A^\dagger A=-\frac{d^2}{dx^2}+\chi^2-2\chi^2\operatorname{sech}^2(\chi x),
\qquad
AA^\dagger=-\frac{d^2}{dx^2}+\chi^2.
$$
Thus the one-soliton Schrodinger operator is paired with the free Schrodinger operator. Its normalizable state annihilated by $A$ is proportional to $\operatorname{sech}(\chi x)$.

= Reflectionless potential
{parent=Quantum mechanics}
{wiki}

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

= Position operator
{parent=Quantum mechanics}
{wiki}

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

= Momentum operator
{parent=Quantum mechanics}
{wiki}

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

= Canonical commutation relation
{parent=Momentum operator}
{wiki}

In position representation, $x$ acts by multiplication and $p_x=-i\hbar\partial_x$, so
$$
[x,p_x]=i\hbar I.
$$

= Ehrenfest theorem
{parent=Quantum mechanics}
{c}
{wiki}

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

For a possibly time-dependent observable $O$,
$$
\frac d{dt}\langle O\rangle
=\frac1{i\hbar}\langle[O,H]\rangle
+\left\langle\frac{\partial O}{\partial t}\right\rangle.
$$

= Proof of Ehrenfest theorem from the Schrodinger equation
{parent=Ehrenfest theorem}

Differentiating $\langle\psi|O|\psi\rangle$ and using $i\hbar|\dot\psi\rangle=H|\psi\rangle$ and its adjoint gives
$$
\frac d{dt}\langle O\rangle
=\frac i\hbar\langle[H,O]\rangle
+\left\langle\frac{\partial O}{\partial t}\right\rangle.
$$

= Classical equations from Ehrenfest theorem
{parent=Ehrenfest theorem}

For $H=p^2/(2m)+U(x)$, the <canonical commutation relation> gives
$$
m\frac d{dt}\langle x\rangle=\langle p\rangle,
\qquad
\frac d{dt}\langle p\rangle=-\langle U'(x)\rangle,
\qquad
\frac d{dt}\langle H\rangle=0.
$$
These are the expectation-value analogues of <Newton's second law> and conservation of energy.

= Correspondence principle
{parent=Ehrenfest theorem}
{wiki}

The correspondence principle states that quantum predictions approach classical mechanics in the appropriate large-scale or semiclassical regime.

= Angular momentum operator
{parent=Quantum mechanics}
{wiki}

= Addition of angular momentum
{parent=Angular momentum operator}
{wiki}

For two subsystems, the total operator is $\mathbf J=\mathbf J_1\otimes I+I\otimes\mathbf J_2$, and $J_\pm=J_{1\pm}+J_{2\pm}$. The coupled basis diagonalizes $\mathbf J^2$ and $J_z$.

= Clebsch-Gordan decomposition
{c}
{parent=Addition of angular momentum}
{wiki=Clebsch%E2%80%93Gordan_coefficients}

The tensor product of spin-$j_1$ and spin-$j_2$ irreducible representations decomposes into one copy of each spin
$$
j=|j_1-j_2|,|j_1-j_2|+1,\ldots,j_1+j_2.
$$
The change-of-basis entries between uncoupled and coupled states are Clebsch--Gordan coefficients.

= Highest-weight states in angular momentum addition
{parent=Addition of angular momentum}

With $J=j_1+j_2$, the maximal state is the product $|J,J\rangle=|j_1,j_1\rangle|j_2,j_2\rangle$. Lowering it once gives
$$
|J,J-1\rangle=
\sqrt{\frac{j_1}{J}}|j_1,j_1-1\rangle|j_2,j_2\rangle
+\sqrt{\frac{j_2}{J}}|j_1,j_1\rangle|j_2,j_2-1\rangle.
$$
The orthogonal combination is the highest-weight state $|J-1,J-1\rangle$.

= Angular momentum singlet state
{parent=Addition of angular momentum}
{wiki=Singlet_state}

Two equal spins $j$ have the unique total-spin-zero state
$$
|0,0\rangle=\frac1{\sqrt{2j+1}}
\sum_{m=-j}^j(-1)^{j-m}|j,m\rangle|j,-m\rangle.
$$
It is invariant under joint rotations.

= Spin-one-half singlet state
{title2=$|0,0\rangle$}
{parent=Angular momentum singlet state}

For two spin-one-half particles, the singlet is
$$
|0,0\rangle=\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2}.
$$
Every component of the total spin annihilates it, and $\mathbf S^2$ has eigenvalue zero.

= Spin-one-half triplet state
{parent=Addition of angular momentum}

The symmetric two-particle states
$$
|\uparrow\uparrow\rangle,
\qquad
\frac{|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle}{\sqrt2},
\qquad
|\downarrow\downarrow\rangle
$$
form the spin-one triplet. On this subspace, $\mathbf S^2$ has eigenvalue $1(1+1)\hbar^2=2\hbar^2$.

= Spin angular momentum
{parent=Angular momentum operator}
{wiki=Spin_(physics)}

Spin operators satisfy $[S_i,S_j]=i\hbar\varepsilon_{ijk}S_k$ and are Hermitian traceless generators of rotations in finite-dimensional irreducible representations.

= Spin one-half along an axis
{title2=$\mathbf n\mathbin{\cdot}\mathbf S$}
{parent=Spin angular momentum}

For $\mathbf n=(\sin\theta,0,\cos\theta)$, the normalized eigenstates of $\mathbf n\cdot\mathbf S$ are
$$
|\uparrow_\theta\rangle
=\cos\frac\theta2|\uparrow\rangle
+\sin\frac\theta2|\downarrow\rangle,
\qquad
|\downarrow_\theta\rangle
=-\sin\frac\theta2|\uparrow\rangle
+\cos\frac\theta2|\downarrow\rangle,
$$
with eigenvalues $+\hbar/2$ and $-\hbar/2$.

= Irreducible spin representation
{parent=Spin angular momentum}

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m\rangle$ for $m=-s,-s+1,\ldots,s$ and Casimir eigenvalue $\hbar^2s(s+1)$.

= Spin ladder operator
{parent=Spin angular momentum}
{wiki=Angular_momentum_operator\#Ladder_operators}

The operators $S_\pm=S_x\pm iS_y$ satisfy
$$
S_\pm|s,m\rangle
=\hbar\sqrt{s(s+1)-m(m\pm1)}|s,m\pm1\rangle.
$$

= Spin-three-halves matrices
{parent=Spin angular momentum}

In descending $S_z$ order, the spin-$3/2$ raising operator has superdiagonal entries $\hbar\sqrt3,2\hbar,\hbar\sqrt3$; $S_x$ and $S_y$ are its Hermitian real and imaginary combinations.

= Spin coherent state
{parent=Spin angular momentum}
{wiki=Spin_coherent_state}

A spin coherent state is obtained by rotating the maximal-weight state $|s,s\rangle$ so that it is an eigenstate of spin along a chosen direction with eigenvalue $s\hbar$.

= Equatorial spin-three-halves coherent state
{parent=Spin coherent state}

At azimuth $\varphi$, the normalized spin-$3/2$ coherent-state components are proportional to $1,\sqrt3e^{i\varphi},\sqrt3e^{2i\varphi},e^{3i\varphi}$.

= Larmor precession of a spin coherent state
{parent=Spin coherent state}
{c}
{wiki=Larmor_precession}

Under $H=-\gamma BS_z$, a coherent spin direction precesses around the $z$-axis with angular velocity $-\gamma B$.

= Time evolution under a spin-Z Hamiltonian
{parent=Larmor precession of a spin coherent state}

The propagator $e^{i\gamma BtS_z/\hbar}$ multiplies each magnetic component $|s,m\rangle$ by $e^{im\gamma Bt}$.

= Infinite square well
{parent=Quantum mechanics}
{wiki}

= Particle in a rectangular box
{parent=Infinite square well}
{wiki=Particle_in_a_box}

Dirichlet separation in a rectangular box gives products of sine waves and energy proportional to $n_x^2/a^2+n_y^2/b^2+n_z^2/c^2$.

= Quantum degeneracy
{parent=Quantum mechanics}
{wiki=Degenerate_energy_levels}

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

= Orbital angular momentum
{parent=Quantum mechanics}
{wiki=Angular_momentum_operator}

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

= Rotation commutators for orbital angular momentum
{parent=Orbital angular momentum}

For $L_i=\varepsilon_{ik\ell}X_kP_\ell$ and $[X_i,P_j]=i\hbar\delta_{ij}$,
$$
[L_i,X_j]=i\hbar\varepsilon_{ijk}X_k,
\qquad
[L_i,P_j]=i\hbar\varepsilon_{ijk}P_k.
$$

= Orbital angular momentum commutation relations
{parent=Orbital angular momentum}

The canonical position-momentum commutators imply
$$
[L_i,L_j]=i\hbar\varepsilon_{ijk}L_k,
\qquad
[L^2,L_i]=0.
$$

= Angular momentum ladder operator
{title2=$L_\pm$}
{parent=Orbital angular momentum}
{wiki=Angular_momentum_operator#Ladder_operators}

The operators $L_\pm=L_x\pm iL_y$ satisfy
$$
[L_z,L_\pm]=\pm\hbar L_\pm,
\qquad
[L^2,L_\pm]=0.
$$
They therefore change the magnetic quantum number $m$ by $\pm1$ while preserving the total angular-momentum quantum number $\ell$.

= Squared orbital angular momentum in position and momentum operators
{parent=Orbital angular momentum}

Operator ordering gives
$$
L^2=X^2P^2-(X\cdot P)^2+i\hbar X\cdot P.
$$

= Spherical Laplacian from orbital angular momentum
{parent=Squared orbital angular momentum in position and momentum operators}

In position representation,
$$
\boxed{L^2=-\hbar^2\nabla_{S^2}^2}.
$$
The radial derivatives cancel after substituting $P=-i\hbar\nabla$ into the Cartesian identity for $L^2$.

= Angular momentum ladder variable
{parent=Orbital angular momentum}

The coordinate combinations $x_1\pm ix_2$ have $L_3$ eigenvalues $\pm\hbar$, so their powers carry magnetic quantum number $\pm m$.

= Angular momentum of a complex-coordinate Gaussian state
{parent=Angular momentum ladder variable}

If $\psi=z e^{-r^2/(2a^2)}$ in two dimensions, then $L_z\psi=\hbar\psi$ and $L_z\psi^*=-\hbar\psi^*$. Orthogonality therefore makes the angular-momentum expectation of $\alpha\psi+\beta\psi^*$ equal to $(|\alpha|^2-|\beta|^2)\hbar$ for normalized coefficients.

= Rotational invariance of a central-potential Hamiltonian
{parent=Orbital angular momentum}

For
$$
H=\frac{P^2}{2m}+U(|X|),
$$
both $P^2$ and $|X|^2$ are rotational scalars, so $[H,L_i]=0$. Consequently $H,L^2,L_3$ commute pairwise and can be simultaneously diagonalized.

= Quantum harmonic oscillator
{parent=Quantum mechanics}
{wiki}

In units with $\hbar=1$, a harmonic oscillator of angular frequency $\omega$ has
$$
H=\omega\left(N+\frac12\right),
\qquad N=A^\dagger A.
$$

= Creation and annihilation operators
{parent=Quantum harmonic oscillator}
{wiki=Creation_and_annihilation_operators}

The bosonic ladder operators satisfy
$$
[A,A^\dagger]=1,
\qquad
A|n\rangle=\sqrt n\,|n-1\rangle,
\qquad
A^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle.
$$

= Number operator
{parent=Creation and annihilation operators}
{wiki}

The number operator $N=A^\dagger A$ satisfies
$$
[N,A]=-A,
\qquad
[N,A^\dagger]=A^\dagger.
$$
It is self-adjoint and positive semidefinite.

= Integer spectrum of the number operator
{parent=Number operator}

If $N|\nu\rangle=\nu|\nu\rangle$, then $A$ lowers the eigenvalue by one and $\|A|\nu\rangle\|^2=\nu\|\nu\rangle\|^2$. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.

= Two commensurate quantum harmonic oscillators
{parent=Quantum harmonic oscillator}

For independent frequencies $1$ and $2$,
$$
H_0=N_A+2N_B+\frac32,
$$
so the eigenspace at energy $k+3/2$ has one basis state $|n,m\rangle$ for each nonnegative solution of $n+2m=k$.

= Three-dimensional isotropic harmonic oscillator
{parent=Quantum mechanics}
{wiki=Quantum_harmonic_oscillator\#N-dimensional_isotropic_harmonic_oscillator}

Its energies are $E_N=\hbar\omega(N+3/2)$, and level $N$ has degeneracy $\binom{N+2}{2}$.

= Two-dimensional isotropic harmonic oscillator
{parent=Quantum mechanics}
{wiki=Quantum_harmonic_oscillator\#N-dimensional_isotropic_harmonic_oscillator}

For unit mass and angular frequency,
$$
H=a_x^\dagger a_x+a_y^\dagger a_y+1.
$$
The level with total occupation $N=n_x+n_y$ has energy $N+1$ and degeneracy $N+1$ in units where $\hbar=1$.

= Oscillator bilinear commutator
{parent=Two-dimensional isotropic harmonic oscillator}

For bosonic modes satisfying $[a_i,a_j^\dagger]=\delta_{ij}$, the number-preserving bilinears $T_{ij}=a_i^\dagger a_j$ obey
$$
[T_{ij},T_{kl}]=\delta_{jk}T_{il}-\delta_{il}T_{kj}.
$$

= Schwinger boson representation
{parent=Two-dimensional isotropic harmonic oscillator}
{c}
{wiki=Schwinger_boson_representation}

Contracting two-mode oscillator bilinears with the Pauli matrices,
$$
T^a=\frac12\sigma^a_{ij}a_i^\dagger a_j,
$$
gives $[T^a,T^b]=i\varepsilon_{abc}T^c$. The subspace with total occupation $N$ is the spin-$N/2$ irreducible representation and has dimension $N+1$.

= Fermi oscillator
{parent=Quantum mechanics}
{c}
{wiki=Fermionic_oscillator}

A Fermi oscillator has nilpotent lowering operator $B$, anticommutator $BB^\dagger+B^\dagger B=1$, and occupation Hamiltonian $B^\dagger B$ with levels zero and one.

= Projection Hamiltonian
{parent=Fermi oscillator}

The fermionic occupation Hamiltonian satisfies $H^2=H$, so its spectrum lies in $\{0,1\}$.

= Fermionic raising and lowering operator
{parent=Fermi oscillator}

The lowering operator sends the occupied state to the vacuum and kills the vacuum; its adjoint performs the reverse and kills the occupied state.

= Tensor product of quantum systems
{parent=Quantum mechanics}
{wiki=Tensor_product\#Quantum_mechanics}

Independent quantum systems combine by tensor product, with local operators acting as $A\otimes I$ or $I\otimes B$.

= Multiparticle quantum state
{parent=Tensor product of quantum systems}

For distinguishable particles with one-particle Hilbert spaces $\mathcal H_i$, the multiparticle state space is $\bigotimes_i\mathcal H_i$. A product state is a tensor product of one-particle states, while a general state is a linear combination of product states.

= Particle exchange operator
{title2=$P_{12}$}
{parent=Multiparticle quantum state}

For two particles, the exchange operator sends $|\alpha\rangle\otimes|\beta\rangle$ to $|\beta\rangle\otimes|\alpha\rangle$. It has eigenvalues $+1$ on symmetric states and $-1$ on antisymmetric states.

= Identical particle
{parent=Particle exchange operator}
{wiki=Identical_particles}

Physical states of identical particles have definite symmetry under every <particle exchange operator>: symmetric for bosons and antisymmetric for fermions.

= Boson
{parent=Identical particle}
{wiki}

Bosons are identical particles whose total multiparticle state is symmetric under exchange. In three-dimensional relativistic quantum theory they have integer spin.

= Fermion
{parent=Identical particle}
{wiki}

Fermions are identical particles whose total multiparticle state is antisymmetric under exchange. In three-dimensional relativistic quantum theory they have half-integer spin.

= Pauli exclusion principle
{c}
{parent=Fermion}
{wiki}

No two identical fermions may occupy the same one-particle quantum state. At zero temperature, noninteracting fermions fill the available one-particle states in increasing order of energy.

= Two identical spin-one bosons
{parent=Identical particle}

For two spin-one particles, the spin tensor product has a six-dimensional symmetric subspace and a three-dimensional antisymmetric subspace. A symmetric spatial state combines with symmetric spin, while an antisymmetric spatial state combines with antisymmetric spin, so that the total bosonic state remains symmetric.

= Degeneracy of two identical spin-one bosons with equally spaced orbital levels
{parent=Two identical spin-one bosons}

If $E_n=E_0+n\Delta$, the two-particle level $2E_0+N\Delta$ has degeneracy
$$
g_N=\begin{cases}
\dfrac92N+6,&N\text{ even},\\[3pt]
\dfrac92(N+1),&N\text{ odd}.
\end{cases}
$$
Each unequal orbital pair contributes six symmetric-spin states and three antisymmetric-spin states; an equal orbital pair exists only for even $N$ and contributes six states.

= Decoupled Fermi oscillators
{parent=Tensor product of quantum systems}

Two decoupled Fermi oscillators have product occupation states and additive energies $E_1n_1+E_2n_2$.

= Kronecker product representation of local operators
{parent=Tensor product of quantum systems}
{c}

In a product basis, an operator local to one subsystem is represented by its matrix Kronecker the identity on every other subsystem.

= Decoupled commuting Hamiltonian terms
{parent=Tensor product of quantum systems}

Commuting Hamiltonian terms on separate factors are simultaneously diagonalized by tensor products of their eigenvectors and have additive eigenvalues.

= Time-dependent perturbation theory
{parent=Quantum mechanics}
{wiki=Time-dependent_perturbation_theory}

Expanding a state in unperturbed energy eigenstates with their free phases converts the time-dependent Schrodinger equation into coupled equations for slowly varying amplitudes. To first order, the perturbation on the right-hand side acts on the initial unperturbed state.

= Continuum transition probability at long times
{parent=Time-dependent perturbation theory}

For a harmonic perturbation and continuum detuning $\Omega(k)$, first-order amplitudes contain
$$
\frac{\sin(\Omega(k)t/2)}{\Omega(k)/2}.
$$
The distributional limit
$$
\frac{\sin^2(\Omega t/2)}{(\Omega/2)^2}
\longrightarrow 2\pi t\,\delta(\Omega)
$$
selects energy-conserving continuum states and produces a transition probability linear in time.

= Time-independent perturbation theory
{parent=Quantum mechanics}
{wiki=Perturbation_theory_(quantum_mechanics)}

Time-independent perturbation theory expands the eigenvalues and eigenstates of $H_0+V$ in powers of a small perturbation $V$.

= First-order energy correction
{title2=$E_n^{(1)}$}
{parent=Time-independent perturbation theory}

The first-order energy correction is the coefficient of the perturbation parameter in the expansion of an energy eigenvalue.

= Nondegenerate energy eigenvalue
{parent=Time-independent perturbation theory}

An <energy eigenvalue> is nondegenerate when its <eigenspace> is one-dimensional. Its normalized <eigenstate> is then unique up to a complex phase.

= First-order nondegenerate perturbation theory
{parent=Time-independent perturbation theory}

For a normalized eigenstate $|n\rangle$ belonging to a <nondegenerate energy eigenvalue>,
$$
E_n^{(1)}=\langle n|V|n\rangle,
\qquad
|n^{(1)}\rangle
=\sum_{m\ne n}|m\rangle
\frac{\langle m|V|n\rangle}{E_n^{(0)}-E_m^{(0)}}.
$$

= Second-order nondegenerate perturbation theory
{parent=Time-independent perturbation theory}

For a normalized nondegenerate eigenstate $|n\rangle$,
$$
E_n^{(2)}
=\sum_{m\ne n}
\frac{|\langle m|V|n\rangle|^2}
{E_n^{(0)}-E_m^{(0)}}.
$$
For a ground state, every denominator is negative, so the correction is nonpositive.

= Imaginary-coupled two-level Hamiltonian
{parent=Time-independent perturbation theory}

For
$$
H=\begin{pmatrix}E_-&-i\lambda\\i\lambda&E_+\end{pmatrix},
\qquad \Delta=E_+-E_->0,
$$
the exact energies are
$$
E_{\pm}^{\rm exact}
=\frac{E_-+E_+}{2}
\pm\sqrt{\frac{\Delta^2}{4}+\lambda^2}.
$$
The perturbation series around $\lambda=0$ has radius $\Delta/2$, set by the branch points $\lambda=\pm i\Delta/2$.

= Degenerate perturbation theory
{parent=Quantum mechanics}
{wiki}

First-order degenerate perturbation theory diagonalizes the perturbing operator within each degenerate eigenspace of the unperturbed Hamiltonian.

= Perturbation diagonal within a degenerate eigenspace
{parent=Degenerate perturbation theory}

If the perturbation is already diagonal in a basis of a degenerate eigenspace, those basis vectors receive its diagonal entries as first-order shifts.

= Exact diagonalization within a degenerate subspace
{parent=Degenerate perturbation theory}

When a perturbation preserves each unperturbed degenerate block and has no inter-block coupling, diagonalizing each block can give the exact spectrum.

= Resonant two-to-one oscillator coupling
{parent=Exact diagonalization within a degenerate subspace}

For oscillators of frequencies $1$ and $2$, the perturbation
$$
H'=A^{\dagger2}B+A^2B^\dagger
$$
preserves $N_A+2N_B$. On the energy-$9/2$ basis $|3,0\rangle,|1,1\rangle$, it is the matrix
$$
\begin{pmatrix}0&\sqrt6\\\sqrt6&0\end{pmatrix},
$$
so the symmetric and antisymmetric combinations are exact eigenstates with shifts $\pm\sqrt6\lambda$.

= Pauli X eigenstate
{parent=Quantum mechanics}
{c}
{wiki=Pauli_matrices}

The Pauli $X$ eigenstates are $(|0\rangle\pm|1\rangle)/\sqrt2$ with eigenvalues $\pm1$.