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The many-body problem asks how a system of many interacting constituents behaves, given their dynamics and interactions. A simple Hamiltonian operator for $N$ nonrelativistic quantum particles contains one-particle terms and pair interactions,Even a compact expression can be difficult to solve: interactions correlate the constituents, and a tensor product of $N$ local state spaces of dimension $d$ has dimension $d^N$ before constraints or symmetries are used.

Mean-field theory replaces part of the correlated environment by an effective average, while Hartree-Fock theory uses a restricted fermionic state description. Their usefulness comes with a question: which correlations have been neglected, and do those correlations control the phenomenon of interest? Electron correlation and collective excitations show why knowing each constituent in isolation need not explain the system.

A particle is an entity described as a localized constituent in an appropriate physical model. In classical mechanics, a point particle has a position and trajectory while its internal size is neglected. This approximation can describe an extended object when the omitted structure does not affect the question being studied.

In quantum mechanics, a quantum particle has a state described by a wave function or a more general state description, with probabilistic measurement outcomes. Quantum field theory describes particle states as excitations of fields. A proton can be treated as a particle in one experiment while its quark and gluon structure matters in another. The particle description therefore specifies what is being resolved, rather than asserting that every particle is a structureless classical point.

A potential barrier is a region of higher potential energy separating accessible regions of a physical system. A classical particle cannot pass through a barrier higher than its total energy along its trajectory, but a quantum particle can have nonzero transmission probability through a finite barrier by quantum tunnelling.

In a one-dimensional region with constant $V_0>E$, the Schrödinger equation gives exponential solutions with tunnelling decay constant $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The wavefunction can therefore penetrate a finite barrier and match an outgoing wave on the other side. The rectangular potential barrier is an exactly solvable model; smooth barriers lead naturally to a semiclassical approximation. Even for $E>V_0$, quantum reflection can occur where the potential changes.

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 quantum particle is a quantum system whose possible measurement outcomes are described by a quantum state rather than a definite classical trajectory. For a single spinless particle, a wavefunction gives a probability amplitude for each position; the Born rule turns its squared modulus into a probability density. A localized detection does not imply that the particle followed a known path before detection.

An electron sent through an interferometer is a useful example: individual arrivals are localized, while many arrivals reveal matter-wave interference. Changing the relative quantum phase of the available paths changes the detection distribution. For several identical quantum particles, bosonic or fermionic exchange symmetry further restricts the joint quantum state.

Quantum transmission is the passage of an incident quantum particle into an outgoing region through a specified interaction or potential. A wave function can have both reflected and transmitted components; the measurable transmission probability compares outgoing and incoming probability currents, rather than comparing wave amplitudes alone.

For a one-dimensional, constant-mass Schrödinger equation, write an incident amplitude of one and transmitted amplitude $t$, with positive asymptotic wavenumbers $k_L$ and $k_R$. The probability current givesOnly when the asymptotic wavenumbers agree does this become $|t{|}^2$. A real, static potential with no additional loss channels has reflection plus transmission probability equal to one. Quantum tunnelling is an important case in which the transmitted component crosses a region forbidden by the corresponding classical energy condition.

Spin is intrinsic angular momentum carried by a quantum particle or another effective quantum degree of freedom. It does not require a literal rotating surface. Spin components generate rotations and obey $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$; a spin-$s$ multiplet has $2s+1$ projections along a chosen quantization axis.

For spin one-half, the operators are $S_i=\hslash {\sigma }_i/2$, with outcomes $\pm \hslash /2$ for a component measurement. A Stern-Gerlach experiment resolves discrete magnetic responses to a field gradient, while magnetic-resonance experiments control and observe Larmor precession. Spin differs from orbital angular momentum, which arises from the spatial dependence of a wavefunction. Combining both is necessary for atomic fine structure and coupling the electronic result to nuclear spin gives hyperfine structure.