Quantum mechanics

Quantum mechanics models states by wavefunctions and observables by operators.

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

Spatial reflection reverses every spatial coordinate.

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

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.

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

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

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

An excited state is an energy eigenstate whose energy exceeds the ground state energy.

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

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

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

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

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

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

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

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

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

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

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

If $U(s)=e^{-isQ/\hslash }$ and $U(s{)}^{†}HU(s)=H$, differentiation at $s=0$ givesIn the Heisenberg picture,

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

In the Heisenberg picture, states are fixed and an observable evolves asFor a time-independent Schrödinger-picture observable,

A time-reversal operator $T$ intertwines forward and backward evolution: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.

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

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

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

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

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

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

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

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

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

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

Because state vectors differing by a quantum phase represent the same ray, a unitary realization of a group action need only satisfy

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.

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

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

For a nonrelativistic particle of mass $m$, the probability current is

For a wavefunction satisfying the Schrodinger equation with a real potential,This continuity equation expresses local conservation of probability.

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.

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.

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

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

For a square potential barrier of width $2a$ and height $U_0=2E$, putThe oscillatory wavenumber outside and decay constant inside are both $k$. Matching the wavefunction and its derivative at both faces gives transmission amplitudeand transmission probability ${\mathrm{sech}}^2(2ka)$.

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 $2(1\pm r)$.

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

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

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

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

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

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

For $V(x)=V_0\delta (x)$, $E={\hslash }^2{k}^2/(2m)$, and $\gamma =m{V}_0/({\hslash }^{2}k)$, continuity and the derivative jump giveThey obey $|r{|}^2+|t{|}^2=1$ and $r^{\ast }t+t^{\ast }r=0$.

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

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

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.

For a reflection-invariant localized potential, order incoming amplitudes from the left and right and outgoing amplitudes toward the left and right. Thenwhere $r$ and $t$ are the reflection and transmission amplitudes. Probability-current conservation makes $S$ unitary for a real potential.

If the left and right travelling-wave coefficients are related bythen

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

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

The $s$-wave scattering length isand the low-energy total cross-section is $4\pi a_s^2$.

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

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 $\hslash$.

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

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

The momentum transfer in a scattering process is the difference between the incoming and outgoing wavevectors, commonly $Q=k-k^{\prime }$ up to a sign convention and a factor of $\hslash$.

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

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

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.

A three-dimensional Bravais lattice iswhere the primitive vectors $a_1,a_2,a_3$ are linearly independent.

A lattice point is a vector $l=n_1{a}_1+n_2{a}_2+n_3{a}_3$ whose coefficients in a chosen primitive basis are integers.

The reciprocal lattice isIf $\Omega =a_1\cdot (a_2\times a_3)$, its primitive vectors areand satisfy $a_i\cdot b_j=2\pi {\delta }_{ij}$.

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.

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt{3}/2)$, reciprocal primitive vectors areThe reciprocal Wigner--Seitz cell is a regular hexagon of area $8{\pi }^2/\sqrt{3}$ and circumradius $4\pi /3$.

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.

Fora reciprocal basis isThus ${\Lambda }^{\ast }=(2\pi /a)\{(h,k,l)\in {\mathbb{Z}}^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt{2}\pi /a$.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

For $S=\{{\sum }_i{l}_i{a}_i:-L_i/2\le l_i\le L_i/2\}$ and ${\alpha }_i=Q\cdot a_i$,For large $L_i$, each factor is sharply peaked when ${\alpha }_i\in 2\pi \mathbb{Z}$, exactly the condition $Q\in {\Lambda }^{\ast }$.

For elastic scattering with $k-k^{\prime }=q\in {\Lambda }^{\ast }$ and $|k|=|k^{\prime }|=k$,The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

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 factors partner Hamiltonians through first-order operators.

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

A Pöschl-Teller potential is a solvable one-dimensional potential built from ${\mathrm{sech}}^{2}x$. The attractive memberis reflectionless when $n$ is a positive integer.

LetThenThus the one-soliton Schrodinger operator is paired with the free Schrodinger operator. Its normalizable state annihilated by $A$ is proportional to $\mathrm{sech}(\chi x)$.

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

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

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

In position representation, $x$ acts by multiplication and $p_x=-i\hslash {\partial }_x$, so

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

For a possibly time-dependent observable $O$,

Differentiating $⟨\psi |O|\psi ⟩$ and using $i\hslash |\dot{\psi }⟩=H|\psi ⟩$ and its adjoint gives

For $H=p^2/(2m)+U(x)$, the canonical commutation relation givesThese are the expectation-value analogues of Newton's second law and conservation of energy.

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

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 $J_z$.

The tensor product of spin-$j_1$ and spin-$j_2$ irreducible representations decomposes into one copy of each spinThe change-of-basis entries between uncoupled and coupled states are Clebsch--Gordan coefficients.

With $J=j_1+j_2$, the maximal state is the product $|J,J⟩=|j_1,j_1⟩|j_2,j_2⟩$. Lowering it once givesThe orthogonal combination is the highest-weight state $|J-1,J-1⟩$.

Two equal spins $j$ have the unique total-spin-zero stateIt is invariant under joint rotations.

For two spin-one-half particles, the singlet isEvery component of the total spin annihilates it, and ${\mathbf{S}}^2$ has eigenvalue zero.

The symmetric two-particle statesform the spin-one triplet. On this subspace, ${\mathbf{S}}^2$ has eigenvalue $1(1+1){\hslash }^2=2{\hslash }^2$.

Spin operators satisfy $[S_i,S_j]=i\hslash {\epsilon }_{ijk}{S}_k$ and are Hermitian traceless generators of rotations in finite-dimensional irreducible representations.

For $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, the normalized eigenstates of $\mathbf{n}\cdot \mathbf{S}$ arewith eigenvalues $+\hslash /2$ and $-\hslash /2$.

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m⟩$ for $m=-s,-s+1,\dots ,s$ and Casimir eigenvalue ${\hslash }^{2}s(s+1)$.

The operators $S_{\pm }=S_x\pm i{S}_y$ satisfy

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

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

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

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

The propagator $e^{i\gamma Bt{S}_z/\hslash }$ multiplies each magnetic component $|s,m⟩$ by $e^{im\gamma Bt}$.

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

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

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

For $L_i={\epsilon }_{ik\ell }{X}_k{P}_{\ell }$ and $[X_i,P_j]=i\hslash {\delta }_{ij}$,

The canonical position-momentum commutators imply

The operators $L_{\pm }=L_x\pm i{L}_y$ satisfyThey therefore change the magnetic quantum number $m$ by $\pm 1$ while preserving the total angular-momentum quantum number $\ell$.

Operator ordering gives

In position representation,The radial derivatives cancel after substituting $P=-i\hslash \nabla$ into the Cartesian identity for $L^2$.

The coordinate combinations $x_1\pm i{x}_2$ have $L_3$ eigenvalues $\pm \hslash$, so their powers carry magnetic quantum number $\pm m$.

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

Forboth $P^2$ and $|X{|}^2$ are rotational scalars, so $[H,L_i]=0$. Consequently $H,L^2,L_3$ commute pairwise and can be simultaneously diagonalized.

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

The bosonic ladder operators satisfy

The number operator $N=A^{†}A$ satisfiesIt is self-adjoint and positive semidefinite.

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

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

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

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 $\hslash =1$.

For bosonic modes satisfying $[a_i,a_j^{†}]={\delta }_{ij}$, the number-preserving bilinears $T_{ij}=a_i^{†}{a}_j$ obey

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

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

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

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

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

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

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

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

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

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

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.

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.

If $E_n=E_0+n\Delta$, the two-particle level $2E_0+N\Delta$ has degeneracyEach 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.

Two decoupled Fermi oscillators have product occupation states and additive energies $E_1{n}_1+E_2{n}_2$.

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

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

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.

For a harmonic perturbation and continuum detuning $\Omega (k)$, first-order amplitudes containThe distributional limitselects energy-conserving continuum states and produces a transition probability linear in time.

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

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

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

For a normalized eigenstate $|n⟩$ belonging to a nondegenerate energy eigenvalue,

For a normalized nondegenerate eigenstate $|n⟩$,For a ground state, every denominator is negative, so the correction is nonpositive.

Forthe exact energies areThe perturbation series around $\lambda =0$ has radius $\Delta /2$, set by the branch points $\lambda =\pm i\Delta /2$.

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

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

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

For oscillators of frequencies $1$ and $2$, the perturbationpreserves $N_A+2N_B$. On the energy-$9/2$ basis $|3,0⟩,|1,1⟩$, it is the matrixso the symmetric and antisymmetric combinations are exact eigenstates with shifts $\pm \sqrt{6}\lambda$.