1 · quantum states and operator mechanics

states and the superposition principle

A pure state of a quantum system is represented by a non-zero vector $\psi$ in a complex separable Hilbert space $\mathcal{H}$. Two vectors that differ by a non-zero complex scalar describe the same physical state. Equivalently, a pure state is a ray $$ [\psi]\in\mathbb{P}(\mathcal{H})=\bigl(\mathcal{H}\setminus\{0\}\bigr)/\mathbb{C}^\times. $$ It is conventional to work with normalized vectors $$ \langle\psi|\psi\rangle=1, $$ so that overall phases of modulus one remain free: $\psi$ and $e^{i\theta}\psi$ determine identical Born-rule probabilities for every observable.

definition (pure state). an equivalence class $[\psi]$ of non-zero vectors in $\mathcal{H}$ under multiplication by $\mathbb{C}^\times$. a normalized representative satisfies $\|\psi\|=1$, and the residual ambiguity is a $U(1)$ phase.

The superposition principle asserts that if $\psi_1$ and $\psi_2$ are admissible state vectors, then any linear combination $$ c_1\psi_1+c_2\psi_2,\qquad c_1,c_2\in\mathbb{C}, $$ is also admissible (subject only to domain and topological restrictions of the particular system). This linear structure has no classical counterpart: classical probability distributions on phase space add as measures and do not interfere.

example (two-path interference). let $\psi_1$ and $\psi_2$ be normalized amplitudes for passage through two spatially separated slits, with $\langle\psi_1|\psi_2\rangle=0$. the normalized superposition $\psi=(\psi_1+\psi_2)/\sqrt{2}$ yields detection probabilities involving the cross term $2\operatorname{Re}\langle\psi_1|\psi_2\rangle_{\mathrm{screen}}$ on a distant screen. replacing the coherent sum by the mixture $\rho=\tfrac12(|\psi_1\rangle\langle\psi_1|+|\psi_2\rangle\langle\psi_2|)$ erases the interference fringes while preserving the marginal slit probabilities.

Mixed states are described by density operators: positive trace-class operators $\rho$ of unit trace, $$ \rho\geq 0,\qquad\operatorname{Tr}\rho=1. $$ A pure state corresponds to a rank-one projector $\rho=|\psi\rangle\langle\psi|$. The expectation value of an observable $A$ in the state $\rho$ is $$ \langle A\rangle=\operatorname{Tr}(\rho A), $$ whenever $\rho A$ is trace-class (or, more carefully, whenever the spectral measure of $A$ pairs with $\rho$ in the usual way).

remark (purity). the quantity $\operatorname{Tr}(\rho^2)$ equals $1$ if and only if $\rho$ is a pure projector, and is strictly smaller for nontrivial mixtures. thus purity is an affine-invariant diagnostic of whether interference between coherent alternatives remains available.

observables as self-adjoint operators

An observable is represented by a self-adjoint operator $A=A^\dagger$ acting on $\mathcal{H}$ (more precisely, on a dense domain $D(A)\subset\mathcal{H}$). Self-adjointness guarantees that the spectrum $\sigma(A)$ is real and that the spectral theorem applies: $$ A=\int_{\sigma(A)}\lambda\,dE_A(\lambda), $$ where $E_A$ is the unique projection-valued measure associated with $A$.

definition (self-adjoint observable). a densely defined operator $A$ equal to its adjoint $A^\dagger$. symmetry ($A\subset A^\dagger$) is not enough: essential self-adjointness or an explicit choice of self-adjoint extension is required before the spectral theorem yields a unique physical observable.

The possible outcomes of a measurement of $A$ are the points of $\sigma(A)$. If the spectrum is discrete and non-degenerate, with orthonormal eigenvectors $\{\psi_n\}$ and eigenvalues $\{a_n\}$, the probability that an ideal projective measurement on a normalized pure state $\psi$ yields $a_n$ is $$ P(a_n)=|\langle\psi_n|\psi\rangle|^2. $$ Immediately after the measurement the state collapses to $\psi_n$. For continuous spectra the corresponding statements are formulated with spectral projections or, more carefully, with the rigged Hilbert-space apparatus of Chapter 3.

example (spin component). on $\mathbb{C}^2$ let $S_z=\tfrac{\hbar}{2}\sigma_z=\tfrac{\hbar}{2}\operatorname{diag}(1,-1)$. for a normalized spinor $\psi=(\alpha,\beta)$ one has $P(+\hbar/2)=|\alpha|^2$ and $P(-\hbar/2)=|\beta|^2$. after a projective measurement yielding $+\hbar/2$, the post-measurement state is $(1,0)$ up to phase.

The uncertainty of an observable in a state $\psi\in D(A)$ is the root-mean-square deviation $$ \Delta A=\sqrt{\langle(A-\langle A\rangle)^2\rangle}, $$ with $\langle A\rangle=\langle\psi|A|\psi\rangle$. For any two self-adjoint operators with $\psi$ in the appropriate domains, the Robertson relation $$ \Delta A\,\Delta B\geq\tfrac12\bigl|\langle[A,B]\rangle\bigr| $$ holds. When $A=Q$ and $B=P$ are the position and momentum operators on $L^2(\mathbb{R})$ (with their usual domains), $[Q,P]=i\hbar I$ on a dense core and one recovers the Heisenberg uncertainty principle $$ \Delta Q\,\Delta P\geq\tfrac{\hbar}{2}. $$

example (minimum-uncertainty packet). Gaussian wave packets saturate $\Delta Q\,\Delta P=\hbar/2$. for $\psi(x)=(2\pi\sigma^2)^{-1/4}\exp\bigl(-(x-x_0)^2/(4\sigma^2)+ip_0 x/\hbar\bigr)$ one finds $\Delta Q=\sigma$ and $\Delta P=\hbar/(2\sigma)$, with means $x_0$ and $p_0$ respectively.

time evolution

The dynamics of a closed system are generated by a self-adjoint Hamiltonian $H$. In the Schrodinger picture the state vector evolves according to $$ i\hbar\frac{d}{dt}\psi(t)=H\psi(t) $$ whenever $\psi(t)$ remains in $D(H)$. When $H$ is time-independent, Stone's theorem supplies a strongly continuous unitary group $U(t)=e^{-itH/\hbar}$ with $$ \psi(t)=U(t)\psi(0). $$

In the Heisenberg picture the state is held fixed and observables evolve by $$ A(t)=U(t)^\dagger A U(t), $$ so that, on a suitable domain, $$ \frac{dA}{dt}=\frac{i}{\hbar}[H,A]+\frac{\partial A}{\partial t}. $$ Expectation values are picture-independent: $$ \langle A\rangle(t)=\langle\psi(t)|A|\psi(t)\rangle=\langle\psi(0)|A(t)|\psi(0)\rangle. $$

remark (conservation). if an observable $A$ is not explicitly time-dependent and commutes with $H$ on a common dense domain in the strong sense required by the Heisenberg equation, then $d\langle A\rangle/dt=0$. in particular the energy itself is conserved when $H$ is time-independent.

example (free evolution on the line). for $H=P^2/(2m)$ on $L^2(\mathbb{R})$, a centered Gaussian packet of width $\sigma$ spreads: the position variance grows as $\sigma(t)^2=\sigma^2+(\hbar t/(2m\sigma))^2$. momentum probabilities are stationary because $P$ commutes with $H$.

spectra and stationary states

Energy eigenstates satisfy $$ H\psi_n=E_n\psi_n $$ whenever $\psi_n\in D(H)$. They are stationary: the corresponding rays are time-independent, and for a configuration-space wave function the probability density $|\psi_n(\mathbf{r})|^2$ does not evolve. The spectrum of $H$ may contain a discrete part (bound states) and a continuous part (scattering states). Bound-state wave functions are square-integrable; continuum generalized eigenfunctions are not, and must be treated distributionally (Chapter 3).

For a particle of mass $m$ in a real potential $V$ that is Kato-small relative to $-\nabla^2$ (or otherwise essentially self-adjoint on a standard core), the time-independent Schrodinger equation reads $$ \Bigl(-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf{r})\Bigr)\psi(\mathbf{r})=E\psi(\mathbf{r}). $$

example (infinite square well). on $(0,a)$ with Dirichlet boundary conditions, $E_n=n^2\pi^2\hbar^2/(2ma^2)$ and $\psi_n(x)=\sqrt{2/a}\sin(n\pi x/a)$ for $n=1,2,\dots$. the spectrum is purely discrete, non-degenerate, and bounded below by $E_1>0$.

example (harmonic oscillator). for $H=P^2/(2m)+\tfrac12 m\omega^2 Q^2$ one obtains $E_n=\hbar\omega(n+\tfrac12)$ with $n=0,1,2,\dots$. the ground state is a minimum-uncertainty Gaussian; excited states are Hermite functions. the creation and annihilation operators $a^\dagger,a$ organize the spectrum algebraically and reappear systematically in Chapter 2.

angular momentum

Orbital angular momentum operators on $L^2(\mathbb{R}^3)$ are defined by $$ \mathbf{L}=\mathbf{r}\times\mathbf{p},\qquad\mathbf{p}=-i\hbar\nabla, $$ on a dense domain (for instance $C_c^\infty(\mathbb{R}^3\setminus\{0\})$ suitably completed). They satisfy $$ [L_i,L_j]=i\hbar\epsilon_{ijk}L_k. $$ The Casimir $\mathbf{L}^2$ and any one component (conventionally $L_z$) may be simultaneously diagonalized on the joint spectral subspaces: $$ \mathbf{L}^2|l,m\rangle=\hbar^2 l(l+1)|l,m\rangle,\qquad L_z|l,m\rangle=\hbar m|l,m\rangle, $$ with $l=0,1,2,\dots$ and $m=-l,\dots,l$. The corresponding angular eigenfunctions are the spherical harmonics $Y_{lm}(\theta,\phi)$.

Addition of angular momenta proceeds via the Clebsch-Gordan decomposition of the tensor product of two representation spaces of $\mathfrak{su}(2)$. Selection rules for matrix elements of vector operators follow from the Wigner-Eckart theorem: reduced matrix elements factor out geometric Clebsch-Gordan coefficients.

example (dipole selection rules). for an electric-dipole operator transforming as a vector under rotations, the Wigner-Eckart theorem forces $\Delta l=\pm 1$ and $\Delta m=0,\pm 1$ between orbital angular-momentum eigenstates (parity forbidding $\Delta l=0$ for a polar vector). these rules organize the gross structure of atomic optical spectra.

spin

In addition to orbital angular momentum, elementary particles carry intrinsic angular momentum: spin. For a particle of spin $s\in\{0,\tfrac12,1,\dots\}$ the spin operators $\mathbf{S}$ act on a $(2s+1)$-dimensional Hilbert space and obey the same commutation relations as $\mathbf{L}$. The total angular momentum is $$ \mathbf{J}=\mathbf{L}+\mathbf{S}, $$ acting on the tensor product of orbital and spin spaces. For electrons $s=\tfrac12$ and $$ \mathbf{S}=\tfrac{\hbar}{2}\boldsymbol{\sigma}, $$ with $\boldsymbol{\sigma}$ the Pauli matrices. The spin-statistics connection and the distinction between bosons and fermions appear naturally once identical particles are considered.

example (stern-gerlach). a spin-$\tfrac12$ beam traversing an inhomogeneous magnetic field coupling to $S_z$ splits into two discrete spots. the apparatus measures a component of $\mathbf{S}$ and prepares the corresponding eigenstate, illustrating both the discrete spectrum of spin and the projective update of Section 1.2.

identical particles

The Hamiltonian of a system of identical particles is invariant under arbitrary permutations of the particle labels. Consequently the physical state space must carry a representation of the symmetric group. For particles in three-dimensional space, the only possibilities consistent with locality and the topology of configuration space are the totally symmetric representation (bosons) and the totally antisymmetric representation (fermions).

The wave function of $N$ identical fermions therefore vanishes whenever two particles occupy the same one-particle orbital: the Pauli exclusion principle. The corresponding Fock-space construction, creation and annihilation operators, and occupation-number representation are developed systematically once second quantization is introduced.

example (two fermions, singlet versus triplet). for two spin-$\tfrac12$ particles in a symmetric spatial orbital, the spin factor must be the antisymmetric singlet $(\lvert\uparrow\downarrow\rangle-\lvert\downarrow\uparrow\rangle)/\sqrt{2}$. a symmetric spin triplet forces an antisymmetric spatial factor and therefore a node whenever the relative coordinate vanishes.

approximation methods

Exact solutions are rare. Two standard perturbative schemes are indispensable.

time-independent perturbation theory. if $H=H_0+\lambda V$ with $H_0$ diagonalized and $\lambda$ small, corrections to energies and eigenstates are expanded in powers of $\lambda$. non-degenerate and degenerate cases must be treated separately: the latter requires diagonalization of the perturbation inside each degenerate subspace of $H_0$ before proceeding order by order.

example (first-order non-degenerate shift). if $E_n^{(0)}$ is simple, $E_n=E_n^{(0)}+\lambda\langle n|V|n\rangle+O(\lambda^2)$. for the linear Stark effect in hydrogen the $n=2$ level is degenerate, so the naive formula fails and one must diagonalize the electric-dipole perturbation inside the four-dimensional $n=2$ subspace.

time-dependent perturbation theory. when the perturbation depends on time, transition amplitudes between unperturbed eigenstates are computed order by order. to first order the transition rate under a weak harmonic drive is governed by Fermi's golden rule. the same formalism yields the interaction of a bound system with classical electromagnetic radiation and the electric dipole selection rules of Section 1.5.

Variational methods and the WKB (semiclassical) approximation supply non-perturbative estimates for ground-state energies and tunneling rates, respectively.

example (variational hydrogenoid trial). a normalized exponential trial $\psi_a(r)=(\pi a^3)^{-1/2}e^{-r/a}$ for the Coulomb Hamiltonian yields $E(a)=\hbar^2/(2ma^2)-e^2/a$. minimizing in $a$ recovers the exact ground-state energy and Bohr radius, illustrating that a well-chosen variational family can be sharp.

atomic systems

The hydrogen atom is solved exactly in the Coulomb potential. Its spectrum, $$ E_n=-\frac{m_e e^4}{2\hbar^2 n^2}\qquad(n=1,2,\dots), $$ together with the degeneracy structure in $\ell$ and $m$ and the form of the radial wave functions, constitutes the prototype of a quantum-mechanical bound-state problem. (Here $m_e$ denotes the reduced electron-proton mass, and the potential is written in Gaussian electrostatic units as in the classical Landau-Lifshitz normalization; the SI transcription replaces $e^4$ by $e^4/(4\pi\varepsilon_0)^2$.)

Multi-electron atoms are treated by the central-field approximation, successive filling of shells according to the Pauli principle, and residual corrections (exchange, spin-orbit coupling, fine and hyperfine structure). The periodic table emerges as a direct consequence of the interplay between Coulomb attraction, centrifugal barriers, and the exclusion principle.

example (shell filling). the $1s$ orbital holds two electrons of opposite spin. the next available orbitals are $2s$ and $2p$; Hund's rules and exchange energetics then organize the ground-term structure of open $p$ and $d$ shells, recovering the broad architecture of the periodic table without yet invoking quantum fields.

scattering

Scattering theory describes the continuous spectrum. An incident plane wave of wave-vector $\mathbf{k}$ is distorted by a short-range potential; the asymptotic form of the wave function defines the scattering amplitude $f(\theta,\phi)$. The differential cross-section is $$ \frac{d\sigma}{d\Omega}=|f|^2. $$ Partial-wave analysis expands the amplitude in Legendre polynomials; the phase shifts $\delta_\ell$ encode the interaction. The optical theorem relates the total cross-section to the imaginary part of the forward amplitude. Resonances appear as rapid variations of phase shifts through $\pi/2$.

For identical particles the scattering amplitude must be symmetrized or antisymmetrized, producing characteristic interference patterns in the cross-section.

example (hard sphere, s-wave). for a hard sphere of radius $a$ at low energy, the $s$-wave phase shift satisfies $\delta_0=-ka+O((ka)^3)$ and the total cross-section approaches $4\pi a^2$, four times the classical geometric value, as a direct quantum-diffraction effect.

summary of physical content

The formalism developed above already determines the measurable predictions of non-relativistic quantum mechanics: discrete spectra $\sigma(H)=\{E_n\}$ for bound systems; transition amplitudes $|\langle n|V|m\rangle|^2$ and the associated selection rules; interference from coherent superpositions $c_1\psi_1+c_2\psi_2$; spin magnetic moments $\boldsymbol{\mu}=-(ge/2m)\mathbf{S}$; shell structure from the Pauli principle on antisymmetric $N$-particle states; and scattering cross-sections $d\sigma/d\Omega=|f|^2$.

All subsequent refinements (relativistic kinematics, quantum fields, renormalization, noncommutative geometry) must reproduce these results in the appropriate low-energy, few-particle limit. The mathematical scaffolding is a separable Hilbert space $\mathcal{H}$, a collection of self-adjoint operators $A=A^\dagger$ with spectral measures $E_A$, and a strongly continuous unitary group $U(t)=e^{-itH/\hbar}$. The remainder of the quantum program investigates how this scaffolding arises from classical geometry by quantization, how it is constrained by symmetry, and how it must be enlarged once variable particle number and locality are required.