## Элементарные частицы в Стандартной модели

Бозон – частица с целым значением спина.

Калибровочные бозоны – бозоны, которые действуют как переносчики фундаментальных взаимодействий.

Фотон – квант электромагнитного излучения, имеет спиновое квантовое число $s=1$, то есть величина спинового момента импульса равна $S=\sqrt{s(s+1)}\hbar=\sqrt2\hbar$.  Самая распространённая по численности частица во Вселенной.

W+, W и Z0-бозоны – переносчики слабого взаимодействия.

Глюон – переносчик сильного взаимодействия, спин 1, безмассовый, несёт цвет-антицвет.

Фермион – частица (или квази-частица) с полуцелым значением спина.  Подчиняется принципу исключения: в одном квантовом состоянии может находиться не более одной частицы.

Элементарная частица – микрообъект субъядерного масштаба, который невозможно расщепить на составные части.

Фундаментальная частица – бесструктурная элементарная частица.

Лептон – фундаментальная частица, фермион, не участвующий в сильном взаимодействии, спин 1/2: электрон $e^-$ и электронное нейтрино $\nu_e$, мюон $\mu^-$ (в 207 раз тяжелее электрона; через 2.2 мкс распадается на электрон, мюонное нейтрино и электронное антинейтрино) и мюонное нейтрино $\nu_{\mu}$, тау-лептон $\tau^-$ (распадается на мюон $\mu^-$, тау-нейтрино$\nu_{\tau}$ и мюонное антинейтрино $\bar{\nu}_{\mu}$) и тау-нейтрино $\nu_{\tau}$, плюс шесть их античастиц.  Нейтрино имеют ненулевую массу.

Адроны – класс элементарных частиц, подверженных сильному взаимодействию. Состоят из кварков.

Кварк – фундаментальная частица, фермион, входящая в состав адронов.  Спин 1/2.  Величина заряда – 1/3 или 2/3 заряда электрона.  Барионное число 1/3 (у антикварков -1/3).  Порождаются глюонами парой кварк-антикварк.

Барионы – адроны, фермионы, состоящие из трёх – красного, зелёного и синего – кварков.

Фундаментальные фермионы – лептоны и кварки.

Мезоны – составные элементарные частицы, адроны, бозоны, состоящие из равного числа кварков и антикварков.  Все мезоны нестабильны.

Пионы (пи-мезоны)$\pi^-$, $\pi^+$ (273 массы электрона), $\pi^0$ (264 массы электрона.  Имеют наименьшую массу среди мезонов и нулевой спин.  $\pi^-$ распадается на $\mu^-$ и антинейтрино,  $\pi^+$ распадается на $\mu^+$ и нейтрино,  $\pi^0$ распадается на два фотона.

## Particle Physics and Cosmology

Fundamental particles:  Each particle has an antiparticle; some particles are their own antiparticles.  Particles can be created and destroyed, some of them (including electrons and positrons) only in pairs or in conjunction with other particles and antiparticles.

Particles serve as mediators for the fundamental interactions.  The photon is the mediator of the electromagnetic interaction.  Yukawa predicted the existence of mesons to mediate the nuclear interaction.  Mediating particles that can exist only because of the uncertainty principle for energy are called virtual particles.

Particle accelerators and detectors:  Cyclotrons, synchrotrons, and linear accelerators are used to accelerate charged particles to high energies to experiment with particle interactions.  Only part of the beam energy is available to cause reactions with targets at rest.  The problem is avoided in colliding-beam experiments.

Particles and interactions:  Four fundamental interactions are found in nature: the strong, electromagnetic, weak, and gravitational interactions.  Particles can be described in terms of their interactions and of quantities that are conserved in all or some of the interactions.

Fermions have half-integer spins; bosons have integer spins.  Leptons, which are fermions, have no strong interactions.  Strongly interacting particles are called hadrons.  They include mesons, which are always bosons, and baryons, which are always fermions.  There are conservation laws for three different lepton numbers and for baryon number.  Additional quantum numbers, including strangeness and charm, are conserved in some interactions.

Quarks:  Hadrons are composed of quarks.  There thought to be six types of quarks.  The interaction between quarks is mediated by gluons.  Quarks and gluons have an additional attribute called color.

Symmetry and the unification of interactions:  Symmetry considerations play a central role in all fundamental-particle theories.  The electromagnetic and weak interactions become unified at high energies into the electroweak interaction.  In grand unified theories the strong interaction is also unified with these interactions, but at much higher energies.

The expanding universe and its composition:  The Hubble law shows that galaxies are receding from each other and that the universe is expanding.  Observations show that the rate of expansion is accelerating due to the presence of dark energy, which makes up 65.8% of energy in the universe.  Only 4.9% of the energy in the universe is in the form of conventional matter; the remaining 26.6% is dark matter, whose nature is poorly understood.

The history of the universe:  In the standard model of the universe, a Big Bang gave rise to the first fundamental particles.  They eventually formed into the lightest atoms as the universe expanded and cooled.  The cosmic background radiation is a relic of the time when these atoms formed.  The heavier elements were manufactured much later by fusion reactions inside stars.

## Nuclear Physics

Nuclear properties:  A nucleus is composed of $A$ nucleons ($Z$ protons and $N$ neutrons).  All nuclei have about the same density.  The radius of a nucleus with mass number $A$ is given approximately by equation $R=R_0\sqrt[3]A$ ($R_0=1.2\times10^{-15}\,\mathrm{m}$).  A single nuclear species of a given $Z$ and $N$ is called a nuclide.  Isotopes are nuclides of the same element (same $Z$) that have different number of neutrons.  Nuclear masses are measured in atomic mass units.  Nucleons have an angular momentum and a magnetic moment.

Nuclear binding and structure:  The mass of a nucleus is always less than the mass of the protons and neutrons within it.  The mass difference multiplied by $c^2$ gives the binding energy $E_{\mathrm{B}}=(ZM_{\mathrm{H}}+Nm_{\mathrm{n}}-\,_Z^AM)c^2$.  The binding energy for a given nuclide is determined by the nuclear force, which is short range and favors pairs of particles, and by the electrical repulsion between protons.  A nucleus is unstable if $A$ or $Z$ is too large or if the ratio $N/Z$ is wrong.  Two widely used models of the nucleus are the liquid-drop model and the shell model; the latter is analogous to the central-field approximation for atomic structure.

Radioactive decay:  Unstable nuclides usually emit an alpha-particle (a $\,_2^4\mathrm{He}$ nucleus) or a beta-particle (an electron) in the process of change to another nuclide, sometimes followed by a gamma-ray photon.  The rate of decay of an unstable nucleus is described by the decay constant $\lambda$, the half-life $T_{1/2}$, or the lifetime $T_{\mathrm{mean}}$: $T_{\mathrm{mean}}=\frac1{\lambda}=\frac{T_{1/2}}{\ln2}=\frac{T_{1/2}}{0.693}$.  If the number of nuclei at time $t=0$ is $N_0$ and no more are produced, the number at time $t$ is given by equation $N(t)=N_0e^{-\lambda t}$.

Biological effects of radiation:   The biological effect of any radiation depends on the product of the energy absorbed per unit mass and the relative biological effectiveness (RBE), which is different for different radiations.

Nuclear reactions:  In a nuclear reaction, two nuclei or particles collide to produce two new nuclei or particles.  Reactions can be exoergic or endoergic.  Several conservation law, including charge, energy, momentum, angular momentum, and nucleon number, are obeyed.  Energy is released by the fission of a heavy nucleus into two lighter, always unstable, nuclei.  Energy is also released by the fusion of two light nuclei into a heavier nucleus.

## Molecules and Condensed Matter

Molecular bonds and molecular spectra:  The principal types of molecular bonds are ionic, covalent, van der Waals, and hydrogen bonds.  In a diatomic molecule the rotational energy levels are given by equation: $E_l=l(l+1)\frac{\hbar^2}{2I}$ $(l=0,1,2,\ldots)$, where $I=m_{\mathrm{r}}r_0^2$ is the moment of inertia of the molecule, $m_{\mathrm{r}}=\frac{m_1m_2}{m_1+m_2}$ is its reduced mass, and $r_0$ is the distance between the two atoms.  The vibrational energy levels are given by equation $E_n=(n+\frac12)\hbar\omega=(n+\frac12)\hbar\sqrt{\frac{k'}{m_{\mathrm{r}}}}$ $n=(0,1,2,\ldots)$, where $k'$ is the effective force constant of the interatomic force.

Solids and energy bands:  Interatomic bonds in solids are of the same types as in molecules plus one additional type, the metallic bond.  Associating the basis with each lattice point gives the crystal structure.

When atoms are bound together in condensed matter, their outer energy levels spread out into bands.  At absolute zero, insulators and semiconductors have a completely filled valence band separated by an energy gap from an empty conduction band.  Conductors, including metals, have partially filled conduction bands.

Free-electron model of metal:  In the free-electron model of the behavior of conductors, the electrons are treated as completely free particles within the conductor.  In this model the density of states is given by equation $g(E)=\frac{(2m)^{2/3}V}{2\pi^2\hbar^3}E^{1/2}$.  The probability that an energy state of energy $E$ is occupied is given by the Fermi-Dirac distribution, $f(E)=\frac1{e^{(E-E_{\mathrm{F}})/kT}+1}$ ($E_{\mathrm{F}}$ is the Fermi energy), which is a consequence of the exclusion principle.

Semiconductors:  A semiconductor has an energy gap of about 1 eV between its valence and conduction bands.  Its electrical properties can be drastically changed by the addition of small concentrations of donor impurities, giving an $n$-type semiconductor, or acceptor impurities, giving a $p$-type semiconductor.

Semiconductor devices:  Many semiconductor devices, including diodes, transistors, and integrated circuits use one or more $p\text{-}n$-junctions.  The current-voltage relationship for an ideal $p\text{-}n$-junction diode is given by equation $I=I_S(e^{eV/kT}-1)$.

## Quantum Mechanics II: Atomic Structure

Three-dimensional problems:  The time-independent Schrödinger equation for three-dimensional problems is given by: $-\frac{\hbar^2}{2m}(\frac{\partial^2\psi(x,y,z)}{\partial x^2}+\frac{\partial^2\psi(x,y,z)}{\partial y^2}+\frac{\partial^2\psi(x,y,z)}{\partial z^2})+U(x,y,z)\psi(x,y,z)=E\psi(x,y,z)$.

Particle in a three-dimensional box:  The wave function for a particle in a cubical box is the product of a function of $x$ only, a function of $y$ only, and a function of $z$ only.  Each stationary state is described by three quantum numbers $(n_X,n_Y,n_Z)$: $E_{n_X,n_Y,n_Z}=\frac{(n_X^2+n_Y^2+n_Z^2)\pi^2\hbar^2}{2mL^2}$, $(n_X=1,2,3,\ldots;n_Y=1,2,3,\ldots;n_Z=1,2,3,\ldots)$.  Most of the energy levels given by this equation exhibit degeneracy: More than one quantum state has the same energy.

The hydrogen atom:  The Schrödinger equation for the hydrogen atom gives the same energy levels as the Bohr model: $E_n=-\frac1{(4\pi\epsilon_0)^2}\frac{m_\mathrm{r}e^4}{2n^2\hbar^2}=-\frac{13.60\,\mathrm{eV}}{n^2}$.  If the nucleus has charge $Ze$, there is an additional factor of $Z^2$ in the numerator.  The possible magnitudes $L$ of orbital angular momentum are given by equation: $L=\sqrt{l(l+1)}\hbar$, $(l=0,1,2,\ldots,n-1)$.  The possible values of the $z$-component of orbital angular momentum are given by equation: $L_z=m_l\hbar$, $(m_l=0,\pm1,\pm2,\ldots,\pm l)$.

The probability that an atomic electron is between $r$ and $r+dr$ from the nucleus is $P(r)\,dr$, given by equation: $P(r)\,dr=|\psi|^2\,dV=|\psi|^2\,4\pi r^2\,dr$.  Atomic distances are often measured in units of $a$, the smallest distance between the electron and the nucleus in the Bohr model: $a=\frac{\epsilon_0h^2}{\pi m_\mathrm{r}e^2}=\frac{4\pi\epsilon_0\hbar^2}{m_\mathrm{r}e^2}=5.29\times10^{-11}\mathrm{m}$.

The Zeeman effect:  The interaction energy of an electron (mass $m$) with magnetic quantum number $m_l$ in a magnetic field $\vec{B}$ along the $+z$-direction is given by equation: $U=-\mu_zB=m_l\frac{e\hbar}{2m}B=m_lm_{\mathrm{B}}B$ $(m_l=0,\pm1,\pm2,\ldots,\pm l)$, where $m_{\mathrm{B}}=\frac{e\hbar}{2m}$ is called the Bohr magneton.

Electron spin:  An electron has an intrinsic spin angular momentum of magnitude $S$, given by equation $S=\sqrt{\frac12(\frac12+1)}\hbar=\sqrt{\frac34}\hbar$.  The possible values of the $z$-component of the spin angular momentum are $S_x=m_s\hbar$ $(m_s=\pm\frac12)$.

An orbiting electron experience an interaction between its spin and the effective magnetic field produced by the relative motion of electron and nucleus.  This spin-orbit coupling, along with relativistic effects, splits the energy levels according to their total angular momentum quantum number $j$: $E_{n,j}=-\frac{13.60\,\mathrm{eV}}{n^2}[1+\frac{n^2}{\alpha^2}(\frac{n}{j+\frac12}-\frac34)]$.

Many-electron atoms:  In a hydrogen atom, the quantum numbers $n$, $l$, $m_l$, and $m_s$ of the electron have certain allowed values given by equation: $n\geq1$, $0\leq l\leq n-1$, $|m_l|\leq l$, $m_s=\pm\frac12$.  In a many-electron atom, the allowed quantum numbers for each electron are the same as in hydrogen, but the energy levels depend on both $n$ and $l$ because of screening, the partial cancellation of the field of the nucleus by inner electrons.  If the effective (screened) charge attracting an electron is $Z_{\mathrm{eff}}e$, the energies of the levels are given approximately by equation: $E_n=-\frac{Z_{\mathrm{eff}}^2}{n^2}(13.6\,\mathrm{eV})$.

X-ray spectra:  Moseley’s law states that the frequency of a $K_{\alpha}$ x ray from a target with atomic number $Z$ is given by equation $f=(2.48\times10^{15}\,\mathrm{Hz})(Z-1)^2$.  Characteristic x-ray spectra result from transition to a hole in an inner energy level of an atom.

Quantum entanglement:  The wave function of two identical particles can be such that neither particle is itself in a definite state.  For example, the wave function could be a combination of one term with particle $1$ in state $A$ and particle $2$ in state $B$ and one term with particle $1$ in state $B$ and particle $2$ in state $A$.  The two particles are said to be entangled, since measuring the state of one particle automatically determines the result of subsequent measurements of the other particle.

## Momentum in Quantum Mechanics

For a particle in state $\Psi$, the expectation value of $x$ is $\langle x\rangle=\int_{-\infty}^{+\infty}x|\Psi(x,t)|^2\,dx$.

$\langle p\rangle=m\frac{d\langle x\rangle}{dt}=-i\hbar\int_{-\infty}^{+\infty}\left(\Psi^*\frac{\partial\Psi}{\partial x}\right)\,dx$.

In general, $\langle Q(x,p)\rangle=\int_{-\infty}^{+\infty}\Psi^*\,Q\left(x,\frac{\hbar}{i}\frac{\partial}{\partial x}\right)\Psi\,dx$.

For example, $T=\frac12mv^2=\frac{p^2}{2m}$, so $\langle T\rangle=-\frac{\hbar^2}{2m}\int_{-\infty}^{+\infty}\Psi^*\frac{\partial^2\Psi}{\partial x^2}\,dx$.

## Quantum Mechanics I: Wave Functions

Wave functions:  The wave function for a particle contains all of the information about that particle.  If the particle moves in one dimension in the presence of a potential energy function $U(x)$, the wave function $\Psi(x,t)$ obeys the one-dimensional Schrödinger equation: $-\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2}+U(x)\Psi(x,t)=i\hbar\frac{\partial\Psi(x,t)}{\partial t}$.  (For a free particle on which no forces act, $U(x)=0$.)  The quantity $|\Psi(x,t)|^2$, called the probability distribution function, determines the relative probability of finding a particle near a given position at a given time.  If the particle is in a state of definite energy, called a stationary state, $\Psi(x,t)$ is a product of a function $\psi(x)$ that depends on only spatial coordinates and a function $e^{-iEt/\hbar}$ that depends on only time: $\Psi(x,t)=\psi(x)e^{iEt/\hbar}$.  For a stationary state, the probability distribution function is independent of time.

A spatial stationary-state wave function $\psi(x)$ for a particle that moves in one dimension in the presence of a potential-energy function $U(x)$ satisfies the time-independent Schrödinger equation: $-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+U(x)\psi(x)=E\psi(x)$.  More complex wave functions can be constructed by super-imposing stationary-state wave functions.  These can represent particles that are localized in a certain region, thus representing both particle and wave aspects.

Particle in a box:  The energy levels for a particle of mass $m$ in a box (an infinitely deep square potential well) with width $L$ are given by the equation: $E_n=\frac{p_n^2}{2m}=\frac{n^2h^2}{8mL^2}=\frac{n^2\pi^2\hbar^2}{2mL^2}$ $(n=1,2,3,\ldots)$.  The corresponding normalized stationary-state wave functions of the particle are given by the equation $\psi_n(x)=\sqrt{\frac2L}\sin\frac{n\pi x}L$ $(n=1,2,3,\ldots)$.

Wave functions and normalization:  To be a solution of the Schrödinger equation, the wave function $\psi(x)$ and its derivative $d\psi(x)/dx$ must be continuous everywhere except where the potential-energy function $U(x)$ has an infinity discontinuity.  Wave functions are usually normalized so that the total probability of finding the particle somewhere is unity: $\int_{-\infty}^{+\infty}|\psi(x)|^2\,dx=1$.

Finite potential well:  In a potential well with finite depth $U_0$, the energy levels are lower than those for an infinitely deep well with the same width, and the number of energy levels corresponding to bound states is finite.  The levels are obtained by matching wave functions at the well walls to satisfy the continuity of $\psi(x)$ and $d\psi(x)/dx$.

Potential barriers and tunneling:  There is a certain probability that a particle will penetrate a potential-energy barrier even though its initial energy is less than the barrier height.  This process is called tunneling.

Quantum harmonic oscillator:  The energy levels for the harmonic oscillator (for which $U(x)=\frac12k'x^2$) are given by the equation: $E_n=(n+\frac12)\hbar\sqrt{\frac{k'}{m}}=(n+\frac12)\hbar\omega$ $(n=1,2,3,\ldots)$.  The spacing between any two adjacent levels is $\hbar\omega$, where $\omega=\sqrt{k'/m}$ is the oscillation angular frequency of the corresponding Newtonian harmonic oscillator.

Measurement in quantum mechanics:  If the wave function of a particle does not correspond to a definite value of a certain physical property (such as momentum or energy), the wave function changes when we measure that property.  This phenomenon is called wave-function collapse.