
Particle Charge Proton +1 Composed of 2 up quarks, 1 down quark, and gluons Neutron 0 Composed of 1 up quark, 2 down quarks, and gluons Electron 1 Up quark +2/3 Down quark 1/3 Photon 0 Carries the electromagnetic force and binds electrons to the nucleus Gluon 0 Carries the strong force and binds quarks, protons, and neutrons
In this plot, the diameter of each particle proportional to Mass$1/3$. This is what the particles would look like if they were uniformdensity spheres.
The electron is exaggerated otherwise it would be invisible.
The blue particles represent the heaviest particle that can be produced by each accelerator.
At this scale, a Planckmass particle has a diameter of 10 km.
Photons, Gluons, and Gravitons are massless.
Particle masses can be expressed in terms of electron Volts.
1 Electron Volt (eV) = The energy gained by an electron upon falling down a potential of 1 Volt. = 1.602$\cdot 1019$ Joules M = Particle rest mass C = Speed of light E = particle rest energy = M C^{2} Electron mass = 9.11$\cdot 1031$ kilograms = 511000 eV 1 keV = $103$ eV 1 MeV = $106$ eV 1 GeV = $109$ eV Mass (GeV) Electron .00051 Proton .9383 Neutron .9396 SLAC limit 45 Heaviest particle that can be produced by the Stanford Linear Accelerator Higgs Boson 125 Discovered at the LHC and Fermilab Fermilab limit 200 Heaviest particle that can be produced by Fermilab LHC limit 700 Heaviest particle that can be produced by the Large Hadron Collider Cosmic rays 10^{12} Highestenergy events observed Big bang energy 10^{19} Energy of particles at the time of the big bang Big bang energy = Planck energy = 1.22$\cdot 1019$ GeV = 1.956$\cdot 109$ Joules = 2.2$\cdot 108$ kg C^{2}
Force Relative Forcecarrying Felt by strength particle Gravity 10^{40} Graviton All particles Weak 10^{5} W & Z bosons All particles except gluons Electromagnetic 1 Photon All particles with electric charge Strong 100 Gluon Quarks and gluons
1687 Earth gravity + Planetary motion → Newtonian gravity Newton 1752 Electric charge + Lightening Franklin 1820 Electric currents + Magnets → Electromagnetism Oersted 1862 Electric force + Magnetic force → Maxwell's equations Maxwell 1864 Electromagnetism + Light → Elecromagnetic waves Maxwell 1905 Electromagnetism + Lorentz transform → Special relativity Einstein, Poincare, Lorentz 1915 Newtonian Gravity + Special relativity → General relativity Einstein 1928 Quantum mechanics + Special relativity → Quantum electrodynamics Dirac, Feynman, Dyson 1967 Electromagnetism + Weak force → Electroweak force Salam, Glashow, Weinberg ? Electroweak force + Strong force → Grand Unified Theory (GUT) ? GUT + General Relativity → Quantum Gravity
"LHC limit" is the heaviest particle that can be produced by the LHC collider.
Weak unification: The electromagnetic force unifies with the weak force Strong unification: The electroweak force unifies with the strong force Gravity unification: Gravity unifies with the electroweak and strong forces
The "Standard Model" describes:
The strong force (quantum chromodynamics)
The electroweak force
The Higgs particle and the mechanism for generating mass
Dark matter, dark energy, and matterantimatter asymmetry are examples of
things that are not explained by the Standard Model.
String theory is an attempt to construct a theory of quantum gravity.
1803 Young discovers the diffraction of light, suggesting that light is a wave 1861 Maxwell develops the "Maxwell's equations", unifying electricity and magnetism 1864 Maxwell finds that light is an electromagnetic wave 1900 Planck solves the blackbody problem by assuming that photon energy is quantized as E = h F 1905 Einstein publishes the "photoelectric effect" experiment, providing the first direct measurement of photon energy and momentum. 1905 Theory of special relativity completed. Einstein publishes the photoelectric experiment 1924 de Broglie postulates that for particles with mass, Momentum * Wavelength = PlanckConstant 1927 Davisson and Germer experimentally verify the de Broglie relation for electrons. F = Photon frequency W = Photon wavelength C = Photon speed = Speed of light = 3.00$34$ Joules/second = F W (Wave equation) E = Photon energy Q = Photon momentum h = Planck constant = 6.63$\cdot 1034$ Joule secondsEinstein's 1905 photoelectric experiment showed that photons have energy and momentum given by
E = h F (Planck equation) E = Q CUsing the wave equation for photons (C=FW) we have
Q W = h (de Broglie equation)Photons have zero rest mass. In 1924 de Broglie found that this equation also applies to particles with finite rest mass. W is the "quantummechanical wavelength" of a particle.
For nonrelativistic electrons (V << C), the energy and momentum are
Q = m V E = ½ m V^{2}The relationship between E and Q is different for photons and electrons.
Photon: E = Q C Electron: E = Q^{2} / (2M)When electrons are relativistic (V ~ C) they behave similar to photons.
F = Frequency W = Wavelength V = Speed C = Speed of light = 3.00$34$ Joules/second = F W (Wave equation) E = Energy Q = Momentum h = Planck constant = 6.63$\cdot 1034$ Joule seconds m = Rest mass Z = Lorentz gamma (only defined for particles with m > 0) = (1V^{2}/C^{2})^{½} M = Relativistic mass = Z m (for particles with positive rest mass) = E/C^{2} (for particles with zero rest mass)The following equations apply for all particles, whether the rest mass is zero or positive.
Q W = h E = (mC^{2})^{2} + (QC)^{2}If the particle has zero rest mass (m=0), such as a photon, then
V = C m = 0 E = Q C M = E / C^{2}If the particle has positive rest mass then
M = Relativistic mass = Z m E = M C^{2} Q = M VIn many cases you can guess the relativistic formula by replacing "m" with "M". For example, in CGS units,
Force = M * Acceleration Electric force = Charge * Electric field Magnetic force = Charge * Velocity * Magnetic fieldIf you try to accelerate a particle to the speed of light,
V/C > 1 M > Infinity Acceleration > 0The particle never reaches the speed of light.
We can define 4 regimes.
Classical: V/C << 1 Relativistic: V/C not close to zero and not close to 1 Ultrarelativistic: V/C close to 1 but not equal to 1 Light: V/C = 1For particles with positive rest mass, the relativistic formulae work for all values of
0 <= V/C < 1In the classical and ultrarelativistic regimes the relativistic formulae can be simplified.
V/C > 0they become the formulas of classical physics.
V/C → 0 Z → 1 + 1/2 V^{2}/C^{2} E → E = m C^{2} + ½ m V^{2} A rest energy plus a kinetic energy Q → m V
V/C < 1 E^{2} = (mC^{2})^{2} + (QC)^{2} E = Z m C^{2} = M C^{2} Q = Z m V = M V
Z > Infinity E > Q CFor particles with V/C << 1, QC << E.
Ultrarelativistic particles behave similar to massless particles in that QC ~ E. In particle colliders, particles are almost always ultrarelativistic.
For particles with zero rest mass, such as photons,
E = Q C = F h
The following table shows example values for the properties of particles.
Rest Kinetic V/C Wavelength Compton radius mass energy (angstroms) (amgstroms) (MeV) (MeV) Infrared photon 0 .0000001 1 107000 107000 Green photon 0 .0000023 1 5550 5550 X ray photon 0 .01 1 1.28 1.28 Gamma ray photon 0 5 1 .0026 .0026 Electron .511 0 0  .0243 Electron from vacuum tube .511 .00001 .00626 12.3 .0243 10 electron Volts Electron from beta decay .511 5 .9957 .0025 .0243 Electron at SLAC .511 45000 .999999999936 .00000028 .0243 Proton 938 0 0  .0000132 Proton in nucleus 938 5 .103 .00013 .0000132 Proton at the LHC 938 7000000 .9999999910 .0000000018 .0000132 Cosmic ray proton 938 e15 nearly 1 1e17 .0000132 Higgs boson 125000 0 0  .0000000099 Baseball pitch e29 e15 .00000015 1e24 e24 146 grams, 45 m/s F22 Raptor, Mach 2.3 e36 3e22 .0000023 5e31 e41 20 tons, 700 m/s Planck particle e22 e22 nearly 1 e25 e25 E = Energy C = Speed of light m = Rest mass Q = Momentum h = Planck constantParticle energy and momentum are related by
E^{2} = (mC^{2})^{2} + (QC)^{2} If Q << mC the particle is nonrelativistic If Q = mC the particle is at the boundary between nonrelativistic and ultrarelativistic If Q >> mC the particle is ultrarlativisticIf the particle is on the boundary (Q=mC) then its wavelength is called the "Compton wavelength". For forcecarrying particles this sets the range of the force.
Compton wavelength = h / Q = h / (mC)A F22 Raptor has the same kinetic energy as a Planck particle.
A baseball pitch has the same Compton wavelength as a Planck particle.
The energy of a particle can be expressed as a rest energy plus a kinetic energy
E = Gamma m C^{2} = m C^{2} + (Gamma  1) m C^{2} = m C^{2} + 1/2 m V^{2} in the limit of V/C << 1If Gamma < 2, a particle's rest energy is larger than its kinetic enegy.
In the Stanford Linear Collider (SLAC), electrons are accelerated to an energy of 45 GeV and collided headon with antielectrons with the same energy.
Rest energy of electron = .0005 GeV Kinetic energy of SLAC electron = 45 GeVSLAC electrons are highly relativistic. When a SLAC electron is collided with a SLAC antielectron, they annihilate and new particles are created. The energy available to create new particles is 90 GeV. This is how a highenergy collison can produce particles that have a larger rest energy than the colliding particles.
For example,
Electron with 45 GeV + Antielectron with 45 GeV > Annihilate > Charm Quark + Anti Charm Quark + 87.4 GeV of kinetic energy Energy in GeV Electron .0005 Proton 1.0 Neutron 1.0 Charm Quark 1.3 Higgs Boson 125 Big Bang particle 10^{19} GeV Energy required for quantum gravity
Quantum Relativistic Strong Planck Example phenomena speeds gravity energy Classical physics Rugby Special relativity * Interstellar spaceships General relativity * * Black holes Quantum mechanics * Atomic sizes & smaller Quantum field theory (QFT) * * Particle colliders Quantum gravity * * * * Big bangQuantum mechanics is relevant if the scale is equal to or less than the quantum wavelength. If an object is larger than the quantum wavelength it behaves classically.
In atoms, the size scale is small enough for quantum phenomena and the characteristic speeds of electrons are nonrelativistic, and so this is the regime of "Quantum mechanics".
Q = Momentum h = Planck Constant = 6.62e34 Joule second W = Wavelength = h/Q meter K = Electric force constant = 8.988e9 Newton meter^{2} / Coulomb^{2} q = Proton charge = 1.602e19 Coulombs G = Gravity constant = 6.67e11 Newton meter^{2} / kg^{2} V = Velocity C = Speed of light R = Distance from the object Re = Object event horizon radius = 2 G M / C^{2} If V << C Special relativity may be neglected If V > 0.1 C Special relativity applies If W << R Quantum mechanics doesn't apply If W >= R Quantum mechanics applies If R >> Re Spacetime is flat enough for general relativity to be neglected If R < 10 Re Gravity warps spacetime and general relativity appliesIf gravity is strong enough to make particles relativistic and if the scale is small enough for quantum mechanics, quantum gravity is relevant. This is the "Planck scale". The space near an event horizon can make particles relativistic but a solar mass black hole is too large for quantum mechanics. Quantum gravity occurs at the centers of black holes and during the big bang.
As the mass of a particle increases, its event horizon radius increases and its quantummechanical wavelength decreases. When they become equal, quantumgravity applies. THis is the "Planck scale".
For an ultrarelativistic particle,
E = Q C = h C / WIf the event horizon radius equals the quantum wavelength,
W = (G h / C^{3})^{½} = 4.1*1035 meters This is the Planck scale
Distance Meter Time Second Mass kg Charge CoulombSI derived units:
Velocity Meter / Second Momentum kg Meter / Second Energy kg Meter^{2} / Second^{2} = Joules Force kg Meter / Second^{2} = Newtons = Joules/meterPhysical constants:
C = Speed of light = 2.998e8 Meters/Second G = Gravity constant = 6.67e11 Newton (Meter/Kg)^{2} h = Planck constant = 6.626e34 Joule Seconds e = Electron charge = 1.602e19 CoulombsWe can define time in terms of distance using C.
Define a "Light meter" as the amount of time it takes for light to travel 1 meter.
1 Light meter = 3.34 Nanoseconds
With the "light meter" we no longer need the "second".
We can also define distance in terms of time.
Define a "light second" as the distance light travels in one second.
1 light second = 2.998e8 meters
With this we no longer need the meter. Distance can be defined in terms of light travel time.
We can define energy in terms of mass using the speed of light.
Energy = Mass C^{2}
1 "kilogram" of energy is equal to C^{2} = 8.99e16 Joules.
Particle masses are usually expressed in terms of energy. For example,
M = Proton rest mass = 1.673e27 kg E = Proton rest energy = M C^{2} = 1.673e27 • 2.998e8^{2} = 1.504e10 JoulesIf a proton and antiproton annihilate they produce 3e10 Joules.
Using the gravity constant we can define mass in terms of distance.
G Mass^{2} / Distance = Energy = Mass C^{2} Mass = C^{2} Distance / G 1 "Gravity mass" = C^{2}/G kilograms = 1.34e27 kilogramsMass can be expressed in terms of "gravity masses".
An object with this mass has an event horizon radius of 1 meter.
X = Distance T = Time M = Mass Using the physical constant C we can express either: X in terms of T T in terms of X Using the physical constants {C,G} we can express either: T and M in terms of X X and M in terms of T X and T in terms of M However we cannot define a universal value for X, T, or M. If we add Planck's constant we can.
E = Photon energy f = Photon frequency h = Planck constant E = h fPlanck used the Planck constant to obtain a universal scale for space, time and mass which are called the "Planck scales".
X = Planck distance T = Planck time M = Planck massPhysical constants:
Speed of light = C = 2.998⋅10^{8} Meters/Second Gravity constant = G = 6.674⋅10^{11} Newton (Meter/Kg)^{2} Electron charge = e = 1.602⋅10^{19} Coulombs Planck constant = h = 6.626⋅10^{34} Joule Seconds Reduced Planck constant= ℏ = h (2π)^{1} = 1.055⋅10^{34} Joule SecondsWhen discussing Planck units, ℏ is used instead of h.
Speed of light: X = C T Gravitational energy: E = G M^{2} / X > C^{2} = G M / X Energy quantization: E T = ℏ > M C^{2} T = ℏSolving for {X, T, M},
Planck length = X = (G ℏ / C^{3})^{½} = 1.62⋅10^{35} meters Planck time = T = (G ℏ / C^{5})^{½} = 5.39⋅10^{44} seconds Planck mass = M = (ℏ C / G)^{½} = 2.18⋅10^{8} kg = 1.22⋅10^{19} GeVThese are the scales for particles at the time of the big bang.
R = Distance between two particles M = Planck mass in kg = (ℏ C / G)^{½} m = Proton mass in kg = 1.673⋅10^{27} kg Q = Planck charge in Coulombs q = Charge on the proton = 1.602e19 Coulombs F_{g} = Gravity force = G M^{2} / R^{2} F_{e} = Electric force = K Q^{2} / R^{2} G = Gravity constant = 6.67e11 Newton Meter^{2} / Kg^{2} K = Electric constant = 8.988e9 Newton Meter^{2} / Coulomb^{2}The Planck charge is defined such that
Electric force between two Planck charges = Gravitational force between two Planck masses G M^{2} = K Q^{2} Q = (ℏ C / K)^{½} = 1.876⋅10^{18} Coulombs q/Q = Proton Charge / Planck charge = .0854 m/M = Proton mass / Planck mass = 7.68⋅10^{20}For two protons,
Gravity force / Electric force = G m m / (K q q) = (m/M)^{2} / (q/Q)^{2} = 8.087⋅10^{37}Gravity is vastly weaker than the electric force. This is because the proton mass is much less than the Planck mass, while the proton charge is similar to the Planck charge.
We can define a measure of the strength of the electric force "Z" by using photons. Suppose two electrons are a distance R apart and a photon has wavelength W = 2 π R. The photon energy is E = hf = hC/W.
R = Distance between two electrons f = Photon frequency W = Photon wavelength = 2 π R h = Planck constant E = Photon energy = h f = h C / W = ℏ C / R q = electron charge E_{e} = Electric energy between the electrons = K q^{2} / R α = Electric energy / Photon energy = (K q^{2} / R) / (hC/W) = 2 π K q^{2} / (hC) = .007297The dimensionless number α appears everywhere in physics and is called the "fine structure constant". For example,
Z = Proton charge / Planck charge = q K^{½} (ℏ C)^{½} = .0854 = α^{½} R_classical / R_compton = α / (2Pi) R_classical / R_quantum = α^{2} / (2π)^{2}
G = Gravitational force constant = 6.67*10^{11} Newton Meter^{2} / Kilogram^{2} C = Speed of light M = Mass of a black hole m = Rest mass of a particle R = Distance between the black hole and the particle F = Force on a particle from the black hole E = Potential energy of the particle with respect to the black hole Re= Black hole event horizon radiusIf the particle is far from the black hole then the Newtonian formulas for force and energy apply
F = G M m / R^{2} E = G M m / RAs the particle falls toward the black hole its velocity approaches C and the gravitational energy approaches the rest energy. General relativity becomes important here, and the characteristic scale R for this to happen is determined by C, G, and M. The only combination of these variables that gives units of length is
R = DimensionlessConstant * C^{2} / (GM)This is an estimate for the radius of a black hole event horizon. Orderofmagnitude estimates like this tend to give the right exponents (C^{2}, G^{1}, M^{1}) but they can't give the dimensionless constant in front. Using general relativity, the exact formula for the event horizon radius is
Re = 2 C^{2} / (GM)The dimensionless constant turns out to be "2".
We can also estimate the event horizon radius by setting the gravitational energy equal to the rest energy.
G M m / R = .5 m C^{2} R = 2 G M / C^{2}This is only an estimate because the Newtonian formulas break down near the black hole.
Plugging in the values for C and G,
Event horizon radius = 2 * 6.67e11 / (3*10^{8} m/s)^{2} * M = 1.49e27 * M = 3000 meters * (M / Mass of sun)If the sun were a black hole the event horizon radius would be 3000 meters. To make the Earth a black hole you would have to squash it to a radius of 7 millimeters.
Neutron stars have a mass between 2 and 4 solar masses and a radius slightly larger than the event horizon radius.
The black hole at the center of the galaxy has a mass of 4 million solar masses and an event horizon radius of 12 million km. This is 1/12 the distance from the Earth to the sun.
In classical physics, an accelerating charge emits synchrotron photons and loses energy. If an electron orbits a proton then the emitted photons cause the electron to inspiral into the proton in 10^{15} seconds. This is an example where a theory predicts a phenomenon that breaks the theory, and this usually points the way to a more fundamental theory. The thing that stops the electron from crashing into the proton is quantum mechanics.
Suppose an electron is on a circular orbit around a proton.
R = Distance between the electron and the proton V = Velocity of the electron C = Speed of light = 3.00e8 m/s M = Electron rest mass = 5.68e31 kg Z = Electric charge on an electron = 1.602e19 Coulombs K = Electric force constant = 8.988*10^{9} Newton Meter^{2} / Coulomb^{2} F = Force between the proton and electron = K Z^{2} / R^{2} E = Electron kinetic energy = .5 M V^{2} Ee = Electric energy between the proton and electron = Integral (Force dR) = K Z^{2} / R Q = Electron momentum = M V h = Planck Constant = 6.62 * 1034 Joule seconds W = Electron quantummechanical wavelength = h / QBalance the electric and the centripetal forces:
K Z^{2} / R^{2} = M V^{2} / RThe kinetic energy is
E = .5 M V^{2} = .5 K Z^{2} / RThe electron becomes relativistic when E ~ M C^2. Define
R_classical = Classical radius of the electron = Radius of a circular orbit for which the kinetic energy equals the rest energy = K Z^{2} / (M C^{2}) = 2.818e15 metersIf R < R_classical, classical physics is guaranteed to fail and so some new physics has to appear. A similar example is a black hole where Newtonian gravity breaks down and general relativity takes over.
The Schwarzschild radius of a black hole is the characteristic distance where infalling matter becomes relativistic.
M = Mass of an electron M_hole = Mass of a black hole E_grav =  G M_hole M / R^{2} R = Distance of an electron from a black hole R_schwarz= Schwarzschild radius of a black hole, the closest distance from which light can escape. = 2 G M / C^{2}Setting the gravitational energy equal to the rest energy gives the gravitational radius of a black hole, which is proportional to the Schwarzschild radius.
As R>0, the scale where quantum mechanics becomes important is the "quantum radius of the electron".
To derive this scale we calculate the electron wavelength as a function of R. We assume that the electron is nonrelativistic and we assume the electron is on a circular orbit around the proton. The balance of electric and centripetal force is
W = Quantummechanical wavelength of the electrom R = Orbital radius of the electronThe parameter that characterizes the importance of quantum mechanics is W/R.
If W/R > 1 Quantum mechanics is important If W/R < 1 Quantum mechanics is unimportant and classical physics can be used W/R = (h/Q) * Q^{2} / (K q^{2} m) = h Q / (K q^{2} m) = Constant * R^{½} As R> Infinity W/R > 0 Classical physics applies As R> 0 W/R > Infinity Quantum mechanics applies The radius where W/R=1 is the quantum radius of the electron. R_quantum = h^{2} / (K q^{2} m) = 2.086e9 metersWe can compare the quantum radius to the classical radius.
R_quantum / R_classical = h^{2} C^{2} / (K^{2} q^{4}) = 740000 Because R_quantum / R_classical >> 1, quantum mechanics becomes important before relativity.R_quantum sets the size of atoms. If you calculate the electron orbital radius in a hydrogen atom using the Bohr theory,
R_bohr = R_quantum / (2 Pi) = 5.29e11 metersThis is the radius of the S=1 orbital in a hydrogen atom.
If a particle is nonrelativistic (V << C),
Q = Momentum = m V E = Energy = ½ m V^{2} Q^{2} = 2 m EIf a particle is relativistic (V~C),
E ~ Q CFor relativistic particles we can define an equivalence between space and time by setting V=C.
Space = C * TimeWe can also define an equivalence between space and energy by setting the space scale equal to the quantum mechanical wavelength.
E = Q C = h C / WIf we set E equal to the particle's rest energy, we call the resulting wavelength the "Compton wavelength". This particle is "barely relativistic".
m C^{2} = h C / R_compton R_compton = h / (m C)All particles with finite mass have a Compton wavelength.
Suppose a photon has an energy equal to a particle's rest energy E.
f = Photon frequency W = Photon wavelength E = h f = h C / W W = h C / E = h / (M C)The wavelength of the photon is equal to the particle's Compton wavelength. If a photon has the same energy as a particle's rest energy the wavelength of the photon is equal to the Compton wavelength.
For an electron,
R_compton = 2.426e12 meters
m = Particle rest mass E = Particle rest energy = m C^{2} T = Particle lifetime X = Characteristic distance a virtual particle travels during its lifetime h = Planck constant Q = Particle momentum C = Speed of lightFrom the Heisenberg uncertainty principle, a virtual particle has a characteristic lifetime such that
E T = hVirtual particles are relativistic and so we assume V ~ C.
The distance the virtual particle travels in its lifetime is
X = T C = h C / EIf we set E equal to the rest energy mC^{2},
X = h / (m C)This is the "Compton wavelength" of a particle.
Force Force carrying Mass of force Compton radius of particle carrying particle forcecarrying particle (GeV) (meters) Gravity Graviton 0 Infinite Weak W 80 1.5e17 Z 91 1.4e17 Electromagnetic Photon 0 Infinite Strong Pion .135 9.2e15Technically the strong force is carried by gluons. For protons and neutrons in a nucleus the strong force can be considered to be carried by pions.
If a force is carried by a massless particle then the force is given by
Force = Constant / R^{2}such a force is said to have "infinite range".
If the force is carried by a massive particle then the force is said to have "finite range". The range of the force is equal to the carrier's Compton wavelength and beyond this range the force is essentially zero.
The size scale of the nucleus is determined by the pion's Compton wavelength.
The forcecarrying particles for the weak force are heavy and hence the weak force has short range, 100 times shorter than the size of a nucleus. This is why the weak force is weak.
The fine structure constant characterizes the strength of the electromagnetic force.
For two electrons, the gravitational force is vastly weaker than the electric force.
Force of gravity / Electric force = G m^{2} / K q^{2} = 2.40e43
Quantum mechanics resolved the electron crisis. The next crisis is the Higgs crisis.
Higgs mass = 125.3 GeV meters Electron quantum radius 2.09e9 Limit for classical physics Atomic scale 2e10 Electron Compton radius 2.43e12 Limit for nonrelativistic quantum mechanics Electron classical radius 2.82e15 Nuclear scale 1e15 Higgs Compton radius 9.9e18Suppose we consider a length scale R.
If R > Electric quantum radius Classical physics applies If R < Electron quantum radius Quantum mechanics apples If R > Electron Compton radius Nonrelativistic quantum mechanics may be used If R < Electron Compton radius Relativistic quantum mechanics must be used If R > Higgs Compton radius The Standard Model applies If R < Higgs Compton radius The Standard Model breaks downThe Standard Model breaks down for scales below the Higgs scale and some new theory must take over for smaller scales. The most promising candidates are "Supersymmetry" and "The Multiverse". The Large Hadron Collider can at present explore down to the Higgs scale but no further. When it is upgraded to higher energies in 2015 it will be able to go beyond the Higgs scale.
There is much drama in the world of supersymmetry. It is predicted that the LHC should already have detected supersymmetric particles and thus far none have been found.
Charge Spin Rest mass Strong Weak Lifetime (GeV) force force (seconds) Up quark +2/3 1/2 ~.0024 * * 882 (Decay of Neutron) Charm quark +2/3 1/2 ~1.25 * * 1e12 (Decay of D+ meson) Top quark +2/3 1/2 ~171 * * 5e25 Down quark 1/3 1/2 ~.0048 * * Strange quark 1/3 1/2 ~.1 * * 1.2e8 (Decay of Kaon+ meson) Bottom quark 1/3 1/2 ~4.2 * * 1.6e12 (Decay of B+ meson) Electron 1 1/2 .000511 * Muon 1 1/2 .1057 * 2.20e6 Tau 1 1/2 1.777 * 2.9e13 Electron neutrino 0 1/2 <2.2e9 * Mixes Muon neutrino 0 1/2 <.00017 * Mixes Tau neutrio 0 1/2 <.016 * Mixes Photon 0 1 0 * Gluon 0 1 0 * 1e23 Timescale of strong force W boson 1 1 80.4 * 3e25 Z boson 0 1 91.2 * 3e25 Graviton 0 2 0 Higgs Boson 0 0 125.3 * 1.6e22 Proton +1 1/2 .9383 * * Neutron 0 1/2 .9396 * * 882All of the above particles are fundamental (not composed of smaller particles) except for the proton and neutron.
All particles with charge feel the electromagnetic force.
All particles feel the gravitational force.
Hadron: Particle composed of quarks, such as a proton, muon, and pion Baryon: Particle composed of 3 quarks, such as a proton and neutron Meson: Particle composed of a quark and an antiquark, such as a pion Lepton: Spin 1/2 particle that does not feel the strong force Bosom: Integer spin, such as {0, 1, 2, ...} Does not obey the Pauli exclusion principle Fermion: Halfinteger spin, such as {1/2, 3/2, ...} Obeys the Pauli exclusion principleThe masses of the quarks are not accurately known.
If no halflife is given, the particle is stable (as far as we know). As neutrinos propagate they can change to other kinds of neutrinos.
The length scale of the strong force is 10^{15} meters. The timescale of the strong force is the length scale divided by the speed of light, which is on the order of 1e23 seconds.
Quarks and gluons are never found in isolation. They are always bound together in the form of a meson (2 quarks) or a baryon (3 quarks), and their identities are continuously mixing.
Gluons interact with themselves and so they have a lifetime equal to the characteristic strong force timescale, 10^{23 seconds. }
Some theories predict that the proton is unstable, with timescales in the range of 10^{36} years. Experiments have found that the proton has a lifetime of at least 10^{34} years.
A blue line indicates that an interaction exists between the given particles. For example, an electron interacts with a photon. The Higgs interacts with all particles with positive rest mass.
Feynman diagrams are a way of illustrating the possible particle interactions.
kg/m^{3} Planck density 5 ⋅10^{96} = PlanckMass / PlanckLength^3 Solar system 2 ⋅10^{8} = Mass of sun / (30 AU)^3 Milky Way 3 ⋅10^{21} = 1.2e12 solar masses / (100000 lightyears)^3 Matter .12⋅10^{27} Mean density of protons & electrons in the universe Dark matter .66⋅10^{27} Mean density of dark matter in the universe Dark energy 1.67⋅10^{27} Mean density of dark energy in the universeAs the universe expands the matter and dark matter density decrease and the dark energy density is constant.
In the early universe the dark matter density was vastly greater than the dark energy density. In the future dark energy will overwhelm dark matter and the universe will expand unchecked.
0 seconds: Big bang Planck Epoch 10^{43} seconds: Grand unification epoch The temperature of the universe is 10^{15} GeV or 10^{27} K The strong, electromagnetic and weak forces are unified as the electronuclear force At this time, gravity separates from the electronuclear force 10^{36} seconds: Strong and electroweak forces separate Beginning of the inflationary epoch 10^{32} seconds: End of the inflationary epoch The inflationary phase transition heats the universe into a quarkgluon plasma The universe is hot enough for W and Z bosons (< 100 GeV) Physics is highly speculative before this epoch 10^{12} seconds: End of the electroweak epoch and beginning of the quark epoch The electroweak force separates into the electromagnetic and weak forces The universe is too cool for W, Z and Higgs bosons The universe is filled with quarks, leptons, and their antiparticles The universe is too hot for baryons or mesons 10^{6} seconds: End of the quark epoch and beginning of the hadron epoch The universe has equal amounts of hadrons and antihadrons 1 second: End of the hadron epoch and beginning of the lepton epoch Hadrons and antihadrons have annihilated, leaving behind only matter Leptons dominate the mass of the universe The universe has equal amounts of leptons and antileptons 2 seconds: Neutrinos decouple from other matter and form the cosmic neutrino background 10 seconds: End of the lepton epoch and beginning of the photon epoch Leptons and antileptons annihilate, leaving behind only leptons Photons dominate the energy of the universe Nucleosynthesis occurs 3 minutes: Nucleosynthesis ends The photon epoch continues The universe is a plasma of free electrons and ions The universe is opaque to photons 300,000 years: The cosmic "Dark age" The universe cools enough for nuclei to combine with electrons to form neutral atoms The universe becomes transparent to photons The photons generated at this time become the cosmic background radiation For the next billion years, the universe is a cold and dark gas 1 Billion years: End of the cosmic dark age. Stars and Galaxies form. Supernovae enrich the universe with heavy elements. 9.2 billion years: Formation of the Sun and the Earth. 9.9 billion years: Uranus and Neptune change places, causing the Earth to be bombarded with comets. 12 billion years: Oxygen appears in the Earth's atmosphere from photosynthesis. 13.1 billion years: Oxygen becomes a major constituent of the atmosphere. First appearance of complex multicellular life. 13.5 billion years: An anoxic extinction event wipes out 95 percent of species. Dinosaurs dominate hereafter. 13.6 billion years: The dinosaurs are wiped out by a 10 kilometer meteor. Mammals emerge hereafter. 10 thousand years ago: Emergence of civilization on the Earth 13.7 billion years: The present. 1 billion years from now: The sun increases in brightness and the Earth's oceans evaporate. 4 billion years from now: The Milky Way and Andromeda galaxies will collide. 5 billion years from now: The sun expands in a nova and consumes the Earth. Several billion years from now: Dark energy causes the universe to expand. Trillions of years from now: The last stars burn out. The universe is dark hereafter. The only remaining objects are white dwarfs, neutrons stars, black holes and cold planets orbiting dead stars. The Big Chill. 10^{40} years: Prootns decay. 10^{70} years: Black holes explode.
Tangible matter = Stuff that interacts by the strong and/or electromagnetic force, such as protons, neutrons, electrons, photons. These particles can be stopped by a meter of lead. Dark matter = Stuff that does not interact by the strong or electromagnetic force but interacts by the weak force and gravity. These particles easily pass through the Earth. Examples include neutrinos. Most of the dark matter in the universe consists of particles that have not yet been discovered. Dark energy = An energy density that has negative pressure.Tangible matter and dark matter have positive pressure and dark energy has negative pressure. All three have positive energy density.
Densities:
kg/m^{3} Planck density 5 *10^{96} = PlanckMass / PlanckLength^3 Black hole 1.8 *10^{19} = Density of a 1 solar mass black hole Neutron star 1 *10^{18} = Upper range for the density at the core Nuclear matter 2.3 *10^{17} = Density of a nucleus White dwarf 1 *10^{9} = White dwarf density Osmium 22.6 *10^{3} = Densest element Water 1 *10^{3} Air 1.22*10^{0} At sea level Solar system 2 *10^{8} = Mass of sun / (30 AU)^3 Milky Way 3 *10^{21} = 1.2e12 solar masses / (100000 lightyears)^3 Ordinary matter .12*10^{27} = Mean density of protons, neutrons, & electrons in the universe Dark matter .66*10^{27} = Mean density of dark matter in the universe Dark energy 1.67*10^{27} = Mean density of dark energy in the universe Sum 2.45*10^{27} = Total density of matter, dark matter, and dark energyAs the universe expands the matter and dark matter density decrease and the dark energy density is constant.
In the early universe the dark matter density was vastly greater than the dark energy density. In the future dark energy will overwhelm dark matter and the universe will expand unchecked.
The Earth's escape velocity is Ve=11.2 km/s. Suppose the Earth had no atmosphere and you launched a cannonball upward with velocity V.
V < Ve Elliptic The cannonball falls back to the Earth V > Ve Hyperbolic The cannonball escapes from the Earth and asymptotes to a positive velocity V = Ve Parabolic The cannonball is on the boundary between escape and falling back. It never returns to the Earth and it asymptotes to zero velocity.If the universe consisted entirely of ordinary matter and dark matter and no dark energy, then there is a critical value of the density such that the expansion of the universe is parabolic. This value is 2.45e27 kg/m^{3}.
d = Density of a parabolic universe = 2.45e27 kg/m^{3} D = 1.0 = Density of all matter and dark energy in the universe / d Dom = .049 = Fraction of ordinary matter in the universe / d Ddm = .27 = Fraction of dark matter in the universe / d Dde = .68 = Fraction of dark energy in the universe / dD, Dom, Ddm, and Dde are scaled relative to the parabolic density d.
If Dde = 0 then If D > 1 The density of the universe is large enough to reverse the expansion from the big bang and the universe collapses in a Big Crunch. The Hubble constant goes from positive to zero to negative. If D > 1 The universe expands forever, ending with a positive Hubble constant If D = 1 The universe stops expanding and the Hubble constant goes to zero. The universe ends in a "Big Chill".If Dde > 0 then dark energy trumps all of the above. If the universe survives long enough to avoid a Big Crunch then dark energy causes the universe to expand unchecked regardless of the matter density. The universe ends in a Big Chill.
Previous to 2005 we knew that the value of D was close to 1 and we couldn't tell if it was larger or smaller than 1. The value of the dark energy density was unknown and the fate of the universe was unknown. The theory of "Inflation" was developed to explain why D is close to 1.
In 2005 measurements of distant supernovae showed that Dde > 0, implying that the universe will end in a Big Chill.
In 2010 the Planck spacecraft measured the precise values of Dom, Ddm, and Dde.
In the plot, Omega_M = Dom + Ddm Omega_Lambda = Dde
You can illustrate the concept of escape velocity with the "My Solar System" simulation at phet.colorado.edu.
Mass Position Velocity X Y X Y Body 1 100. 0 0 0 0 Sun Body 2 1. 100 0 Vx Vy PlanetIf Vx=100 and Vy=0 the planet orbits the sun on a circular orbit. What do the orbits look like if you vary Vx?
The escape velocity from the sun at X=100 is Ve=100*Squareroot(2).
If Vx=0 and Vy > Ve the planet escapes. If Vx=0 and Vy < Ve the planet crashes into the sun.
Scale Constituents of matter (meters) 1 Materials, gases, chemicals, pizza 10^{9} Molecules 10^{10} Elements (hydrogen, helium, ...) 10^{14} Nuclei & electrons 10^{15} Protons, neutrons, pions, electrons, photons, neutrinos, dark matter 10^{16} Quarks, electrons, photons, gluons, neutrinos, W, Z, Higgs, dark matter
Number of Parameter parameters 6 Quark masses. Up, down, charm, strange, top, bottom 3 Lepton masses. Electron, muon, tau 3 Neutrino masses. Electron neutrino, muon neutrino, tau neutrino 1 Z mass. The W mass is determined by the Z mass. 1 Higgs mass 1 Electric force constant 1 Gravitational force constant 1 Strong force constant 1 Weak force constant 4 3 Neutrino mixing angles and one phase 6 Cosmological parameters. These are: Density of tangible matter in the universe (nuclei, electrons, etc) Density of dark matter in the universe Dark energy density (cosmological constant) Scalar spectral index of the universe Curvature fluctuation amplitude of the universe reionization optical depth of the universeDark matter is likely a source of new parameters, such as the masses of the dark matter particles.
The masses of the proton and neutron can be calculated from the masses of the up and down quark.
At present the origin of these parameters is unknown. Ideally, a future physics theory will explain the origin of the parameters based on a compact set of principles, and the number of parameters will decrease. This was the hope of Einstein. Since special relativity and general relativity can be generated from compact principles it was hoped that particle physics could as well, but at present no successful principles have been found.
An alternative to principles is The Multiverse, where there are multiple universes, each with different parameters, and we live in a universe where the parameters allow for the existence of intelligent life. This is similar to the Anthropic Principle.
If you change the parameters of the universe there are extreme consequences. For example,
If the electron mass increases, protons will consume the electrons to produce neutrons, leaving behind a boring universe with no nuclei.
If the dark energy density is increased, the universe expands too fast for galaxies to form.
If you increase the electric force relative to the strong force then nuclei can't form.
Possibilities for the laws of physics:
Natural The parameters of the universe will be found to originate from compact principles and no fine tuning is required for life. Fine tuned The parameters of physics require fine tuning to be amenable to life. Intelligent design The parameters of the universe were designed to be amenable to life. Multiverse There are multiple universes with different laws of physics, most of them dull and lifeless, but the probability of one of them accommodating life is unity. Fortuitous There is only one universe. The laws of physics require fine tuning and we are lucky that they are amenable to life. Matrix The universe is a computer simulation. Darwin universe Universes beget universes and the laws of physics evolve by natural selection.
An analogue of The Multiverse is the Drake equation.
P = Probability that a star has a planet with intelligent life N = Number of stars in the universe. If: P ~ 1 Life is natural P << 1 Life requires fine tuning P << 1 and PN >> 1 Life requires fine tuning but life is probable in the universe. This is the "Anthropic principle" or the "Multiverse" scenario. P << 1 and PN << 1 Life is improbable in the universe. Either life is "Lucky" or we live in The Matrix or the Earth was intelligently designed.
The rules of soccer are "natural" in the sense that they flow from a single premise (don't use your arms) and they lead to a rich game. The rules of football are "unnatural" in that the rulebook is thick and it takes a squad of referees to enforce.
The notes for 12tone equal temperament coincide well with the note of just intonaton.
The most resonant notes in the 12tone equal temperament scale are the fourth and the fifth and these are particularly close to their justintonation counterparts.
The frequency ratio between a fourth and a fifth in justtemperament is
R = (3/2) / (4/3) = 9/8 = 1.125In a 12tone equaltempered scale the frequency ratio of a whole step is
R = 2^{(2/12)} = 1.122which is nearly the same as the ratio between a fourth and a fifth. This is why the 12tone scale works so well. If you try any number other than 12 it doesn't work. This is why the 12tone scale is the most useful for writing harmony.
Tunings exist that use numbers different from 12, such as for Indian, Thai, and Arabic music. These tunings can generate exotic melodic structure but they are less useful for harmony than the 12tone scale.
The 12tone scale is natural in the sense that it doesn't have any "free parameters". The choice of the number "12" emerged naturally from the positions of the resonant notes. It is also "fortuitous" in that the values of Z are so small.
Soccer is an example of a "natural sport". The rules are simple and if you change the parameters (such as field size, number of players, etc) the game is essentially the same.
American football requires "fine tuning". In order for the sport to make sense you need a large rulebook. It also has lots of "free parameters" because there are many different ways the rules could be constructed.
The chess player Edward Lasker once said:
"While the Baroque rules of Chess could only have been created by humans, the rules of Go are so elegant, organic, and rigorously logical that if intelligent life forms exist elsewhere in the universe, they almost certainly play Go."
The rules of chess are an example of "fine tuning" and there are lots of free parameters (the moves allowed by each piece).
An alien planet could conceivably have formed as early as 1 billion years after the big bang, meaning that there are likely aliens with a head start on us by billions of years.
An alien civilization could easily build a fission or fusion rocket that travels at 1/10 the speed of light, which would take 1 million years to cross the galaxy. The aliens have plenty of time to get here.
Millions of years ago Big bang 13700 First planets formed 13000 Earth formed 4500 Photosynthesis 3000 Oxygen atmosphere 600 Multicellular life 600 Vertebrates 480 Tetrapod vertebrates 400 Mammals, birds, and reptiles are all tetrapods Mammals 170 Dinosaur extinction 66 Cats 25 Cheetahs 6 Fastest land animal Tigers 1.8 Humans 1 Lions .9 Agriculture .01 Civilization .005 Calculus .0004 Smartphones .00001
We presently possess the technology to build a fission and fusion rocket, each of which can reach a speed of .1 times the speed of light, and such a rocket can cross the Milky Way galaxy in a time that is a small fraction of the age of the universe. If aliens had built such a rocket they could easily have already colonized the galaxy.
Speed of light = C Speed of a fission or fusion rocket = V = .1 C Diameter of the Milky Way = X = .1 million light years Time to cross the galaxy = T = X/V = 1 million years Age of the universe = 13800 million light years
The divinity hypothesis becomes persuasive if there is a physical mechanism allowing it to happen, a mechanism that obeys the known laws of physics. Such a mechanism exists. An advanced alien civilization is equivalent to a diety.