Review of Basic Physics Review of Basic Physics BackgroundBackground

Basic physical quantities & unitsBasic physical quantities & units• Unit prefixesUnit prefixes• Basic quantitiesBasic quantities• Units of measurementUnits of measurement• Planck unitsPlanck units• Physical constantsPhysical constants

Unit PrefixesUnit Prefixes• See See http://www.bipm.fr/enus/3_SI/si-prefixes.hhttp://www.bipm.fr/enus/3_SI/si-prefixes.htmltml

for the official international standard unit prefixes.for the official international standard unit prefixes.• When measuring physical things, these prefixes When measuring physical things, these prefixes

always stand for powers of 10always stand for powers of 1033 (1,000). (1,000).• But,But, when measuring digital things (bits & bytes) they when measuring digital things (bits & bytes) they

often stand for powers of 2often stand for powers of 21010 (1,024). (1,024).– See also alternate kibi, mebi, See also alternate kibi, mebi, etc.etc. system at system at http://http://physics.nist.gov/cuu/Units/binary.htmlphysics.nist.gov/cuu/Units/binary.html

• Don’t get confused!Don’t get confused!

Three “fundamental” quantitiesThree “fundamental” quantities

QuantityTypicalsymbols Some Units Planck Units

position,length,distance,radius

x, L, , d, r m, Å, in, ft,yd, mi, au,ly, pc

LP = (G/c3)1/2

= 1.610 35 m

time t, T yr, hr, sec TP = (G/c5)1/2

= 5.410 44 s

mass m, M g, lb, amu MP = (c/G)1/2

= 22 g

Some derived quantitiesSome derived quantitiesQuantity

TypicalSymbols

SomeUnits

SomeFormulas Dimensions

area A acre L2

volume V liter L3

frequency f Hz 1/Tvelocity v c v=dxdt L/Tmomentum p p=mv ML/Tangularmomentum

L L=pd ML2/T

acceleration a a=dvdt L/T2

force F N F=ma ML/T2

energy, work,heat, torque

U, E, W,G, H,

J, eV W=FdE=mc2

ML2/T2

power P W P=dEdt ML2/T3

pressure,energy density

p, E Pa,atm,psi

p=F/A

E=E/VM/LT2

Electrical QuantitiesElectrical Quantities

QuantityTypicalSymbols

SomeUnits

SomeFormulas Dimensions

charge Q, q C, qe Qcurrent I, i A i=dqdt Q/Tvoltage V,v V V=U/Q E/Qelectric fieldstrength

E V/m E=V/d F/Q

currentdensity

J J=I/A Q/TL2

resistance R R=V/I ET/Q2

capacitance C F C=dq/dv Q2/Einductance L H L=E/(di/dt)

• We’ll skip magnetism & related quantities this semester.

Information, Entropy, Information, Entropy, TemperatureTemperature

• These are important physical quantities alsoThese are important physical quantities also• But, are different from other physical quantitiesBut, are different from other physical quantities

– Based on statistical correlationsBased on statistical correlations

• But, we’ll wait to explain them till we have a But, we’ll wait to explain them till we have a whole lecture on this topic later.whole lecture on this topic later.

• Interestingly, there have been attempts to Interestingly, there have been attempts to describe all physical quantities & entities in describe all physical quantities & entities in terms of information (e.g., Frieden, Fredkin).terms of information (e.g., Frieden, Fredkin).

Unit definitions & conversionsUnit definitions & conversions• See See http://www.cise.ufl.edu/~mpf/physlim/units.txthttp://www.cise.ufl.edu/~mpf/physlim/units.txt

for definitions of the above-mentioned units, for definitions of the above-mentioned units, and more. (Source: and more. (Source: Emacs calcEmacs calc software.) software.)

• Many mathematics applications have built-in Many mathematics applications have built-in support for physical units, unit prefixes, unit support for physical units, unit prefixes, unit conversions, and physical constants.conversions, and physical constants.– Emacs calc package (by Dave Gillespie)Emacs calc package (by Dave Gillespie)– MathematicaMathematica– Matlab - ?Matlab - ?– Maple - ?Maple - ?

Fundamental physical constantsFundamental physical constants• Speed of light Speed of light c c = 299,792,458 m/s= 299,792,458 m/s• Planck’s constant Planck’s constant h h = 6.6260755×10= 6.6260755×103434 J s J s

– Reduced Planck’s constant Reduced Planck’s constant = = hh / 2 / 2• hh : circle :: : circle :: : radian : radian

• Newton’s gravitational constant Newton’s gravitational constant GG = = 6.67259×106.67259×101111 N m N m22 / kg / kg

• Others: permittivity of free space, Boltzmann’s Others: permittivity of free space, Boltzmann’s constant, Stefan-Boltzmann constant to be constant, Stefan-Boltzmann constant to be introduced later as we go along.introduced later as we go along.

Physics you should already knowPhysics you should already know• Basic Newtonian mechanicsBasic Newtonian mechanics

– Newton’s laws, motion, energy, etc.Newton’s laws, motion, energy, etc.

• Basic electrostaticsBasic electrostatics– Ohm’s law, Kirchoff’s laws, etc.Ohm’s law, Kirchoff’s laws, etc.

• Also helpful, but not prerequisite (we’ll Also helpful, but not prerequisite (we’ll introduce them as we go along):introduce them as we go along):– Basic statistical mechanics & thermodynamicsBasic statistical mechanics & thermodynamics– Basic quantum mechanicsBasic quantum mechanics– Basic relativity theoryBasic relativity theory

Generalized Classical MechanicsGeneralized Classical Mechanics

Generalized MechanicsGeneralized Mechanics• Classical mechanics can be expressed most Classical mechanics can be expressed most

generally and concisely in the generally and concisely in the LagrangianLagrangian and and HamiltonianHamiltonian formulations. formulations.

• Based on simple functions of the system state:Based on simple functions of the system state:– Lagrangian: Kinetic minus potential energy.Lagrangian: Kinetic minus potential energy.– Hamiltonian: Kinetic plus potential energy.Hamiltonian: Kinetic plus potential energy.

• The dynamical laws can be derived from either The dynamical laws can be derived from either energy function.energy function.

• This framework generalizes to quantum This framework generalizes to quantum mechanics, quantum field theories, mechanics, quantum field theories, etc.etc.

Euler-Lagrange EquationEuler-Lagrange Equation

Where:Where:• LL((qqii, , vvii) is the system’s ) is the system’s LagrangianLagrangian function. function.

• qqii : :≡≡ Generalized position coordinate indexed Generalized position coordinate indexed ii..

• vvii : :≡ Velocity of generalized coordinate ≡ Velocity of generalized coordinate ii,,

• (as appropriate)(as appropriate)• tt :≡ Time coordinate :≡ Time coordinate

– In a given frame of reference.In a given frame of reference.

.

ii v

L

q

L.

ii v

L

q

L.

ii v

L

q

L

ii v

L

q

L

ii qv :tftff /or d/d:

ii pF or just

Note the over-dot!

Euler-Lagrange exampleEuler-Lagrange example• Let Let qq = = qqii be the ordinary be the ordinary xx, , yy, , zz coordinates of a coordinates of a

point particle with mass point particle with mass mm..• Let Let LL = = ½½mvmvii

22 − − VV((qq). (Kinetic minus potential.)). (Kinetic minus potential.)

• Then, ∂Then, ∂LL/∂/∂qqii = − ∂ = − ∂VV/∂/∂qqii = = FFii

– The force component in direction The force component in direction ii. .

• Meanwhile, ∂Meanwhile, ∂LL/∂/∂vvii = = ∂(½∂(½mvmvii22)/∂)/∂vvii = mv = mvii = p = pii

– The momentum component in direction The momentum component in direction ii..

• And,And,– Mass times acceleration in direction Mass times acceleration in direction ii..

• So we get So we get FFii = = mamaii (Newton’s 2 (Newton’s 2ndnd law) law)

iiiiii mavmmvpvLt

)()/)(/(

Least-Action PrincipleLeast-Action Principle• The The actionaction of an energy means the integral of of an energy means the integral of

that energy over time.that energy over time.• The trajectory specified by the Euler-Lagrange The trajectory specified by the Euler-Lagrange

equation is one that locally extremizes the equation is one that locally extremizes the action of the Lagrangian:action of the Lagrangian:– Among trajectories Among trajectories ss((tt))

between specified pointsbetween specified pointsss((tt00) and ) and ss((tt11).).

• Infinitesimal deviations from this trajectory Infinitesimal deviations from this trajectory leave the action unchanged to 1leave the action unchanged to 1stst order. order.

1

0

d)(t

t

tsLA

A.k.a.Hamilton’sprinciple

Hamilton’s EquationsHamilton’s Equations• The Hamiltonian is defined as The Hamiltonian is defined as HH : :≡ ≡ vviippii − − LL..

– Equals Equals EEkk + + EEpp if if LL = = EEkk − − EEpp and and vviippii = 2 = 2EEkk = = mvmvii22..

• We can then describe the dynamics of (We can then describe the dynamics of (qqii, , ppii) states ) states using the 1using the 1stst-order -order Hamilton’s equationsHamilton’s equations::

• These are equivalent to but often easier to solve than These are equivalent to but often easier to solve than the 2the 2ndnd-order Euler-Lagrange equation.-order Euler-Lagrange equation.

• Note that any Hamiltonian dynamics is Note that any Hamiltonian dynamics is bideterministicbideterministic– Meaning, deterministic in Meaning, deterministic in bothboth the forwards and reverse the forwards and reverse

time directions.time directions.

qHp

pHq

/

/

Implicitsummationover i.

Field TheoriesField Theories• Space of indexes Space of indexes ii is continuous, thus is continuous, thus

uncountable. A topological space uncountable. A topological space TT, , e.g.e.g., , RR33..• Often use Often use φφ((xx) notation in place of ) notation in place of qqii..• In In locallocal field theories, the Lagrangian field theories, the Lagrangian LL((φφ) is ) is

the integral of a the integral of a Lagrange density function Lagrange density function ℒℒ((xx) ) over the entire space over the entire space TT..

• This This ℒℒ((xx) depends only ) depends only locallylocally on on φφ, , e.g.e.g.,, ℒℒ((xx) = ) = ℒℒ((φφ((xx), (∂), (∂φφ/∂/∂xxii)()(xx), (), (xx)) ))

• All successful physical theories can be All successful physical theories can be explicitly written down as local field theories!explicitly written down as local field theories!– There is no instantaneous action at a distance.There is no instantaneous action at a distance.

Special Relativity and the Special Relativity and the Speed-of-Light LimitSpeed-of-Light Limit

The Speed-of-Light LimitThe Speed-of-Light Limit• No form of information No form of information (including quantum (including quantum

information)information) can propagate through space at a velocity can propagate through space at a velocity (relative to its local surroundings)(relative to its local surroundings) that is greater than that is greater than the speed of light, the speed of light, cc, ~3×10, ~3×1088 m/s. m/s.

• Some consequences:Some consequences:– No closed system can propagate faster than No closed system can propagate faster than cc..

• Although you can Although you can definedefine open systems that do by definition open systems that do by definition– No given piece of matter, energy, or momentum can No given piece of matter, energy, or momentum can

propagate faster than propagate faster than cc. . – All of the fundamental forces All of the fundamental forces (including gravity)(including gravity) propagate propagate

at at (at most)(at most) cc..– The probability mass that is associated with a quantum The probability mass that is associated with a quantum

particle flows in an entirely local fashion, no faster than particle flows in an entirely local fashion, no faster than cc. .

Early History of the LimitEarly History of the Limit• The principle of locality was anticipated by NewtonThe principle of locality was anticipated by Newton

– He wished to get rid of the “action at a distance” aspects of his law of He wished to get rid of the “action at a distance” aspects of his law of gravitation.gravitation.

• The finiteness of the speed of light was first observed by The finiteness of the speed of light was first observed by Roemer in 1676.Roemer in 1676.– The first decent speed estimate was obtained by Fizeau in 1849.The first decent speed estimate was obtained by Fizeau in 1849.

• Weber & Kohlrausch derived a velocity of Weber & Kohlrausch derived a velocity of cc from empirical from empirical electromagnetic constants in 1856.electromagnetic constants in 1856.– Kirchoff pointed out the match with the speed of light in 1857.Kirchoff pointed out the match with the speed of light in 1857.

• Maxwell showed that his EM theory implied the existence of Maxwell showed that his EM theory implied the existence of waves that always propagate at waves that always propagate at cc in 1873. in 1873.– Hertz later confirmed experimentally that EM waves indeed existedHertz later confirmed experimentally that EM waves indeed existed

• Michaelson & Morley (1887) observed that the SoL was Michaelson & Morley (1887) observed that the SoL was independent of the observer’s state of motion!independent of the observer’s state of motion!– Maxwell’s equations apparently valid in all inertial reference frames!Maxwell’s equations apparently valid in all inertial reference frames!– Fitzgerald (1889), Lorentz (1892,1899), Larmor (1898), PoincarFitzgerald (1889), Lorentz (1892,1899), Larmor (1898), Poincaréé

(1898,1904), & Einstein (1905) explored the implications of this...(1898,1904), & Einstein (1905) explored the implications of this...

Relativity: Non-intuitive but TrueRelativity: Non-intuitive but True• How can the How can the speedspeed of something be a of something be a

fundamental constant? Seemed broken...fundamental constant? Seemed broken...– If I’m moving at velocity If I’m moving at velocity vv towards you, and I shoot towards you, and I shoot

a laser at you, what speed does the light go, relative a laser at you, what speed does the light go, relative to me, and to you?to me, and to you? Answer: both Answer: both cc!! ((NotNot vv++cc.).)

• Newton’s laws were the same in all Newton’s laws were the same in all frames of frames of referencereference moving at a constant velocity. moving at a constant velocity.– Principle of Relativity (PoR): Principle of Relativity (PoR): All laws of physics All laws of physics

are invariant under changes in velocityare invariant under changes in velocity

• Einstein’s insight: The PoR Einstein’s insight: The PoR isis consistent w. consistent w. Maxwell’s theory! Maxwell’s theory! Change def. of space+time.Change def. of space+time.

Some Consequences of RelativitySome Consequences of Relativity• Measured Measured lengthslengths and and time intervalstime intervals in a system in a system

vary depending on the system’s velocity relative to vary depending on the system’s velocity relative to observers.observers.– Lengths are shortened in direction of motion.Lengths are shortened in direction of motion.– Clocks run slower.Clocks run slower.

• Sounds paradoxical, but isn’t!Sounds paradoxical, but isn’t!– Mass is amplified.Mass is amplified.

• Energy and mass are the same quantity measured in Energy and mass are the same quantity measured in different units: different units: EE==mcmc22..

• Nothing (incl. energy, matter, information, Nothing (incl. energy, matter, information, etc.etc.) can ) can go faster than light! (SoL limit.)go faster than light! (SoL limit.)

Three Ways to Understand Three Ways to Understand cc limitlimit

• Energy of motion contributes to mass of object.Energy of motion contributes to mass of object.– Mass approaches Mass approaches as velocity as velocitycc..– Infinite energy needed to reach Infinite energy needed to reach cc..

• Lengths, times in a faster-than-light moving Lengths, times in a faster-than-light moving object would become imaginary numbers!object would become imaginary numbers!– What would that mean?What would that mean?

• Faster than light in one reference frame Faster than light in one reference frame Backwards in time in another reference frameBackwards in time in another reference frame– Sending info. backwards in time violates causality, Sending info. backwards in time violates causality,

leads to logical contradictions!leads to logical contradictions!

The The cc limit in quantum physics limit in quantum physics• Sometimes you see statements about “nonlocal” Sometimes you see statements about “nonlocal”

effects in quantum systems. effects in quantum systems. Watch out!Watch out!– Even Einstein made this mistake.Even Einstein made this mistake.

• Described a quantum thought experiment that seemed to require Described a quantum thought experiment that seemed to require “spooky action at a distance.”“spooky action at a distance.”

• Later it was shown that this experiment did Later it was shown that this experiment did notnot actually violate actually violate the speed-of-light limit for information.the speed-of-light limit for information.

• These “nonlocal” effects are only illusions, emergent These “nonlocal” effects are only illusions, emergent phenomena predicted by an entirely phenomena predicted by an entirely locallocal underlying underlying theory respecting SoL limit..theory respecting SoL limit..– Widely-separated systems can maintain quantum Widely-separated systems can maintain quantum

correlationscorrelations, but that isn’t true non-locality., but that isn’t true non-locality.

The “Lorentz” TransformationThe “Lorentz” Transformation• Lorentz, PoincarLorentz, Poincaré: All the laws of physics é: All the laws of physics

remain unchanged relative to the reference remain unchanged relative to the reference frame (frame (xx′,′,tt′) of an object moving with constant ′) of an object moving with constant velocity velocity v = v = ΔΔxx//ΔΔtt in another reference frame in another reference frame ((xx,,tt) under the following conditions:) under the following conditions:

/)/(

/)(

cxtt

vtxx

Where:

21:

/:

cv

Actually it was written down earlier; e.g., one form by Voigt in 1887

Note: our γ here is the reciprocal of the quantity denoted γ by other authors.

Consequences of Lorentz TransformConsequences of Lorentz Transform• Length contraction (Fitzgerald, 1889, Lorentz 1892):Length contraction (Fitzgerald, 1889, Lorentz 1892):

– An object having length An object having length in its rest frame appears, when in its rest frame appears, when measured in a relatively moving frame, to have the (shorter) measured in a relatively moving frame, to have the (shorter) length length γγ. (For lengths parallel to direction of motion.). (For lengths parallel to direction of motion.)

• Time dilation (PoincarTime dilation (Poincaréé, 1898):, 1898):– If time interval If time interval ττ is measured between two co-located events is measured between two co-located events

in a given frame, a larger time in a given frame, a larger time t = t = ττ//γγ will be measured will be measured between those events in a relatively moving frame.between those events in a relatively moving frame.

• Mass expansion (Einstein’s fix for Newton’s Mass expansion (Einstein’s fix for Newton’s FF==mama))::– If an object has mass If an object has mass mm00 in its rest frame, then it is seen to in its rest frame, then it is seen to

have the larger mass have the larger mass m m = = mm00//γγ in a relatively moving frame. in a relatively moving frame.

Light-like

Isochrones(space-like)

Isospatials(time-like)

In this example:v = Δx/Δt = 3/5γ = Δt′/Δt = 4/5vT = v/γ = Δx/Δt′ = 3/4

Lorentz Transform VisualizationLorentz Transform Visualization

Line colors:

x

t

x′

t′ Original x,t(“rest”) frame

New x′,t′(“moving”) frame

x′=0

t′=0

The “tourist’s velocity.”

An Alternative View: Mixed FramesAn Alternative View: Mixed Frames

t′

x

t

x′

x

tStandardFrame #1

StandardFrame #2

x′

t′

x

t′

t

x′

MixedFrame #1

MixedFrame #2

In this example:v = Δx/Δt = 3/5vT = Δx/Δt′ = 3/4 γ = Δt′ /Δt = 4/5

Note that (Δt)2 = (Δx)2 + (Δt′)2

by the PythagoreanTheorem!

Note the obvious complete symmetryin the relation between the two mixed frames.

(Light pathsshown ingreen here.)

Relativistic Kinetic EnergyRelativistic Kinetic Energy• Total relativistic energy Total relativistic energy EE of any object is of any object is EE = = mcmc22..• For an object at rest with mass For an object at rest with mass mm00, , EErestrest = = mm00cc22..

• For a moving object, For a moving object, mm = = mm00//γγ– Where Where mm00 is the object’s mass in its rest frame. is the object’s mass in its rest frame.

• Energy of the moving object is thus Energy of the moving object is thus EEmovingmoving = = mm00cc22//γγ..

• Kinetic energy Kinetic energy EEkinkin :≡ :≡ EEmovingmoving − − EErestrest

= = mm00cc22//γγ − − mm00cc22 = = mm00cc22(1 − (1 − γγ))

• Substituting Substituting γγ = (1− = (1−ββ22))1/21/2 and Taylor-expanding gives: and Taylor-expanding gives:)( 61654

832

21

restkin EE

Pre-relativistic kinetic energy ½ m0v2

Higher-orderrelativistic corrections

Spacetime IntervalsSpacetime Intervals• Note that the lengths and times between two events are Note that the lengths and times between two events are not not

invariant under Lorentz transformations.invariant under Lorentz transformations.• However, the following quantity However, the following quantity isis an invariant: an invariant:

The The spacetime interval sspacetime interval s, where:, where:ss22 = ( = (ctct))22 − − xxii

22

• The value of The value of ss is also the is also the proper timeproper time ττ::– The elapsed time in rest frame of object traveling on a straight line The elapsed time in rest frame of object traveling on a straight line

between the two events. (Same as what we were calling between the two events. (Same as what we were calling tt′ earlier.)′ earlier.)• The sign of The sign of ss22 has a particular significance: has a particular significance:

ss22 > 0 - Events are > 0 - Events are timelike separated timelike separated ((ss is real) is real)May be causally connected.May be causally connected.

ss22 = 0 - Events are = 0 - Events are lightlike separated lightlike separated ((ss is 0) is 0)Only 0-rest-mass signals may connect them.Only 0-rest-mass signals may connect them.

ss22 < 0 - Events are < 0 - Events are spacelike separated spacelike separated ((ss is imaginary) is imaginary)Not causally connected at all.Not causally connected at all.

Relativistic MomentumRelativistic Momentum• The relativistic momentum The relativistic momentum pp = = mmvv

– Same as classical momentum, except that Same as classical momentum, except that mm = = mm00//γγ..

• Relativistic energy-momentum-rest-mass relation:Relativistic energy-momentum-rest-mass relation:EE22 = ( = (ppcc))22 + ( + (mm00cc22))22

If we use units where If we use units where cc = 1, this simplifies to just: = 1, this simplifies to just: EE22 = = pp22 + + mm00

22

• Note that if we solve for Note that if we solve for mm0022, we get:, we get:

mm0022 = = EE22 − − pp22

• This is another relativistic invariant! This is another relativistic invariant! – Later we will show how it relates to the spacetime interval Later we will show how it relates to the spacetime interval

ss22 = = tt22 − − xx22, and to a computational interpretation of physics., and to a computational interpretation of physics.