Formula Sheet: First midterm 2203, Fall Formula Sheet: First midterm 2203, Fall 2012 1) Relativity:...

download Formula Sheet: First midterm 2203, Fall Formula Sheet: First midterm 2203, Fall 2012 1) Relativity: space and time Transforms from S to S’: : γ = 1/(1 v o 2 /c)1/2 v o in along

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Transcript of Formula Sheet: First midterm 2203, Fall Formula Sheet: First midterm 2203, Fall 2012 1) Relativity:...

  • 1

    FormulaSheet:Firstmidterm2203,Fall2012

    1) Relativity:spaceandtimeTransformsfromStoS:: =1/(1vo2/c2)1/2voinalongthexaxis.x=(xvot) vx=(vxvo)/(1vxvo/c2)y=y vy= vy/(1vxvo/c2)z=z vz= vz/(1vxvo/c2)z=zt=(t(vo/c2)x)ChangesigntogofromStoS:Timedilationt=t0:t0Propermeasurementoftime.Lengthcontraction=L0/ :L0propermeasurementoflength.2) RelativisticMechanics

    E = [E(vo/c)(pc)] p=[p(vo/c2)/E]p=mvpc=mc2(v/c) EK(kineticenergy)=mc2(1)F=dp/dt E(totalenergy)=mc2

    E2=p2c2+(mc2)2: E2 mc2( )2 E=pcformasslessparticlev=pc2/E E=EK+mc2

    RelativisticDopplershiftf/f=[1(v/c)cos] orf/f=[(1v/c)/(1+v/c)]1/2(for=0o)

    3) AtomsNotation

    ZAZN WhereA=Z+NwithZbeingthenumberofprotonsandNthenumberof

    neutrons.AnexampleisCarbonwith6neutronsand6protons

    612C6

    KinetictheorypV=nRTorpV=NkBT,withkBtheBoltzmannsconstantandRtheuniversalgasconstant. kB=R/NANAisAvogadrosnumber

    averagekineticenergy

    12m v 2 = 3

    2kBT

    4) Quantizationoflight:

    PhotonhasenergyhfE=hf,

    E = ,with

    = 2f E=hc/ =1240/eVnm E=pc Photoelecticeffect ComptonScatteringKmax=hfe =c(1cos) eisworkfunction c=h/mec=0.00246nm

    Groupvelocity

    vg =ddk

    ,Phasevelocity

    v p =c 2

    vand

    v = vg

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    BraggScattering2dsin=n 5) QuantizationofAtomicEnergylevels:BohrAtom

    Rydbergseriesseries

    1

    = R[ 1n2 f

    1n2i] ,iistheinitialstateandfthefinalstate:Ris

    Rydbergconstant.BohrModel:Theangularmomentumisquantized

    L = mvr = n or

    2rn = n yieldingthefollowingequations:

    rn =n22

    ke2mwheretheBohrorbitradiusisdefined

    a0 =2

    ke2m= 0.05292nm

    En = m2n2

    (ke2

    )2 giving

    En = 13.6eVn2

    velocity:

    vn =ke2

    n

    Forarealsystemmshouldbethecenterofmassmwhere

    m'=mMm +M

    =M

    1+ M /m

    Then

    r'n =n22

    ke 2m'= n2

    mem'

    a0 and

    E 'n = m'2n2

    ( ke2

    )2 = m'

    me

    E1n2

    Hydrogenlikeions:oneelectronboundtoZenucleus:

    vn =ke2

    n,

    En = mZ 2

    2n2(ke

    2

    )2 ,

    rn =n22

    kZe2m

    6) Particlesaswaves

    p=h/ or=h/p

    ke2 =1.44nm(eV )

    =hc2mc2K

    =1240eV inm2mc2K

    deBrogliewavelength=h/p

    Uncertaintyprinciple

    kx 1/2 and

    t 1/2 whichcanalsobewrittenas

    px /2 and

    Et /2

    7) SchrdingerEquationinoneDimensionGeneralpropertiesofSchrdingersEquation:QuantumMechanics

    SchrdingerEquation(timedependent)

    2

    2m 2x 2

    +U = it

    Standingwave

    (x, t) = (x)eit

    SchrdingerEquation(timeindependent)

    2

    2m 2x2

    +U = E

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    Normalization(onedimensional) * (x,t)(x,t)dx =1

    +

    ForaconstantpotentialE>U

    (x) = Aeikx + Be+ikx with

    k = 2m(E U)2

    E

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    Constants

    c=2.998x10+8m/s h=6.626x1034J.sec=4.136x1015eV.sec

    =1.055x1034 J.s = 6.582x1016eV .s

    k =140

    = 8.988x109N .m 2 /C2

    me=9.109xx1031kg mec2=0.511MeVmp=1.673x1027kg mpc2=938.28MeVmn=1.675x1027kg mnc2=939.57MeVmp= 1836me mn=1839me 1u=931.5MeV/c2 nm=109me=1.6x1019coul kB=8.617x105eV/KeV=1.6x1019J NA=6.022x1023objects/molebinomialexpansion:(1+x)n=1+nx+n(n1)x2/2!+n(n1)(n2)x3/3!m=106m,nm=109m,pm=1012m,fm=1015mK=103,M=106,G=109

    Usefulrelationships:

    hc = 1240eV inmc = 197eV inmke2 = 1.44eV inmke2

    c=1137

    R=m kee( )2

    4c3=mc2 kee( )2

    4 c( )3

    R = 0.011nm1