Formula Sheet: First midterm 2203, Fall 2012 Formula Sheet: First midterm 2203, Fall 2012 1)...
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1 Formula Sheet: First midterm 2203, Fall 2012 1) Relativity: space and time Transforms from S to S’:: γ = 1/(1- vo 2 /c 2 ) 1/2 vo in along the x axis. x’ = γ(x – vot) v’ x = (vx –vo)/(1 – vxvo/c 2 ) y’ =y v’ y = vy/γ (1 – vxvo/c 2 ) z’ =z v’ z = vz/γ (1 – vxvo/c 2 ) z’ =z t’ = γ (t – (vo/c 2 )x) Change sign to go from S’ to S: Time dilation Δt=γΔt0: Δt0 Proper measurement of time. Length contraction Λ=ΔL0/γ :L0 proper measurement of length. 2) Relativistic Mechanics E’ = γ [E – (vo/c)(pc)] p’ = γ [p – (vo/c 2 )/E] p= γ mv pc= γ mc 2 (v/c) EK(kinetic energy) = mc 2 (γ –1) F=dp/dt E (total energy) = γ mc 2 E 2 =p 2 c 2 + (mc 2 ) 2 : E 2 − mc 2 ( ) 2 E=pc for massless particle v = pc 2 /E E=EK + mc 2 Relativistic Doppler shift f’/f = γ [1 – (v/c)cosθ] or f’/f = [(1‐v/c)/(1+v/c)] 1/2 (for θ =0 o ) 3) Atoms Notation Z A Z N Where A=Z+N with Z being the number of protons and N the number of neutrons. An example is Carbon with 6 neutrons and 6 protons 6 12 C 6 Kinetic theory pV=nRT or pV=NkBT, with kB the Boltzmann’s constant and R the universal gas constant. kB=R/NA NA is Avogadro’s number average kinetic energy 1 2 mv 2 = 3 2 k B T 4) Quantization of light: Photon has energy hf E = hf, E = ω , with ω = 2πf E = hc/λ = 1240/λ eV nm E=pc Photoelectic effect Compton Scattering Kmax = hf –eφ λ’ ‐ λ = λc(1 – cos θ) eφ is work function λc = h/mec = 0.00246 nm Group velocity v g = dω dk , Phase velocity v p = c 2 v and v = v g
Transcript of Formula Sheet: First midterm 2203, Fall 2012 Formula Sheet: First midterm 2203, Fall 2012 1)...
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FormulaSheet:Firstmidterm2203,Fall2012
1) Relativity:spaceandtimeTransformsfromStoS’::γ =1/(1vo2/c2)1/2voinalongthexaxis.x’=γ(x–vot) v’x=(vx–vo)/(1–vxvo/c2)y’=y v’y= vy/γ(1–vxvo/c2)z’=z v’z= vz/γ(1–vxvo/c2)z’=zt’=γ(t–(vo/c2)x)ChangesigntogofromS’toS:TimedilationΔt=γΔt0:Δt0Propermeasurementoftime.LengthcontractionΛ=ΔL0/γ :L0propermeasurementoflength.2) RelativisticMechanics
E’ =γ [E–(vo/c)(pc)] p’=γ[p–(vo/c2)/E]p=γmvpc=γmc2(v/c) EK(kineticenergy)=mc2(γ–1)F=dp/dt E(totalenergy)=γmc2
E2=p2c2+(mc2)2: E2 − mc2( )2 E=pcformasslessparticle
v=pc2/E E=EK+mc2
RelativisticDopplershiftf’/f=γ[1–(v/c)cosθ] orf’/f=[(1‐v/c)/(1+v/c)]1/2(forθ=0o)
3) AtomsNotation
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ZAZN WhereA=Z+NwithZbeingthenumberofprotonsandNthenumberof
neutrons.AnexampleisCarbonwith6neutronsand6protons
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612C6
KinetictheorypV=nRTorpV=NkBT,withkBtheBoltzmann’sconstantandRtheuniversalgasconstant. kB=R/NANAisAvogadro’snumber
averagekineticenergy
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12m v 2 =
32kBT
4) Quantizationoflight:
PhotonhasenergyhfE=hf,
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E = ω ,with
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ω = 2πf E=hc/λ =1240/λeVnm E=pc Photoelecticeffect ComptonScatteringKmax=hf–eφ λ’‐ λ=λc(1–cosθ) eφisworkfunction λc=h/mec=0.00246nm
Groupvelocity
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vg =dωdk
,Phasevelocity
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v p =c 2
vand
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v = vg
2
BraggScattering2dsinθ=nλ 5) QuantizationofAtomicEnergylevels:BohrAtom
Rydbergseriesseries
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1λ
= R[ 1n2 f
−1n2i] ,iistheinitialstateandfthefinalstate:Ris
Rydbergconstant.BohrModel:Theangularmomentumisquantized
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L = mvr = n or
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2πrn = nλ yieldingthefollowingequations:
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rn =n22
ke2mwheretheBohrorbitradiusisdefined
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a0 =2
ke2m= 0.05292nm
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En = −m2n2
(ke2
)2 giving
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En = −13.6eVn2
velocity:
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vn =ke2
n
Forarealsystemmshouldbethecenterofmassm’where
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m'=mMm +M
=M
1+ M /m
Then
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r'n =n22
ke 2m'= n2
me
m'a0 and
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E 'n = −m'2n2
( ke2
)2 = −
m'me
E1n2
Hydrogenlikeions:oneelectronboundtoZenucleus:
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vn =ke2
n,
En = −mZ 2
2n2(ke
2
)2 ,
rn =n22
kZe2m
6) Particlesaswaves
p=h/λ orλ=h/p
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ke2 =1.44nm(eV )
λ =hc2mc2K
=1240eV inm2mc2K
deBrogliewavelengthλ=h/p
Uncertaintyprinciple
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ΔkΔx ≥1/2 and
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ΔωΔt ≥1/2 whichcanalsobewrittenas
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ΔpΔx ≥ /2 and
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ΔEΔt ≥ /2
7) SchrödingerEquationinoneDimensionGeneralpropertiesofSchrödinger’sEquation:QuantumMechanics
SchrödingerEquation(timedependent)
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−2
2m∂ 2Ψ∂x 2
+UΨ = i∂Ψ∂t
Standingwave
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Ψ(x, t) = Ψ(x)e−iωt
SchrödingerEquation(timeindependent)
−2
2m∂ 2ψ∂x2
+Uψ = Eψ
3
Normalization(onedimensional) Ψ * (x,t)Ψ(x,t)dx =1−∞
+∞
∫
ForaconstantpotentialE>U
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ψ(x) = Ae−ikx + Be+ikx with
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k =2m(E −U)2
E<U
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ψ(x) = Ae−ηx + Be+ηx with
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η =2m(U − E)2
Operators:
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p =−i ∂∂x,
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E = −i ∂
∂t,
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K = −
2
2m∂ 2
∂x2,
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H = −
2
2m∂ 2
∂x 2+U
EigenfunctionsandEigenvalues:
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p Ψ = pψ thenpistheeigenvalueandΨ isaneigenfunction.
71)OneDimensionalQuantumSystems
a)Particleinaone‐dimensionalbox,withU=0for0<x<L;
Wavefunctions
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ψ(x) =2Lsin(nπx /L) withn=1,2,3,‐‐‐‐
Energies
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En =n2π22
2mL2
b) SimpleHarmonicOscillator:U=kx2/2
AllowedEnergies
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En = n +1/2( )ωc with
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ωc =km
Firstfewwavefunctions
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Ψ0(x) = A0e−x 2
2b
Ψ1(x) = A1
xbe−x 2
2b
Ψ2(x) = A21− 2x
2
b2
e
−x 2
2b
with
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b =
mωc
c) Tunneling:BarrierofheightU0withKineticEnergyE,wherewithinthebarrierE<U0 Withintheclassicallyforbiddenregionthewavefunctionis
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Ψ(x) = Ae+αx + Be−αx
where
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α =2m(U0 − E)2
TheTunnelingprobability
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T ≈ e−αL d) Generalpropertiesofwavefunctions Thewavefunctionanditsderivativemustbecontinuous. Intheclassicallyforbiddenregion(E<U)thewavefunctioncurvesawayfromtheaxis,
exponentiallyaswavepenetratestheclassicallyforbiddenregion. Intheclassicallyallowedregion(E<U)thewavefunctioncurvestowardaxisand
oscillates
!
"# 0
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Constants
c=2.998x10+8m/s h=6.626x10‐34J.sec=4.136x10‐15eV.sec
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=1.055x10−34 J.s = 6.582x10−16eV .s
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k =14πε0
= 8.988x109N .m 2 /C2
me=9.109xx10‐31kg mec2=0.511MeVmp=1.673x10‐27kg mpc2=938.28MeVmn=1.675x10‐27kg mnc2=939.57MeVmp= 1836me mn=1839me
1u=931.5MeV/c2 nm=10‐9me=1.6x10‐19coul kB=8.617x10‐5eV/KeV=1.6x10‐19J NA=6.022x1023objects/molebinomialexpansion:(1+x)n=1+nx+n(n‐1)x2/2!+n(n‐1)(n‐2)x3/3!µm=10‐6m,nm=10‐9m,pm=10‐12m,fm=10‐15m
K=103,M=106,G=109
Usefulrelationships:
hc = 1240eV inmc = 197eV inmke2 = 1.44eV inmke2
c=1137
R=m kee( )2
4πc3=mc2 kee( )2
4π c( )3
R = 0.011nm−1
