formulario
DESCRIPTION
.TRANSCRIPT
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EUF
Joint Entrance Examination
for Postgraduate Courses in Physics
For the first semester 2015
14-15 October 2014
LIST OF CONSTANTS AND FORMULAE
Do not write anything on this list. Return it at the end of the first exam day.
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Physical constants
Speed of light in a vacuum c = 3.00108 m/sPlancks constant h = 6.631034 J s = 4.141015 eV s
hc = 1240 eV nm
Wiens constant W = 2.898103 m KPermeability of free space (Magnetic constant) 0 = 4pi107 N/A2 = 12.6107 N/A2
Permittivity of free space (Electric constant) 0 =1
0c2= 8.851012 F/m
1
4pi0= 8.99109 Nm2/C2
Gravitational constant G = 6.671011 N m2/kg2Elementary charge e = 1.601019 CElectron mass me = 9.111031 kg = 511 keV/c2Compton wavelength C = 2.431012 mProton mass mp = 1.6731027 kg = 938 MeV/c2Neutron mass mn = 1.6751027 kg = 940 MeV/c2Deuteron mass md = 3.3441027 kg = 1.876 MeV/c2Mass of particle m = 6.6451027 kg = 3.727 MeV/c2Rydberg constant RH = 1.10107 m1 , RHhc = 13.6 eVBohr radius a0 = 5.291011 mAvogadros number NA = 6.021023 mol1Boltzmanns constant kB = 1.381023 J/K = 8.62105 eV/KGas constant R = 8.31 J mol1 K1
Stefan-Boltzmann constant = 5.67108 W m2 K4
Radius of Sun = 6.96108 m Mass of Sun = 1.991030 kgRadius of Earth = 6.37106 m Mass of Earth = 5.981024 kgDistance from Earth to Sun = 1.501011 m
1 J = 107 erg 1 eV = 1.601019 J
Numerical constants
pi = 3.142 ln 2 = 0.693 cos(30) = sin(60) =
3/2 = 0.866e = 2.718 ln 3 = 1.099 sin(30) = cos(60) = 1/2
1/e = 0.368 ln 5 = 1.609log10 e
= 0.434 ln 10 = 2.303
1
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Classical Mechanics
L = r p dLdt
= r F Li =j
Iijj TR =ij
1
2Iijij I =
r2 dm
r = rer v = rer + re a =(r r2
)er +
(r + 2r
)e
r = e + zez v = e + e + zez a =( 2) e + (+ 2) e + zez
r = rer v = rer + re+r sin e
a =(r r2 r2 sin2
)er
+(r + 2r r2 sin cos
)e
+(r sin + 2r sin + 2r cos
)e
E =1
2mr2 +
L2
2mr2+ V (r) V (r) =
rr0
F (r)dr Veffective =L2
2mr2+ V (r)
RR0
drE V (r) L
2mr2
=
2
m(t t0) = L
mr2
d2u
d2+ u = m
L2u2F (1/u) , u =
1
r;
(du
d
)2+ u2 =
2m
L2[E V (1/u)]
d
dt
(L
qk
) Lqk
= 0, L = T V ddt
(T
qk
) Tqk
= Qk
Qk =Ni=1
Fixxiqk
+ Fiyyiqk
+ Fizziqk
Qk = Vqk(
d2r
dt2
)rotating
=
(d2r
dt2
)inertial
2 v ( r) r
H =
fk=1
pkqk L; qk = Hpk
; pk = Hqk
;H
t= L
t
2
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Electromagnetism
Ed~`+
t
BdS = 0 E+ B
t= 0
BdS = 0 B = 0DdS = Q =
dV D =
Hd~` t
DdS = I =
JdS H D
t= J
D = 0E+P = E B = 0(H+M) = H
PdS = QP P = P
Md~`= IM M = JM
V = Ed~` E = V dH = Id
~`er4pir2
B = A
dE =1
4pi0
dQ
r2er dV =
1
4pi0
dQ
rF = q(E+ vB) dF = Id~`B
J = E J+ t
= 0
u =1
2(DE+BH) S = EH A = 0
4pi
JdV
r
( = 0, J = 0) 2E = 2E
t2n1 sin 1 = n2 sin 2
F =1
4pi0
qQ
r2er U =
1
4pi0
qQ
rE =
1
4pi0
Q
r2er V =
1
4pi0
Q
r
Relativity
=1
1 V 2/c2 x = (x V t) t = (t V x/c2)
vx =vx V
1 V vx/c2 vy =
vy (1 V vx/c2) v
z =
vz (1 V vx/c2)
E = mc2 = m0c2 = m0c
2 +K E =
(pc)2 + (m0c2)2
3
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Quantum Mechanics
i~(x,t)
t= H(x,t) H =
~22m
1
r
2
r2r +
L2
2mr2+ V (r)
px =~i
x[x, px] = i~
a =
m
2~
(x+ i
p
m
)a|n = n|n 1 , a|n = n+ 1|n+ 1
L = Lx iLy LY`m(,) = ~l(l + 1)m(m 1) Y`m1(,)
Lz = x py y px Lz = ~i
, [Lx,Ly] = i~Lz
E(1)n = n|H|n E(2)n =m 6=n
|m|H|n|2E
(0)n E(0)m
, (1)n =m6=n
m|H|nE
(0)n E(0)m
(0)m
S =~2~ x =
(0 11 0
), y =
(0 ii 0
), z =
(1 00 1
)
(~p) =1
(2pi~)3/2
d3r ei~p~r/~ (~r) (~r) =
1
(2pi~)3/2
d3p ei~p~r/~ (~p)
Modern Physics
p =h
E = h =
hc
En = Z2 hcRH
n2
RT = T4 maxT = b L = mvr = n~
= hm0c
(1 cos ) n = 2d sin x p ~/2
4
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Thermodynamics and Statistical Mechanics
dU = dQ dW dU = TdS pdV + dN
dF = SdT pdV + dN dH = TdS + V dp+ dN
dG = SdT + V dp+ dN d = SdT pdV Nd
F = U TS G = F + pV
H = U + pV = F N
(T
V
)S
= (p
S
)V
(S
V
)T
=
(p
T
)V
(T
p
)S
=
(V
S
)p
(S
p
)T
= (V
T
)p
p = (F
V
)T
S = (F
T
)V
CV =
(U
T
)V
= T
(S
T
)V
Cp =
(H
T
)p
= T
(S
T
)p
Ideal gas: pV = nRT, U = ncT, pV = const., = (c+R)/c
S = kB lnW
Z =n
eEn Z =deE() = 1/kBT
F = kBT lnZ U =
lnZ
=N
ZNeN = kBT ln
fFD =1
e() + 1fBE =
1
e() 1
5
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Mathematical results
x2nex2
dx =1.3.5...(2n+1)
(2n+1)2nn
(pi
) 12
k=0
xk =1
1 x (|x| < 1)
du
u(u 1) = ln(1 1/u) ei = cos + i sin
dz
(a2 + z2)1/2= ln
(z +z2 + a2
)lnN ! N lnN N
du
1 u2 =1
2ln
(1 + u
1 u)
exp(t2)dt =
pi
1
a2 + y2dy =
1
aarctan
y
a
x
a2 + x2dx =
1
2ln(a2 + x2)
0
zx1
ez + 1dz = (1 21x) (x) (x) (x > 0)
0
zx1
ez 1 dz = (x) (x) (x > 1)
(2) = 1 (3) = 2 (4) = 6 (5) = 24
(2) =pi2
6= 1,645 (3) = 1,202 (4) =
pi4
90= 1,082 (5) = 1,037
pipi
sin(mx) sin(nx) dx = pim,n
pipi
cos(mx) cos(nx) dx = pim,n
dx dy dz = d d dz dx dy dz = r2dr sin d d
Y0,0 =
1
4piY1,0 =
3
4picos Y1,1 =
3
8pisin ei
Y2,0 =
5
16pi
(3 cos2 1) Y2,1 = 15
8pisin cos ei Y2,2 =
15
32pisin2 e2i
P0(x) = 1 P1(x) = x P2(x) = (3x2 1)/2
General solution to Laplaces equation in spherical coordinates, with azimuthal symmetry:
V (r,) =l=0
(Alrl +
Blrl+1
) Pl(cos )
6
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AdS =
(A) dV
Ad~`=
(A) dS
Cartesian coordinates
A = Axx
+Ayy
+Azz
A =(Azy Ay
z
)ex +
(Axz Az
x
)ey +
(Ayx Ax
y
)ez
f = fx
ex +f
yey +
f
zez 2f =
2f
x2+2f
y2+2f
z2
Cylindrical coordinates
A =1
(A)
+
1
A
+Azz
A =[
1
Az A
z
]e +
[Az Az
]e +
[1
(A)
1
A
]ez
f =f
e +1
f
e +
f
zez 2f = 1
(f
)+
1
22f
2+2f
z2
Spherical coordinates
A = 1r2(r2Ar)
r+
1
r sin
(sin A)
+
1
r sin
(A)
A =[
1
r sin
(sin A)
1r sin
A
]er
+
[1
r sin
Ar 1r
(rA)
r
]e +
[1
r
(rA)
r 1r
Ar
]e
f =fr
er +1
r
f
e +
1
r sin
f
e
2f = 1r2
r
(r2f
r
)+
1
r2 sin
(sin
f
)+
1
r2 sin2
2f
2
7