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.

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

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

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

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

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

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

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