LIST OF CONSTANTS AND FORMULAE

EUF

Joint Entrance Examination

for Postgraduate Courses in Physics

For the second semester of 2017

April 04-05, 2017

LIST OF CONSTANTS AND FORMULAE

Physical constants

Speed of light in vacuum c= 3.00_×108 m/s

Planck’s constant h= 6.63_×10−34 J s = 4.14_×10−15 eV s

~_{=h/2π= 1.06×10}−34 _{J s = 6.58×10}−16 _{eV s}

hc_≃1240 eV nm = 1240 MeV fm

~_{c≃200 eV nm = 200 MeV fm}

Wien’s constant W = 2.898_×10−3 _{m K}

Permeability of free space (Magnetic constant) µ0 = 4π×10−7 N/A2 = 12.6×10−7 N/A2

Permittivity of free space (Electric constant) ǫ0 = 1

µ0c2

= 8.85_×10−12 F/m

1 4πǫ0

= 8.99_×109 Nm2/C2

Gravitational constant G= 6.67_×10−11 _{N m}2_/kg2

Elementary charge e= 1.60_×10−19 _C

Electron mass me= 9.11×10−31 kg = 511 keV/c2

Compton wavelength λC= 2.43×10−12 m

Proton mass mp= 1.673×10−27 kg = 938 MeV/c2

Neutron mass mn= 1.675×10−27 kg = 940 MeV/c2

Deuteron mass md= 3.344×10−27 kg = 1.876 MeV/c2

Mass of an α particle mα= 6.645×10−27 kg = 3.727 MeV/c2

Rydberg constant RH = 1.10×107 m−1 , RHhc= 13.6 eV

Bohr radius a0 = 5.29×10−11 m

Avogadro’s number NA= 6.02×1023 mol−1

Boltzmann’s constant kB= 1.38×10−23 J/K = 8.62×10−5 eV/K

Gas constant R= 8.31 J mol−1_K−1

Stefan-Boltzmann constant σ= 5.67_×10−8 _{W m}−2_K−4

Radius of the Sun = 6.96_×108 m Mass of the Sun = 1.99_×1030 kg Radius of the Earth = 6.37_×106 m Mass of the Earth = 5.98_×1024 kg Distance from the Earth to the Sun = 1.50_×1011 m

1 J = 107 _{erg 1 eV = 1.60×10}−19 _{J 1 ˚A = 10}−10 _{m 1 fm = 10}−15 _m

Numerical constants

π ∼= 3.142 ln 2∼= 0.693 cos(30◦_{) = sin(60}◦_{) =} √_3/2∼

= 0.866 e∼= 2.718 ln 3∼= 1.099 sin(30◦_{) = cos(60}◦_{) = 1/2}

Error propagation rules

If the error ofX is σX (i. e., measurements of X are given as X±σX), then

F =f(a,b)_⇒σF =

s ∂f ∂a 2 σ2 a+ ∂f ∂b 2 σ2 b

S =a+b, D=a₋b_⇒σS =σD =

q σ2

a+σb2

P =ab, Q= a

b ⇒ σP P = σQ Q = r σ_a a 2

+σb

b 2

Classical Mechanics

L=r_×p dL

dt =r×F Li = X

j

Iijωj TR =

X

ij

1

2Iijωiωj I =

Z

r2 dm

r =rˆer v= ˙rˆer+rθ˙ˆeθ a=

¨

r₋rθ˙2ˆer+

rθ¨+ 2 ˙rθ˙ˆeθ

r =ρˆeρ+zˆez v= ˙ρˆeρ+ρϕ˙ˆeϕ+ ˙zˆez a= ¨ρ−ρϕ˙2

ˆeρ+ (ρϕ¨+ 2 ˙ρϕ˙) ˆeϕ+ ¨zˆez

r =rˆer v= ˙rˆer+rθ˙ˆeθ

+rϕ˙sinθˆeϕ

a= r¨₋rθ˙2 _−rϕ˙2_sin2_θˆe

r

+rθ¨+ 2 ˙rθ˙₋rϕ˙2_{sinθcosθˆe}

θ

+rϕ¨sinθ+ 2 ˙rϕ˙sinθ+ 2rθ˙ϕ˙cosθˆeϕ

E = 1 2mr˙

2₊ L2

2mr2 +V(r) V(r) = −

Z r

r0

F(r′)dr′ Veffective =

L2

2mr2 +V(r)

Z R

R0

dr q

E₋V(r)_{− 2mr}L2

=

r

2

m(t−t0) θ˙= L mr2 d dt ∂L ∂q˙k

− _∂q∂L k

= 0, L=T ₋V d

dt

∂T ∂q˙k

− _∂q∂T k

=Qk

Qk = N X i=1 Fix ∂xi ∂qk

+Fiy

∂yi

∂qk

+Fiz

∂zi

∂qk

Qk =−

∂V ∂qk

d2_r dt2

inertial =

d2_r dt2

rotating

+ 2ω×

dr

dt

rotating

+ω×(ω×r) + ˙ω×r

H =

f

X

k=1

pkq˙k−L; q˙k=

∂H ∂pk

; p˙k =−

∂H ∂qk

; ∂H

∂t =− ∂L

Electromagnetism

I

E_·dl+ d

dt Z

B_·dS= 0 _∇×E+ ∂B

∂t = 0 I

B_·dS = 0 _∇·B= 0

I

E_·dS =Q/ǫ0 = 1/ǫ0

Z

ρdV _∇·E=ρ/ǫ0

I

B_·dl₋µ0ǫ0

d dt

Z

E_·dS=µ0I =µ0

Z

J_·dS _∇×B₋µ0ǫ0

∂E

∂t =µ0J

F=q(E+v_×B) dF=Idl_×B

E= q 4πǫ0

ˆer

r2 E=−∇V V =−

Z

E_·dl V = q 4πǫ0

ˆer

r

F2→1 =

q1q2 4πǫ0

(r1 −r2)

|r1−r2|3

U12 = 1 4πǫ0

q1q2

|r₁₋r_2|

dB= µ0I 4π

dl_׈er

r2 B(r) =

µ0 4π

Z _J(r′_)×(r−r′₎

|r₋r′_|3 dV ′

B=_∇×A A(r) = µ0 4π

Z _J(r′_)dV′

|r₋r′_|

J=σE _∇·J+ ∂ρ

∂t = 0

u= ǫ0

2E·E+ 1 2µ0

B_·B S= 1

µ0

E_×B

∇·P=₋ρP P·nˆ =−σP ∇×M=JM M×nˆ =KM

D=ǫ0E+P=ǫE B =µ0(H+M) = µH

Relativity

γ = p 1

1₋V2_/c2 x

′

=γ(x₋V t) t′ =γ t₋V x/c2

v_x′ = vx−V 1₋V vx/c2

v′_y = vy

γ(1₋V vx/c2)

v_z′ = vz

γ(1₋V vx/c2)

E =γm0c2 p=γm0V T =T0

s

1 +V /c

Quantum Mechanics

i~ ∂Ψ(x,t)

∂t =HΨ(x,t) H =

−~2

2m

1

r ∂2

∂r2 r+ ˆ

L2

2mr2 +V(r)

px =

~

i ∂

∂x [x, px] =i~

ˆ

a =

r mω

2~

ˆ

x+i pˆ mω

ˆ

a_|n_i=√n_|n₋1_i, aˆ†_|n_i=√n+ 1_|n+ 1_i

L± =Lx±iLy L±Yℓm(θ,ϕ) = ~

p

l(l+ 1)₋m(m_±1) Yℓm±1(θ,ϕ)

Lz =x py−y px Lz =

~

i ∂

∂ϕ , [Lx,Ly] =i~Lz E_n(1) =_hn_|δH_|n_i E_n(2) = X

m6=n

|hm_|δH_|n_i|2 En(0)−Em(0)

, φ(1)_n = X

m6=n

hm_|δH_|n_i En(0)−Em(0)

φ(0)_m

ˆ

S = ~

2~σ σx =

0 1 1 0

, σy =

0 ₋i i 0

, σz =

1 0

0 ₋1

¯

ψ(~p) = 1 (2π~₎3/2

Z

d3_{r e}−i~p·~r/~

ψ(~r) ψ(~r) = 1 (2π~₎3/2

Z

d3_{p e}i~p·~r/~ _¯

ψ(~p)

eAˆ _≡

+∞

X

n=0 ˆ

An

n!

Modern Physics

p= h

λ E =hν =

hc

λ En=−Z

2 hcRH

n2 =−Z 2 13,6

n2 eV

RT =σT4 λmaxT =W L=mvr =n~

λ′₋λ= h

m0c

(1₋cosθ) nλ= 2dsinθ ∆x∆p_≥~_{/2 ∆E ∆t≥}~_/2

hE_i= PEnP(En)

P

Thermodynamics and Statistical Mechanics

dU =dQ₋dW dU =T dS₋pdV +µdN

dF =₋SdT ₋pdV +µdN dH =T dS+V dp+µdN

dG=₋SdT +V dp+µdN dΦ =₋SdT ₋pdV ₋N dµ

F =U ₋T S G=F +pV

H =U+pV Φ = F ₋µN

∂T ∂V

S,N

=₋

∂p ∂S

V,N

∂S ∂V

T,N

=

∂p ∂T

V,N

∂T

∂p

S,N

=

∂V ∂S

p,N

∂S ∂p

T,N

=₋

∂V ∂T

p,N

p=₋

∂F ∂V

T,N

S =₋

∂F ∂T

V,N

CV =

∂U ∂T

V,N

=T

∂S ∂T

V,N

Cp =

∂H

∂T

p,N

=T

∂S ∂T

p,N

Ideal gas: pV =nRT, U =CVT =ncVT,

Adiabatic process: pVγ = const., γ =cp/cV = (cV +R)/cV

S =kBlnW

Z =X

n

e−βEn

Z =

Z

dγe−βE(γ) β = 1/kBT

F =₋kBT lnZ U =−

∂

∂β lnZ S = ∂

∂T (kBTlnZ)

Ξ =X

N

ZNeβµN Φ = −kBTln Ξ

fFD=

1

eβ(ǫ−µ)_{+ 1} fBE =

1

Mathematical results

Z ∞

−∞

x2ne−ax2dx= 1.3.5...(2n+1) (2n+1)2n_an

π a

12

(n= 0,1,2, . . .)

∞

X

k=0

xk= 1

1₋x (|x|<1) e

iθ _{= cosθ+isinθ}

Z _dx

(a2_+x2₎1/2 = ln

x+√x2_+a2 _lnN!∼_{=NlnN −N}

Z _dx

(a2_+x2₎3/2 =

x a2√_x2_+a2

Z _x2_dx

(a2_+x2₎3/2 = ln

x+√x2 _+a2_{− √} x

x2_+a2

Z _dx

1₋x2 = 1 2ln

1 +x

1₋x

Z _dx

x(x₋1) = ln(1−1/x)

Z

1

a2_+x2 dx= 1

a arctan x a

Z x

a2_+x2 dx= 1 2 ln(a

2_+x2₎

Z ∞

0

zx−1

ez_{+ 1} dz = (1−2

1−x_{) Γ(x)ζ(x) (x >0)}

Z ∞

0

zx−1

ez₋₁ dz = Γ(x)ζ(x) (x >1)

Γ(2) = 1 Γ(3) = 2 Γ(4) = 6 Γ(5) = 24 Γ(n) = (n₋1)!

ζ(2) = π 2

6 ∼= 1.645 ζ(3)∼= 1.202 ζ(4) =

π4

90 ∼= 1.082 ζ(5) ∼= 1.037

Z π

−π

sin(mx) sin(nx)dx=πδm,n

Z π

−π

cos(mx) cos(nx)dx=πδm,n

dxdydz =ρdρdφdz dxdydz =r2drsinθdθdφ

Y0,0 =

r

1

4π Y1,0 =

r

3

4π cosθ Y1,±1 =∓ r

3

8π sinθe

±iφ

Y2,0 =

r

5

16π 3 cos

2_θ

−1

Y2,±1 =∓

r

15

8πsinθcosθe

±iφ _Y

2,±2 =∓

r

15 32πsin

2_θe±2iφ

P0(x) = 1 P1(x) = x P2(x) = (3x2₋1)/2

General solution to Laplace’s equation in spherical coordinates, with azimuthal symmetry:

V(r,θ) =

∞

X

l=0

(Alrl+

Bl

∇·(_∇×V) = 0 _∇×∇f = 0

∇×(_∇×V) = _∇(_∇·V)_{− ∇}2V

I

A_·dS=

Z

(_∇·A) dV

I

A_·dl=

Z

(_∇×A)_·dS

Cartesian coordinates

∇·A = ∂Ax

∂x + ∂Ay

∂y + ∂Az

∂z

∇×A =

∂Az ∂y − ∂Ay ∂z

ˆex+

∂Ax ∂z − ∂Az ∂x

ˆey+

∂Ay ∂x − ∂Ax ∂y ˆez

∇f = ∂f

∂xˆex+ ∂f ∂yˆey +

∂f

∂zˆez ∇

2_{f =} ∂2f

∂x2 +

∂2_f

∂y2 +

∂2_f

∂z2

Cylindrical coordinates

∇·A=1

ρ

∂(ρAρ)

∂ρ + 1 ρ ∂Aϕ ∂ϕ + ∂Az ∂z

∇×A=

1 ρ ∂Az ∂ϕ − ∂Aϕ ∂z

ˆeρ+

∂Aρ ∂z − ∂Az ∂ρ

ˆeϕ+

1

ρ

∂(ρAϕ)

∂ρ − 1 ρ ∂Aρ ∂ϕ ˆez

∇f =∂f

∂ρˆeρ+

1

ρ ∂f ∂ϕˆeϕ+

∂f

∂zˆez ∇

2_{f =} 1

ρ ∂ ∂ρ ρ∂f ∂ρ + 1 ρ2

∂2_f

∂ϕ2 +

∂2_f

∂z2

Spherical coordinates

∇·A=1

r2

∂(r2_A

r)

∂r +

1

rsinθ

∂(sinθAθ)

∂θ +

1

rsinθ

∂(Aϕ)

∂ϕ

∇×A=

1

rsinθ

∂(sinθAϕ)

∂θ −

1

rsinθ ∂Aθ ∂ϕ ˆer + 1

rsinθ ∂Ar

∂ϕ −

1

r

∂(rAϕ)

∂r

ˆeθ+

1

r

∂(rAθ)

∂r − 1 r ∂Ar ∂θ ˆeϕ

∇f =∂f

∂rˆer+

1

r ∂f ∂θˆeθ+

1

rsinθ ∂f ∂ϕˆeϕ

∇2f =1

r2

∂ ∂r

r2∂f

∂r

+ 1

r2_sinθ

∂ ∂θ

sinθ∂f ∂θ

+ 1

r2_sin2_θ

∂2_f