Mathematics for Physicists and Astronomers
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1 PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester ASTRONOMY DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Spring 2017 Mthemtics for Physicists nd Astronomers Answer question ONE (Compulsory) nd TWO other questions, one ech from section A nd section B. Instructions: All questions Answer ALL re questions. mrked out of ten. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. There re 120 possible mrks for this pper. 2 hours The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. TURN OVER 1
2 Unit 7: Vectors nd Differentition - 50 mrks totl 1. () For the sclr field φ = x 2 y cosh z, clculte i. the grdient, φ, ii. the lplcin, 2 φ. (b) For the vector field F = e z cos (x y) i + e x cos (yz) j, clculte i. the divergence, F, ii. the curl, F, iii. the lplcin, 2 F. [3] [3] [4] 2. () By using plne polr coordintes, (r, θ), show tht the ccelertion ( = r) of [7] prticle in two dimensions is You my ssume the results = ( r r θ 2 ) ˆr + (2ṙ θ + r θ) ˆθ. d ˆr d t = θ ˆθ nd d ˆθ d t = θ ˆr. (b) Consider prticle moving on two dimensionl pth, so tht its position s function of time, t, is given by r(t) = e 2t θ(t) = t 2. i. Obtin expressions for the rdil (ˆr ) nd trnsverse (ˆθ ) components of the [4] ccelertion. ii. At wht time(s) is the force on the prticle purely trnsverse? 3. Consider the two dimensionl vector field F = 1 1 2x 2 i + y x j. () Find n eqution for the field lines of this field, giving n explicit expression for y [5] in terms of x. (b) Sketch the fields lines. CONTINUED 2
3 4. () Show tht, in sphericl polr coordintes, the unit vectors in the direction of in- [6] cresing r, θ nd φ re ˆr = sin θ cos φ i + sin θ sin φ j + cos θ k ˆθ = cos θ cos φ i + cos θ sin φ j sin θ k ˆφ = sin φ i + cos φ j. (b) Consider the vector field F = xz i + yz j (x 2 + y 2 ) k. i. Express this field in sphericl polr coordintes using the bsis vectors in (). [3] ii. Obtin the curl of the field in sphericl polr coordintes. [5] iii. Convert your nswer bck to crtesin coordintes. Note: You my wish to use the formul sheets provided on pges 6 onwrds. TURN OVER 3
4 Unit 8: Vector Integrtion - 50 mrks totl 5. A prticle moves through force field F = x 3 i + x y j + xz 2 k. Find the work done on the prticle if it moves from the origin O = (0, 0, 0) to the point P = (1, 1, 1) long: () the curve described by the prmeteristion x = t, y = t 2, z = t 3, [3] (b) the stright line OP. 6. The mss density of thick sphericl shell is given by ρ(r) = ρ 0 where r is the distnce [8] r3 from the centre of the sphere nd ρ 0 is constnt. The inner nd outer rdii of the shell re nd b respectively. Clculte the moment of inerti of the shell when centred on the origin nd rotting bout the z-xis. 7. Evlute the double integrl I = x 2 y dx dy where R is the region described by 0 x 2 nd x 2 y 4. R [7] 8. () Stte Stokes Theorem, explining the mening of ll terms. [6] (b) Clculte the circultion of the vector field V = 2 yi + 3x j + 0k round the circle [9] x 2 + y 2 = 2, z = 0 9. A force field F = x yi + y 2 j moves prticle round closed loop in the x y plne. [8] The prticle strts t the origin, moves long the x-xis to (1, 0), then follows the curve y = 1 x 2 to (0, 1) nd finlly returns to the origin long the y-xis. Use Green s theorem in the plne to clculte the work done by the force on the prticle. 10. Show tht the vector field V = (x + y + 3z)i + (x 2 y + 2z)j + (3x + 2 y 4z)k is [7] conservtive, nd find the sclr potentil φ such tht V = φ. CONTINUED 4
5 Unit 9: Probbility nd Sttistics - 20 mrks totl 11. You re presented with three coins, two of which re fir nd one counterfeit tht lwys lnds heds. You choose coin t rndom nd flip it three times nd ech time it lnds heds. Use Byes theorem to find the probbility tht the chosen coin is the counterfeit one. [5] 12. Consider rndom vrible A tht cn tke three vlues A = { 1, 0, 1} with probbilities P A ( 1) = 0.2, P A (0) = 0.3, nd P A (1) = 0.5. A function of the rndom vrible A is defined s F() =. Find the probbility distribution of F(). [5] 13. The probbility distribution function p(x) of rndom vrible is positive definite, such tht the probbility of its vlue lying between x nd x + d x is given by p(x)d x. The rndom vrible is constrined between x = 0 nd x = 1 nd p(x) = ke x. () Sketch the probbility distribution function p(x). (b) Find the constnt k. (c) Find the men vlue of x. (d) Find the vrince of x. [4] TURN OVER 5
6 Cylindricl Polr Coordintes For sclr field V (r, θ, z), For vector field F(r, θ, z), Supplementry Formule V = V r ˆr + 1 V r θ ˆθ + V z ẑ 2 V = 1 r V V r r r r 2 θ + 2 V 2 z. 2 F = 1 r r (r F r) + 1 F θ r θ + F z z F = 1 ˆr r ˆθ ẑ r / r / θ / z F r r F θ F z 1 F z = r θ F θ Fr ˆr + z z F z ˆθ + 1 r r r (r F θ) F r ẑ θ Sphericl Polr Coordintes For sclr field V (r, θ, φ), V = V r ˆr + 1 V r θ ˆθ + 1 V r sin θ φ ˆφ. 2 V = 1 r 2 V 1 + r 2 r r r 2 sin θ θ For vector field F(r, θ, φ), r (r2 F r ) + 1 r sin θ θ (sin θ F θ) + 1 sin θ V θ F φ F = 1 r 2 r sin θ φ 1/(r 2 sin θ) ˆr 1/(r sin θ) ˆθ 1/r ˆφ F = / r / θ / φ F r r F θ r sin θ F φ = 1 r sin θ θ (sin θ F φ) F θ ˆr φ r sin θ + 1 r 2 sin 2 θ 2 V φ 2. F r φ r (r F φ) ˆθ + 1 r r (r F θ) F r ˆφ. θ CONTINUED 6
7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s
8 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x x 2 2 +x 2 1 x x x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx
9 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A
10 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk
11 END OF EXAMINATION PAPER 11
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