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Mathematical Physics1. Describe the mapping produced by the functions: 1 (a) W (z) = p(z2+1)(z 2) (b) W (z) = pz 11 ip21 2. Sum the series: P 1 R0 1 2 n= 1 n +n+1 (ln x)2 x2 +a2
3. Do the integral: dx
4. For large positive x, nd approximately: dt ext0
1 R
xt
5. Show that: 1 1 x1 x2 ... ... n n x1 1 x2 1
::: 1 Y : : : xn (xj xi) ... = ::: 1 i 0. (c) The mass is uniformly distributed in a circular cylinder of in nite length, with axis along the z axis. (d) The mass is uniformly distributed in a circular cylinder of nite length, with axis along the z axis. (e) The mass is uniformly distributed in a right cylinder of elliptical cross section and in nite length, with axis along the z axis. (f) The mass is uniformly distributed in a dumbbell whose axis is oriented along the z axis. (g) The mass is in the form of a uniform wire wound in the geometry of an in nite helical solenoid with axis along the z axis. 4. Goldstein 2.17 (2nd edition) A point mass is constrained to move on a massless hoop of radius a xed in a vertical plane that is rotating about the vertical with constant angular speed !. Obtain the Lagrange equations of motion assuming the only external forces arise from gravity. What are the constants of motion? Show that if ! is greater than a critical value !0, there can be a solution in which the particle remains stationary on the hoop at a point other than the bottom, but that if ! < !0, the only stationary point for the particle is at the bottom of the hoop. What is the value of !0? 5. Goldstein 2.22 (2nd edition) 1 _ The one dimensional harmonic oscillator has the Lagrangian L = 2 mx2 1 kx2 . Suppose you did 2 not know the solution to the motion, but realized that the motion must be periodic and therefore 3
could be described by a Fourier series of the form x(t) = Pj=0 aj cos(j!t), (taking t = 0 at a turning point), where ! is the (unknown) angular frequency of the motion. This representation for x(t) de nes a many-parameter path for the system point in con guration space. Consider the action integral I for two points t1 and t2 separated by the period T = 2 =!. Show that with this form for the system path, I is an extremum for nonvanishing x only if aj = 0 for j 6= 1, and only if !2 = k=m. 6. Goldstein 3.4 (2nd edition) Consider a system om which the total forces acting on the particles consist of conservative ~ ~ forces Fi and frictional forces fi proportional to the velocity. Show that for such a system the 1 ~ r virial theorem holds in the form T = 2 Pi Fi ~i, providing the motion reaches a steady state and is not allowed to die down as a result of the frictional forces. 7. Two identical satellites are 45 km apart and move in a common circular orbit around the earth. Each satellite can change its speed (by ring engines in negligible time) by 8 km/hr in the instantaneous direction of motion (i.e. tangent to its orbit). Find a sequence of manoeuvres of the lagging satellite so that it can \dock" with the other one. (In order to dock, both satellites must be at the same place with the same velocity.) Hint: The lagging satellite will decelerate, go into an elliptical orbit, and then accelerate back in the circular orbit. 8. Goldstein 3.14 (2nd edition) Show that the motion of a particle in the potential eld V (r) = k + rh2 , is the same as r the motion under the Kepler potential alone when expressed in terms of a coordinate system rotating or precessing around the centre of force. For negative total energy, show that if the additional potential term is very small compared to the Kepler potential, then the angular speed of precession of the elliptical orbit is _ = 2 l2mh . The perihelion of Mercury is observed to precess (after correction for known planetary perturbations) at the rate of about 4000 of arc per century. Show that this precession could be accounted h for classically, if the dimensionless quantity = ka (which is a measure of the perturbing inverse square potential relative to the gravitational potential) were as small as 7 10 8. (The eccentricity of Mercury's orbit is 0:206, and its period is 0:24 year. 9. Goldstein 3.23 (2nd edition) A magnetic monopole is de ned (if one exists) by a magnetic eld singularity of the form ~ b~ B = rr , where b is a constant (a measure of the magnetic charge, as it were). Suppose a 3 particle of mass m moves in the eld of a magnetic monopole and a central force eld derived from the potential V (r) = k . r ~ (a) Find the form of Newton's equation of motion, using the Lorentz force given by F = _ ~ cv ~ q E + 1 ~ B ]. By looking at the product ~ p, show that while the mechanical angular r ~ momentum is not conserved (the eld of force is non-central), there is a conserved vector ~ ~ cr D = L qb ~ . r (b) By paralleling the steps leading from Eq.(3.79) to Eq.(3.82), show that for some f (r) there ~ is a conserved vector analogous to the Laplace-Runge-Lenz vector in which D plays the same ~ role as L in the pure Kepler force problem.
4
10. Goldstein 3.27 (2nd edition) A central force potential frequently encountered in nuclear physics is the rectangular well, de ned by the potential
V (r) = V0 ; for r a = 0; otherwise: Show that the scattering produced by such a potential in classical mechanics is identical with the q refraction of light rays by a sphere of radius a and relative index of refraction n = (E + V0 )=E . (This equivalence demonstrates why it is possible to explain refraction phenomena both by Huygens' waves and by Newton's mechanical corpuscles.) Show that the di erential crosssection is 2) n2 2 ( ) = 4 cos(a =2) (n cos(+=n2 1)(n cos( 2=2)) : (1 2n cos( =2) What is the total cross section?11. Goldstein 4.22 (2nd edition) A particle is thrown up vertically with initial speed v0, reaches a maximum height and falls back to the ground. Show that the Coriolis de ection when it again reaches the ground is opposite in direction, and four times greater in magnitude, than the Coriolis de ection when it is dropped at rest from the the same maximum height. 12. Goldstein 5.15 (2nd edition) An automobile is started from rest with one of its doors initally at right angles. If the highes of the door are toward the front of the car, the door will slam shut as the automobile picks up speed. Obtain a formula for the time needed for the door to close if the acceleration f is constant, the radius of gyration of the door about the axis of rotation is r0, and the centre of mass is at a distance a from the hinges. Show that if f is 1 ft=sec2 and the door is a uniform rectangle 4 ft wide, the time will be approximately 3:04 seconds. 13. Goldstein 5.17 (2nd edition) (a) Express in terms of Euler's angles the constraint conditions for a uniform sphere rolling without slipping on a at horizontal surface. Show that they are nonholonomic. (b) Set up the Lagrangian equations for this problem by the method of Lagrange multipliers. Show that the translational and rotational parts of the kinetic energy are separately conserved. Are there any other constants of motion? 14. Goldstein 7.25 (2nd edition) ~ A particle of rest mass m, charge q, and initial velocity ~0 enters a uniform electric eld E v perpendicular to ~0. Find the subsequent trajectory of the particle and show that it reduces to v a parabola as the limit c becomes in nite. 15. Goldstein 7.27 (2nd edition) Starting from the equation of motion, dpi = Fi, derive the relativistic analog of the virial dt theorem, which states that for motions bounded in space and such that the velocities involved ~ r do not approach c inde nitely close, then L0 + T = F ~, where L0 is the form the Lagrangian takes in the absence of external forces. Note that although neither L0 nor T corresponds exactly to the kinetic energy in the nonrelativistic mechanics, their sum, L0 + T , plays the same role ~ r as twice the kinetic energy in the nonrelativistic virial theorem, T = 1 Pi F ~. 2 5
16. Goldstein 8.11 (2nd edition) (a) The point of suspension of a plane simple pendulum of mass m and length l is constrained to move along a horizontal track and is connected to a point on the circumference of a uniform y wheel of mass M and radius a through a massless connecting rod also of length a, as shown in the gure. The y wheel rotates about a centre xed on the track. Find a Hamiltonian for the combined system and the Hamilton's equations of motion. (b) Suppose the point of suspension were moved along the track according to some function of time x = f (t), where x reverses at x = 2a (relative to the centre of the y wheel). Again nd a Hamiltonian and the Hamilton's equations of motion. 17. A mass m moves in a circular orbit under the in uence of a central potential km . Show rn that the circular orbit is stable under small oscillations, i.e. the mass will oscillate about the circular orbit, if n < 2. 18. Goldstein 9.14 (2nd edition) By any method you choose, show that the following transformation is canonical: q q x = 1 ( 2P1 sin Q1 + P2); px = 2 ( 2P1 cos Q1 Q2 ); q q y = 1 ( 2P1 cos Q1 + Q2 ); px = 2 ( 2P1 sin Q1 P2);
where is some xed parameter. Apply this transformation to th eproblem of a particle of charge q moving in a plane that's ~ perpendicular to a constant magnetic eld B . Express the Hamiltonian for this problem in the (Qi; Pi) coordinates letting the parameter take the form 2 = qB=c. From this Hamiltonian obtain the motion of the particle as a function of time. 19. Goldstein 9.39 (2nd edition) (a) Show from the Poisson bracket condition for conserved quantities that the Laplace-Runge~ ~ ~ rr Lenz vector, A = p L mk~ , is a constant of motion for the Kepler problem. ~ (b) Verify the Poisson bracket relations for the components of A as given by Ai; Lj ] = ijk Ak . 20. A gyroscope spins about its axis with angular velocity !. The moment of inertia about this axis is C , and that about a transverse axis is A. The gyroscope oats on a pool of mercury so that the only torque acting on it is one constraining its axis to remain in a horizontal plane. The gyroscope is placed at the earth's equator, with earth's angular velocity !. Show that the axis of the gyroscope will oscillate about the north-south direction, and nd the period for small oscillations. 21. Goldstein 10.6 (2nd edition) A charged particle is constrained to move in a plane under the in uence of a central force 1 ~ potential (nonelectromagnetic) V = 2 kr2, and a constant magnetic eld B perpendicular to ~ 1 ~r the plane, so that A = 2 B~. Set up the Hamilton-Jacobi equation for Hamilton's characteristic function in plane polar coordinates. Separate the equation and reduce it to quadratures. Discuss the motion if the canonical momentum p is zero at time t = 0. 6
22. Goldstein 10.21 (2nd edition) (a) Consider three dimensional isotropic harmonic oscillator (all the frequencies equal) so that the motion is completely degenerate. Transform to the \proper" action-angle variables, expressing the energy in terms of only one of the action variables. (b) Solve the problem of the isotropic oscillator in action-angle variables using spherical polar coordinates. Transform again to proper action-angle variables and compare with the result of part (a). Are the two sets of proper variables the same? What are their physical signi cances? This problem illustrates the feasibility of separating a degenerate motion in more than one set of coordinates. The non-degenerate oscillator can be separated in Cartesian coordinates, not in polar coordinates. 23. If neutrons from a cosmic-ray interaction one light-year from the earth were to reach 1 here with a probability of 2 or greater, what must their minimum energy be? If they then decay, what is the maximum angle to the ight path at which their decay electrons could be produced? What is the maximum angle for the decay neutrinos? At this maximum angle, what is the maximum energy of the neutrino? 24. A point mass m, moves smoothly on a wire stretched out along the curve y = f (x) in the vertical xy plane. y is the vertical direction, and the curve y = f (x) has a minimum at the origin. Another point mass m hangs from a light rigid rod of length l suspended from the rst point mass, and is free to move in the xy plane. Both masses are acted upon by the earth's (uniform) gravitational eld ~ vertically downwards. g (a) Set up the Lagrangian for this system. (b) Write down the equations of motion for small oscillations about the equilibrium con guration of the system. 3 (c) Henceforth specialize to the case where the curve y = f (x) is a circle of radius 4 l. Find the eigenfrequencies of small oscllations, !1 and !2. (d) Let the matrices T and V de ne the kinetic and potential energy coe cients in the small 1 oscillations Lagrangian L = 2 Tij q_i q_j 1 Vij qi qj , where qi are your chosen generalized coordinates 2 of the system. Find the normal modes of the system and the matrix A such that AT TA = 1 2 2 and AT V A = diag(w1 ; w2 ). (e) Starting from rest in the equilibrium position, an impulse I is given to the mass that is sitting on the wire, which instantaneously changes its momentum to a nite value without changing its position. Find the subsequent motion of both the masses. 25. Consider a system with a single degree of freedom, with the Hamiltonian H (q; p) = ap2 + bq, where q; p are the canonical coordinate and momentum and a; b are constants. Consider the function S (q; t; ) which is a complete integral of the Hamilton-Jacobi equation, @S +H (q; @S ) = @t @q 0, where is an arbitrary constant in the complete integral. (a) Consider S of the form S (q; t; ) = W (q; ) t, and substitute it in the Hamilton-Jacobi equation. Solve for W (q; ) by performing the integral, and hence obtain S (q; t; ). (b) Use the solution S (q; t; ), so obtained, as a generating function of the (q; P ) type (i.e. treat as the new momentum P ), and de ne the new coordinate Q = @S . Show that the @ transformation from (q; p) to ( ; ) (Q; P ) is a canonical transformation. (c) Solve for q; p in terms of ; (and t), and verify that you have thus obtained the general solution of the Hamilton equations of the original Hamiltonian. 7
(d) What is the interpretation of and , if the Hamiltonian represents the system of a particle of mass m in a uniform gravitational eld g (i.e. a = 21 ; b = mg)? m 26. Consider a linear chain in which points of two di erent masses (m1 and m2 ) are connected alternately with springs of sti ness k. Equilibrium spacing of masses is L, and motion of all masses is purely longitudinal. Find the relation between frequency and wave vector for small amplitude travellng waves along the chain. Show that no wavelike motion can exist in the k k frequency range m1 < !2 < m2 for m1 > m2. 27. Goldstein 12.1 (2nd edition) The transverse vibrations of a stretched string can be approximated by a discrete system consisting of equally spaced mass points located on a weightless string.R hShow that if ithe @ 1 spacing is allowed to go to zero, the Lagrangian approaches the limit L = 2 _ 2 T @x dx for the continuous string, where T is the xed tension. What is the equation of motion if the density is a function of position? 28. Goldstein 12.4 (2nd edition) Show that if and are taken as two independent eld variables, the Lagrangian density leads to the Schrodinger equation
~ ~ L = 8 h2m r r + V2
+ h( 4 i
_
_)
h2 r2 + V = ih @ ; 8 2m 2 @t and its complex conjugate. What are the canonical momenta? Obtain the Hamiltonian density corresponding to L.29. Find the kinematically allowed region in the s t plane for two body elastic collisions. The Mandelstam variables are s = (p1 + p2 )2 and t = (p1 p01)2 , where pi refer to 4 momenta.
8
Electromagnetic Theory1. Jackson 2.9 (2nd edition) (a) Two halves of a long hollow conducting cylinder of inner radius b are separated by small lengthwise gaps on each side, and are kept at di erent potentials V1 and V2. Show that the potential inside is given by ( ; ) = V1 + V2 + V1 V2 tan 1( 22b 2 cos ) 2 2 b (b) Calculate the surface-charge density on each half of the cylinder. 2. Jackson 4.7 (2nd edition) A localised distribution of charge has a charge density 1 (r) = 64 r2e r sin2 (a) Make a multipole expansion of the potential due to this charge density and determine all the nonvanishing multipole moments. Write down the potential at large distances as a nite expansion in Legendre polynomials. (b) Determine the potential explicitly at any point in space, and show that near the origin, correct to r2 inclusive, 1 r2 (r) ' 4 120 P2 (cos ) Read part (c) to see the physical context. 3. A charge q is placed at a distance r from a semi-in nite medium of dielectric constant . Find the magnitude and direction of the force felt by the charge q. 4. Jackson 5.2 (2nd edition) (a) For a solenoid wound with N turns per unit length and carrying a current I , show that the magnetic- ux density on the axis is given for N ! 1 by NI Bz = 2 c (cos 1 + cos 2 ) where the angles are de ned on page 205 of Jackson(2nd.Ed.). (b) For a long solenoid of length L and radius a, show that near the axis and near the center of the solenoid the magnetic induction is mainly parallel to the axis, but has a small radial component 2 B ' 96 cNI ( aLz4 ) correct to order a2 =L2 and for z L, a. The coordinate z is measured from the center point of the axis, with the ends of the solenoid at z = L=2. (c) Show that at the end of a long solenoid the magnetic induction near the axis has components NI Bz ' 2 c ; B ' NI ( a ) c 9
5. Jackson 6.9 (2nd edition) A dielectric sphere of dielectric constant and radius a is located at the origin. There is a uniform applied electric eld E0 in the x direction. The sphere rotates with an angular velocity ! about the z axis. Show that there is a magnetic eld H = r M , where 3 1 )E ! ( a )5:xz M= ( 5 + 1 0 c r> where r> is the larger of r and a. The motion is non-relativistic. You may use the results of Section 4.4 for the dielectric sphere in an applied eld. 6. Jackson 6.12 (2nd edition) A transverse plane wave is incident normally in vacuum on a perfectly absorbing at screen. From the law of conservation of linear momentum, show that the pressure (called radiation pressure) exerted on the screen is equal to the eld energy per unit volume in the wave. 7. Jackson 6.18 (2nd edition) Consider the Dirac expression
~x A(~ ) = g
Z d~0 (~ x0 ) l x ~ L j~ x0 j3 x ~
for the vector potential of a magnetic monopole and its associated string L. Suppose for de niteness that the monopole is located at the origin and the string along the negative z axis. ~ (a) Calculate A explicitly and show that in spherical coordinates it has components Ar = 0, A = 0, and cos A = g(1r sin ) = g tan( 2 ) r ~ ~ ~ (b) Verify that B = r A is the Coulomb-like eld of a point charge, except perhaps at = . ~ (c) With the B determined in (b), evaluate the total magnetic ux passing through the circular loop of radius R sin shown in the gure on Pg.267. Consider < =2 and > =2 separately, but always calculate the upwards ux. H~ l (d) From A d~ around the loop, determine the total magnetic ux through the loop. Compare the result with that found in part (c). Show that they are equal for 0 < < =2, but have a constant di erence for =2 < < . Interpret this di erence. 8. Jackson 7.3 (2nd edition) Two plane semi-in nite slabs of the same uniform, isotropic, nonpermeable, lossless dielectric with index of refraction n are parallel and separated by an air gap (n = 1) of width d. A plane electromagnetic wave of frequency ! is incident on the gap from one of the slabs with angle of incidence i. For linear polarisation both parallel to and perpendicular to the plane of incidence (a) calculate the ratio of the power transmitted into the second slab to the incident power and the ratio of re ected to incident power; (b) for i greater than the critical angle for total internal re ection, sketch the ratio of transmitted power to incident power as a function of d measured in units of wavelength in the gap. 9. Jackson 7.4 (2nd edition) A plane polarised electromagnetic wave of frequency ! in free space is incident normally on the at surface of a nonpermeable medium of conductivity and dielectric constant . 10
(a) Calculate the amplitude and phase of the re ected wave relative to the incident wave for arbitrary and . (b) Discuss the limiting cases of a very poor and a very good conductor and show that for a good conductor the re ection coe cient (ratio of re ected to incident intensity) is approximately R ' 1 2! c 10. Jackson 7.14 (2nd edition) Use the Kramers-Kronig relation, Eq.(7.120), to calculate the real part of (!), given the imaginary part of (!) for positive ! as (a)
Im =(b)
(! ! 1 )
(! !2)];
!2 > !1 > 0
Im = (! 2 !2)! + 2 !2 (1) 2 0 In each case sketch the behaviour of Im (!) and the result for Re (!) as functions of !. Comment on the reasons for similarities or di erences of your results as compared with the curves in Fig. 7.8.11. Jackson 8.8 (2nd edition) A right circular cylinder of nonpermeable dielectric with dielectric constant and radius a serves as a dielectric wave guide in vacuum. (a) Discuss the propagation of waves along such a guide, assuming that the azimuthal variation of the elds is eim . (b) For m = 1, determine the mode with the lowest cut-o frequency and discuss the properties of its elds (cut-o frequency, spatial variation, etc.), assuming that 1. 12. Jackson 9.2 (2nd edition) A radiating quadrupole consists of a square of side a with charges q at alternate corners. The square rotates with angular velocity ! about an axis normal to the plane of the square and through its center. Calculate the quadrupole moments, the radiation elds, the angular distribution of radiation and the total radiated power all in the long wavelength approximation. 13. Jackson 10.5 (2nd edition) The force equation for an electronic plasma, including a phenomenological collision term, but neglecting the hydrostatic pressure (zero temperature approximation) is @~ + (~ r)~ = e (E + ~ B ) ~ v v ~ v ~ v ~ v @t m c where is the collision frequency. (a) Show that in the presence of static, uniform, external, electric and magnetic elds, the linearised steady-state expression for Ohm's Law becomes
Ji =
Xj
ij Ej
(2)
11
where the conductivity tensor is 4 (1 + !B22 )
!p2
0 1 !B B !B 1 @0
0 C 0 A 2 !B ) 0 (1 + 2
1
~ and !p(!B ) is the electronic plasma (precession) frequency. The direction of B is chosen as the z axis. (Steady state means that the velocity ~ is independent of ~ and t.) v r14. Jackson 11.6 (2nd edition) (a) Use the relativistic velocity addition law and the invariance of phase to discuss the Fizeau experiments on the velocity of propagation of light in moving liquids. Show that for liquid ow at a speed v parallel or antiparallel to the path of the light the speed of light, as observed in the laboratory, is given to rst order in v by 1 u = n(c!) v(1 n2 + ! dn(!) ) n d! where ! is the frequency of the light in the laboratory (in the liquid and outside it), n(!) is the index of refraction of the liquid. Because of the extinction theorem, it is assumed that the light travels with speed u0 = c=n(!0) relative to the moving liquid. (b) If you can, read the paper: W.M. Macek, J.R. Schneider and R.M. Salamon, J. Appl. Phys. 35 (1964) 2556. 15. Jackson 11.11 (2nd edition) An in nitely long straight wire of negligible cross-sectional area is at rest and has a uniform linear charge density q0 in the inertial frame K 0 . The frame K 0 (and the wire) move with a velocity ~ parallel to the direction of the wire with respect to the laboratory frame K . v (a) Write down the electric and magnetic elds in cylindrical coordinates in the rest frame of the wire. Using the Lorentz transformation properties of the elds, nd the components of the electric and magnetic elds in the laboratory. (b) What are the charge and current densities associated with the wire in its rest frame? In the laboratory? (c) From the laboratory charge and current densities, calculate directly the electric and magnetic elds in the laboratory. Compare with the results of (a). 16. Jackson 11.14 (2nd edition) ~ ~ In the rest frame of a conducting medium, the current density satis es Ohm's Law J 0 = E 0 , where is the conductivity and primes denote quantities in the rest frame. (a) Taking into account the possibility of convection current as well as conduction current, show that the covariant generalisation of Ohm's law is J c12 (U J )U = c F U where U is the 4-velocity of the medium. (b) Show that if the medium has a velocity ~ = c ~ with respect to some inertial frame that the v 3-vector current in that frame is ~ ~ ~ ~ J = E + ~ B ~ ( ~ E )] + ~ v 12
where is the charge density observed in that frame. (c) If the medium is uncharged in its rest frame ( 0 = 0), what is the charge density and the ~ expression for J in the frame of part (b)? This is the relativistic generalisation of Eq.(10.8). 17. Jackson 12.10 (2nd edition) Consider the precession of the spin of a muon, initially longitudinally polarised, as the muon ~ moves in a circular orbit in a plane perpendicular to a uniform magnetic eld B . (a) Show that the di erence of the spin precession frequency and the orbital gyration frequency is = eBa mc independent of the muon's energy, where a = (g 2)=2 is the magnetic moment anomaly. (Find equations of motion for the components of spin along the mutually perpendicular directions de ned by the particle's velocity, the radius vector from the center of the particle and the magnetic eld.) (b) For the CERN Muon Storage Ring referred to in Section 11.1(c)], the orbit radius is R = 2:5 meters and B = 17 103 gauss. What is the momentum of the muon? What is the time dilation factor ? How many periods of precession T = 2 = occur per observed laboratory mean lifetime of the muons? m = 105:66MeV, 0 = 2:2 10 6sec, a ' =2 ]. (c) Express the di erence frequency in units of the orbital rotation frequency and compute how many precessional periods (at the di erence frequency) occur per rotation for a 300MeV muon, a 300MeV electron, a 5GeV electron (this last value is typical of e+e storage rings at Stanford and Hamburg). 18. Jackson 12.14 (2nd edition) Consider the Proca equations for a localised steady state distribution of current that has only a static magnetic moment. This model can be used to study the observable e ects of a nite ~ x photon mass on the Earth's magnetic eld. Note that if the magnetisation is M (~ ), the current ~ ~ ~ density can be written as J = c(r M ). ~ = mf (~ ), where m is a xed vector and f (~ ) is a localised scalar function, (a) Show that if M ~ x ~ x the potential is0
Z j~ x j x ~ ~x ~ A(~ ) = m r f (x0 ) e ~0 d3x0 ~ ~ j~ x j x (b) If the magnetic dipole is a point dipole at the origin, show that the magnetic eld is 2 2 r 2 r ~x B (~ ) = 3^(^ m) m](1 + r + 3r ) er3 3 2m e r rr ~ ~ ~ (c) The result of (b) shows that at xed r = R (on the surface of the Earth), the Earth's magnetic eld will appear as a dipole angular distribution, plus an added constant magnetic eld (an apparently external eld) antiparallel to m. Satellite and surface observations lead ~ to the conclusion that this \external" eld is less than 4 10 3 times the dipole eld at the magnetic equator. Estimate a lower limit on 1 in Earth radii and an upper limit on the photon mass in grams from this datum. This method of estimating is due to E.Schrodinger, Proc.Roy.Irish Acad. A49,135(1943). See A.S.Goldhaber and M.M.Nieto,Phys. Rev.Letters 21,567(1968).13
19. Jackson 14.2 (2nd edition) Using the Lienard-Wiechert elds, discuss the time average power radiated per unit solid angle in the nonrelativistic motion of a particle with charge e, moving (a) along the z axis with instantaneous position z(t) = a cos !0t, (b) in a circle of radius R in the x y plane with constant angular frequency !0. Sketch the angular distribution of the radiation and determine the total power radiated in each case. 20. Jackson 14.10 (2nd edition) Bohr's correspondence principle states that in the limit of large quantum numbers, the classical power radiated in the fundamental is equal to the product of the quantum energy (h!0) and the reciprocal mean lifetime of the transition from the principal quantum number n to (n 1). (a) Using nonrelativistic approximations, show that in a hydrogen-like atom the transition probability (reciprocal mean lifetime) from a circular orbit of principal quantum number n to (n 1) is 1 = 2 e2 ( Ze2 )4 mc2 1 3 hc hc h n5 (b) For hydrogen, compare the classical value from (a) with the correct quantum-mechanical results for the transition 2p ! 1s (1:6 10 9sec), 4f ! 3d(7:3 10 8sec) and 6h ! 5g(6:1 10 7sec).
14
Quantum Mechanics1. For a normalised wave packet (x; t) centred about the point x0 , we de ne the width x(t), R by ( x(t))2 = 1 dx (x x0 )2 j (x; t)j2. Suppose that at t = 0, we have the square wave 1 packet: (x; 0) = p1 ; for x0 a < x < x0 + a 2a = 0; otherwise Use the free particle Schrodinger equation to see how this wave function evolves in time, and nd an expression for the width x(t). 2. Use the WKB approximation to obtain the energy levels En as a function of n, for the potential V (x) = cjxj where c > 0. 3. Consider an electron of charge e moving on a circle of circumference L, which encloses a magnetic ux . Find the expectation value hevi in the ground state, as a function of , where ^ 1 (P e A) is the velocity operator of the electron. (hev i is time-independent, so it is called ^ c^ v=m ^ ^ the persistent current if it is non-zero.) 4. The Hamiltonian governing the rotational motion of some system with angular momentum l = 1 is given by: H = h12 (aL2 + bL2 + cL2 ). Find all the energy levels. x y z 5. Find the ground state wave function and energy of the helium atom variationally, using some simple variational wave function. 6. For the scattering potential: V (~) = h~22c (with c > 0), calculate the phase shift l (k). r r 7. Consider solutions of the Dirac equation inside a sphere of radius R, which satisfy the boundary conditions = 0 for j~j > R and n r ^ = i at the surface j~j = R (^ is the unit r n normal vector). (a) Show that the Hamiltonian, H = c 0~ p + mc2 0, is hermitian, i.e. if 1 and R 2 are two wave ~ R r y functions satisfying the above boundary conditions, show that d3~ 1 H 2 = d3~ (H 1)y 2 . r (b) Find all the energy levels and wave functions. (This problem describes the bag model in nuclear physics. Note that if we had chosen a di erent boundary condition, = 0 at the surface j~j = R, there would have been no solutions at all apart from = 0 everywhere). r 8. Find the positive energy eigenvalues of a Dirac electron in a constant and uniform magnetic ~ eld B . 9. Solve the one-dimensional Dirac equation with eA0 (x) = V (x), where
V (x) = V0 ; for jxj < a = 0; otherwisewith 0 < V0 < mc2 . Find a bound state with spin down, after explaining the correct boundary conditions at x = a. Find the minimum value of V0 which just binds the nth bound state, i.e. En = mc2 ; what is the corresponding eigenfunction? 15
10. Consider the inelastic scattering: e + H (1s) ! e + H (2s). Calculate this scattering cross-section in the Born approximation. How important is the contribution of the exchange process as compared to the direct process ? ^2 ^ 11. Virial theorem: For a one-dimensional problem described by H = 2pm + ajxjn (where ^ ^ , the expectation values satisfy a; n > 0), the virial theorem states that in any eigenstate of H 2 n ^ 2 ^ ^ h 2p^m i = n+2 hH i and hajxjni = n+2 hH i. ^ ^ ^ ^ ^ hp^ (a) Consider the hermitian operator: D = xp2+^x , and compute D; H ]. Use the result to prove the virial theorem. ^ (b) How does the unitary operator U = exp(i D) transform x and p, i.e. what are U xU 1 and ^ ^ ^ 1 equal to? (D is sometimes called the scaling operator.) ^ U pU ^ 12. Coherent states: (a) For the simple harmonic oscillator with the time independent wave functions n(x), satisfy1 1 ^ ing H n(x) = h!(n + 2 ) n(x), consider the superposition at t = 0, (x; 0) P cn n (x). How n=0 should the coe cients cn be chosen so that (x; 0) is an eigenstate of the lowering operator a, i.e. a (x; 0) = (x; 0), where the eigenvalue is some given complex number? Using the ^ ^ expression for a, nd the explicit form of the wave function (x; 0). (Make sure that (x; 0) is ^ correctly normalised.) Eigenstates of a are called coherent states. ^ @ ^ (b) Now let (x; 0) evolve in time according to the equation ih @t (x; t) = H (x; t). Show that (x; t) remains a coherent state at all times, except that the eigenvalue of a changes with time. ^ How does it change? (c) The mean position hxi and width x of the wave function (x; t) are de ned as: hxi = 1 R dx x , and ( x)2 = 1 dx(x hxi)2 , assuming that (x; t) is normalised. Show that R 1 1 hxi varies with time according to the classical equation of motion, while x does not change at all. (d) Calculate the mean momentum hpi and p, and show that they have similar properties as hxi and x in (c). All these are important properties of coherent states. 13. Angular momentum: (a) A two-dimensional representation of the angular momentum algebra is provided by the three matrices: ! ! ! h 0 1 ; J = h 0 i and J = h 1 0 : Jx = 2 1 0 y z 2 i 0 2 0 1
Given a unit vector n = (nx; ny ; nz ), where P n2 = 1, compute the rotation matrix ^ i=x;y;z i R = exp(i n J=h), which rotates by an angle about the direction n. What is the value of R ^ ~ ^ for = 2 ? (b) A three-dimensional representation of angular momentum is given by:
1 0 1 0 1 0 0 i 0 0 0 i 0 0 0 Jx = h B 0 0 i C ; Jy = h B 0 0 0 C and Jz = h B i 0 0 C : A @ A @ A @ 0 0 0 i 0 0 0 i 0 (This can be written in a compact notation as (Ji)jk = ih ijk .) Compute the rotation matrix R = exp(i n J=h). ^ ~16
Hint: The P matrix elements Rij can only be a linear combination of the three matrices: ni nj and ijk nk . Find the three coe cients.k
ij ,
14. Holstein-Primako transformation: There is a striking similarity between the equal spacing of the eigenvalues of Jz (in any particular representation of angular momentum labelled by an integer or half-integer j ) and the raising and lowering operators J+ and J on the one hand, and the equal spacing of the energy levels of a simple harmonic oscillator and the raising and lowering operators ay and a on the other q q hand. Show that if we de ne: Jz = h(j aya); J+ = h( 2j aya)a and J = hay 2j aya, ~ then they satisfy the SO(3) commutations relations and J 2 = j (j + 1)h2 ), if a; ay] = 1. (The square root of an operator is de ned as a Taylor series expansion, e.g. q 2 p 1 2j aya = 2j 1 1 a2ja 8 a2ja : : :]. 2 What are the maximum and minimum possible eigenvalues of aya in this representation?y y
15. Landau Levels: (a) Consider a spinless particle of charge e and mass m moving in a uniform magnetic eld, eB ~ say, B = B z. Find the energy levels in terms of the cyclotron frequency !c = mc . (Use the ^ ~ 1~ r `symmetric gauge' A = 2 B ~, and cylindrical coordinates.) (b) Let us now concentrate on the ground states. What do the probabilities (i.e j j2) look like physically? Suppose that the particle is constrained to move inside a large circular disc; large hc means that the area of the disc R2 >> the `Landau area' 2eB . What is the degeneracy of the ground state in terms of these two areas? 1 (c) Finally, assume that the particle is a spin 2 electron. Find the energy levels and degeneracies in that case. 16. Hyper ne structure: The hyper ne splitting in the H atom is caused by the interaction between the proton's mage ~ pe ~ netic dipole moment, ~ p = 2gMc Sp, and the electron's dipole moment ~ e = mc Se. (M and ~ ~ m denote the proton and electron masses, e is the electron charge, Sp and Se are the proton and electron spin operators, and gp = 5:59 is the Lande g-factor for the proton). The vec~ tor potential A produced by the proton's dipole moment (located at ~ = 0) is given to be r 3. (~ p ~)=r r (a) Show that the resultant magnetic eld felt by the electron is: ~ B = ~ 3p + 3~ ~rp5~ + 83 ~ p 3 (~). r r r r ~ (b) The hyper ne interaction is then given by ~ e B . Use rst-order perturbation theory to calculate the energy splitting between the j = 0 and j = 1 states in the 1s state of the H atom. What is the wavelength of light emitted in a transition between the two states? (Use the values M = 1840, a = 0:53A and = 1 .) 0 m 137 17. Calculate the geometric phase for a general representation of angular momentum labelled by j . ~ ~ ~ ~ (a) Namely, for H = B J , where J 2 = j (j + 1)h2 and B = B (sin cos ; sin sin ; cos ). Show that r normalised ground state is: the 2j P (2j)! (cos )2j p(sin )peip jj = j pi. j 0i = p=0 p!(2j p)! 2 z 2 (b) Now show that ih 0 jd 0i = 2j (sin 2 )2d . Hence the geometric phase for a closed loop enclosing the solid angle is j . 17
18. What is the Wigner function at nite temperature, W (x; p), for a simple harmonic oscillator? Show that one obtains the classical expression (upto a factor of h), if h! > 1, use the Thomas-Fermi method to calculate the ground state energy E0 , and the particle density (~), as a function of N . (Ignore internal degrees of freedom like spin in r this problem.) (b) Now use the exact expression for the energy levels to verify that the Thomas-Fermi value of E0 is correct to leading order in N . 20. Thomson scattering: Consider the elastic scattering of a high-energy photon from an electron in an atom. If the photon energy is much larger than the atomic binding energies (but much smaller than the rest mass energy of the electron), then it is su cient to do rst-order perturbation theory with the e2 ~ e ~ p term 2mc2 A2 . (The second-order perturbation amplitude due to mc A ~ is negligible due to the large energy denominator.) (a) If ~p; and ~p ; denote the initial and nal polarisation vectors, show that the di erential ~ ~ d 2 scattering cross-section is: d = r0 j~p ; ~p; j2, which is independent of p; p0 and the atomic ~ ~ e2 state of the electron. Here r0 = mc2 is called the classical radius of the electron. What is its numerical value? (b) Show that for 90 scattering, the nal photon is linearly polarised along the direction p p0 . ^ ^ Finally, compute the total unpolarised scattering cross-section.0 0 0 0
21. Bogoliubov transformation: (a) Given two bosonic operators a and ay (with a; ay] = 1), consider the Hamiltonian H = aya + aa + ayay, where > 0 but may be complex. Diagonalise the Hamiltonian, i.e. write it in the form H = 0byb + constant, where b; by] = 1 and 0 is a function of and ^ . Hence nd the ground state energy and wave function. Calculate hN i in the ground state, ^ where N = aya. Is this problem always well-de ned? If not, why not? (b) Given four fermionic operators a1, ay , a2 and ay (with fai; ayg = ij and all other anticom1 2 j y a + ay a ) + a a + ay ay . Here must be real mutators equal to zero), consider H = (a1 1 2 2 1 2 2 1 but may not be positive. Diagonalise H and answer the same questions as in part (a), except ^ 1 that N = ay a1 + ay a2 here. 2
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