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6.1: Problems

( \newcommand{\kernel}{\mathrm{null}\,}\)

  1. (a) Establish all the abstract groups having an order 2N6. Compute typical products. Which groups are Abelian? Indicate at least two isomorphic realizations for each group.

    (b) Identify the subgroups. Which are invariant?

  2. Write down the permutations of n=3 and n=4 objects. Arrange the result in a compact fashion. Consider at first the subgroup of even permutations (the alternating group). Make use of cycles.
  3. Find the joint effect of two mirror planes (see Figure B.1). Consider also parallel mirrors.

    Screen Shot 2020-08-07 at 3.38.51 PM.png

  1. A spherical wave pulse diverges from the space-time point (0, 0, 0, 0) in the inertial frame . Consider a frame moving along the z direction with the velocity β=tanhμ. The observer in sees also spherical wave fronts. However, the space-time points making up a surface r=ct=const do not look synchronous, hence spherical in . Show that the surfaces are ellipsoids of revolution with one common focus. Find the major and minor axes a, b, and the eccentricity in terms of r and β. Find also the lengths of the perihelion and the aphelion. Use polar coordinates.
  2. Consider the composition of rotations in the SU(2) formalism: U=UU where U=l0=ilσ, with

l0=cosϕ2,l=sinϕ2ˆu

(a) Express {l0,l} in terms of {l0,l} and {l0,l}.

(b) Refer to the Rodriues-Hamilton theorem (Figure 2.1) and obtain the cosine law of spher­ ical trigonometry.

(c) Obtain the sine law.

6. Check your general expressions by applying the special cases:

(a) U=UU=U2

(b) \(\hat{u} = \frac{1}{\sqrt{3}} (1, 1, 1),}&{\phi = \frac{2 \pi}{3}\)

\(\hat{u} = \frac{1}{\sqrt{3}} (1, 0, 0),}&{\phi = \frac{\pi}{2}\)

Note that U and U' generate symmetry operations on the cube.

7. Consider the one-dimensional motion of a particle of rest mass m, under the influence of a force eEz. At t=0 the particle is at rest. Show that the trajectory is represented in the z, ct plane as a hyperbola and find the semi-diameter. Develop the analogy with the cyclotron problem as far as you can. Discuss the significance of the approximation

γ1=1β21

8. Consider an electromagnetic field

f=E+iB

in a small space-time region. The Lorentz invariant of the field is:

f2=E2B2+2iEB=I1+iI2=g2exp(2iψ)

(a) Consider the case f20. In this case, a canonical frame exists in which EcanBcan and ζ=Bcan/Ecan, the pitch, is a real number (which could be 0 or ). Discuss the possible values of ζ according to the signs of I1 and I2. Summarize your conclusions in a table such as that shown in Table B.1.

Screen Shot 2020-08-07 at 4.01.57 PM.png

Table B.1: Table for Problem 8

(b) Express Ecan,Bcan,ζ in terms of I1,I2 and g,ψ.

(c) Assume ζ0,. Take ˆx along Ecan. Consider a passive Lorentz transformation in the ˆz direction, to a frame of velocity v(β=v/c=tanhμ) with respect to the canonical frame. Find tanθE,tanθB,tan(θEθB) in terms of β,ζ and also μ,ψ, where θE and θB are the angles by which the electric and magnetic fields rotate under the Lorentz transformation, as shown in Figure B.2.

Screen Shot 2020-08-07 at 4.05.24 PM.png

Figure B.2: Problem 8 coordinate frame and angles.

(d) Consider now the cases ζ=0;ζ=. Take ˆx in the direction of the non-vanishing canonical field. Discuss the effect of a Lorentz transformation similar to that considered in (c). Give the ratio of the magnitudes of the electric and magnetic fields after the Lorentz transformation.

9. (a) Find the polar decomposition of the matrix

(1ζ01)

Verify the relation (11b) on p. II-53. Consider the cases δ=1 and δ<<1.

(b) Find

Pˆa(pσ)Pˆa

where

Pˆa=12(1+ˆaσ)

10. Verify Eq’s (23) - (26) on II-42, 43.

11. Show that the field matrix F=(E+iB)σ can be derived from the matrix equivalent of the four-potential. What, if any, conditions are to be imposed on the latter?

12. (a) Express the reflection of a four-vector K=k01+kσ in a moving plane. The normal of the plane is ˆa. Its velocity is v=vˆa with v/c=tanhμ. (Hint:transform to the rest frame of the mirror.)

(b) Show that the combination of two mirrors v1=v1ˆa1, and v2=v2ˆa2 yields a Lorentz transformation.

13. Verify the equivalence of Equations (4) and (5) in Section 4.2 by transforming each factor from space-to the body-frame.

14. Show that the relation

|ξξ|=12(1+ˆkσ)

can be obtained through stereographic projection.

Hint: Project the sphere k21+k22+k23=1 from the south pole to the equatorial plane interpreted as the complex z-plane. Express k1,k2,k3 in terms of z,z and set z=ξ1/ξ0 with |ξ0|2+|ξ1|2=1.

15. Find the unitary matrix U that connects two given set of spinors with each other:

(|η,|ˉη)=(|ξ,|ˉξ)U

Express first its elements, then its components in terms of ξ0,ξ1,η0,η1.

16. The Pauli algebra can be considered as a generalization of elementary vector algebra and the knowledge of the latter is helpful in matrix manipulation.

However, one can approach the problem also from the converse point of view and derive the vector relations through matrix operations. Define

A=aσ,B=bσ,C=cσ

and associate

abwith12{A,B}=12(AB+BA)

a×bwith12i{A,B}=12i(ABBA)

Consider the Jacobi identity

[[A,B],C]+[[B,C],A]+[[C,A],B]=0

and the condition for associativity:

A(BC)(AB)C=0

(Equation B.1.5 is easily verified for commutators. For its significance see [Hal74] .)

Translate Equation B.1.5 and B.1.6 by means of Equations B.1.3 and B.1.4, and obtain the familiar relations for triple vector products.

17. Give explicit spinorial expressions for the following polarization forms: |x (linear polariza­ tion along the x-axis); |θ/2 (polarized at the angle θ/2 with the x-axis); |R (right circularly polarized).

(a) Use the ˆκ(ϕ,θ,ψ) scheme and assign ϕ=ψ=θ=0 to |x=(1,0). Express |θ/2,|θ/2,|R,|ˉR in terms of |x and |ˉx.

(b) Use the ˆs(α,β,γ) scheme. Assign β=0,α=γ=π/2 to |R. Express the above mentioned spinors in terms of |R and |ˉR. Note that the results of (a) and (b) are consistent with each other.

18. Give the matrix representations of a quarter wave, plate, a half wave plate, a rotator and a plane polarizer in both the ˆk and the ˆs schemes.

19. (a) We know of an optical instrument only that it transforms |R into |ˉR and vice versa. Find the most general matrix operator consistent with this fact

(b) Sharpen this answer by using the additional information that the instrument passes a beam |x unchanged. What is the name of this device?

20. Consider an arbitrary Hermitian 2×2 matrix: S=s0+sσ with s20s20 in general.

(a) Show that it is possible to decompose S into a sum of two matrices with determinant zero. That is:

S=K+K

where

K=k0+kσk20k2

K=k0+kσk20k2

(b) Show that if one imposes:

k=kˆkk=kˆkkandkparallel

the decomposition becomes unique. Find k0,k0,k,k,hatk.

21. Consider an approximately monochromatic beam of unpolarized light, it has been suggested that such a beam be considered as a random sequence of elliptically polarized light, whereby the parameters of ellipticity α,β vary slowly compared to 1/ω but fast compared to the time of observation (see [Hur45]). This author shows that the average ellipticity is given by the median value

(a2a1)m=tan(15)

This result can be obtained very simply. Assume that all representative points of the Poincare ́ sphere are equally probable. Consider the quantity:

S=2a1a2a21+a22

for an arbitrary point on the sphere.

Take the average of |S| over the Poincare ́ sphere, using the statistical assumption above.

Deduce the value

(a2a1)0

corresponding to |S|.


This page titled 6.1: Problems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by László Tisza (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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