ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 12

(March 31, 2009)

(51) For light from some galaxy the Balmer spectrum of the hydrogen atom is found at wavelengths λ = (729.2 nm) m^{2}/(m^{2}−4), m = 3, 4, 5, …. Neglect expansion of space and find the speed at which the galaxy moves away or towards us. Due April 1 before class (4 points).

(52) (1) Taylor expand \sqrt{x + ϵ} to leading order in ϵ. (2) Substitute x = c^{4}m_{0}^{2} and ϵ = c^{2}p^{2}. (3) Compare with mv^{2}/2. Due April 1 in class (4 points).

(53) Derive for the momentum in the rest frame p^{0} = c m_{0}, where m_{0} is the rest mass, from the relativistic transformation laws and the known definition of momentum in the non-relativistic limit. Due April 1 in class (4 points).

(54) (1) Write down the definition of ∂_{α}. Due April 3 in class (1 point).

(2) Let A^{α} be a vector field. Write down the scalar form this and ∂_{α} (no scalar operators). Due April 3 in class (1 point).

(3) Use the quantities of assignments (1) and (2) to write down a rank two tensor T^{αβ} field (no tensor operator). Due April 3 in class (1 point).

(4) Write the equation

∂_{α} F^{αβ} = \frac{4π}{c} J^{β}

explicitly out for β = 0, 1, 2, 3. Due April 6 in class (4 points).

(5) Compare the β = 0 result with ∇E⃗ = 4πρ and the β = i = 1, 2, 3 results with ∇ × B⃗ − c^{−1} (∂E⃗/∂t) = 4πJ⃗/c. Due April 6 in class (4 points).

(55) An electron and a positron, each with mass equal to 0.511 MeV, annihilate at rest into two photons. Chose the rest frame of the positronium, such that one of the photons moves in positive x = x^{1} direction.

1. For each photon find the momentum 4-vector.

2. Find the 4-momentum in a frame that moves at a velocity cβx̂ with respect to the rest frame of the positronium.

3. Suppose that the annihilation took place 10^{9} years ago in a galaxy that is receding from us at β = 4/5. What is the energy of the photon that we observe?

Due April 6 before class (10 points).