B2 Symmetry and Relativity Revision 1 TT 2020
Revision notes “Highlights” → basic things to remember (for material on syllabus)
2019 Revision Lectures 2019 TT revision lectures available on Panopto. Timings given here so you can skip the material which has dropped from the syllabus
2017 Q2 (a) ● Write down the components of the 4-wave vector K of an electromagnetic wave (you may assume that it is a 4-vector).
2017 Q2 (a) ● Show that the phase φ of the wave is Lorentz invariant. wave phase Space-time event 4-wave vector (already know it’s a 4-vector) Phase is therefore a contraction of two 4-vectors → a scalar Lorentz invariant
2017 Q2 (a) ● Show that a single, isolated electron, propagating in vacuum, cannot emit a photon. invalid
2017 Q2 (b) ● Two events in the laboratory frame S are characterised by the following 4-coordinates where x d,g are 3-vectors. Write down the condition for these events to be connected by a space-like interval.
2017 Q2 (b) ● Can we find an inertial frame S’ where these two events are occurring simultaneously? Find the answer without drawing diagrams. – Show that H can be transformed to H’ with H’ 0 =0 Choose axes such that y,z components zero Lorentz transformation H space-like
2017 Q2 (c) ● Two photons of the same angular frequency ω and with 4-momenta P 1 and P 2 move in the lab frame S. The first photon moves along the x direction, while the second photon moves along the y direction. Find the rest mass of the system as a function of ω.
2017 Q2 (c) ● Find the velocity of the centre of mass frame relative to the lab frame. – Geometric – 4-momentum total mass of 2-photon system, calculated before
2017 Q2 (d) ● In the lab frame S an electron is injected with initial 3-velocity v =(0,0,0) into a region with uniform, static and orthogonal magnetic and electric fields B =(0,0,B) (magnetic field) and E =(0,E,0) (electric field). Find the trajectory exactly in a suitably chosen frame in the given cases. Qualitatively describe, discuss and sketch in the lab frame.
2017 Q2 (d) z ● |E/B| > c B – Note invariants y E x – Choose frame S’ with B’ = 0
2017 Q2 (d) z ● Field transformations B y E x ● B || unchanged and need to subtract B in z direction, so choose S’ velocity v in xy plane ● Choose v in x direction to keep E simple as well
2017 Q2 (d) z ● Select frame by B calculating v, then find E’ y v E x
2017 Q2 (d) ● Now find E’
2017 Q2 (d) ● In S’, constant force due to E – Initial velocity is in the -x’ direction – Picks up -y’ velocity (q < 0!) due to E → parabolic? – Careful: can’t accelerate beyond c – Recall constant force problem → hyperbolic motion ● In S, note that electron starts from rest – Picks up -y velocity, bends toward +x
2017 Q2 (d) ● |E/B| < c ● Choose frame with E’’=0
2017 Q2 (d) ● In S’’, pure B field → circular motion – Larmor radius ● In S, electron drifts in +x direction while oscillating in y – As before, starts from rest, so starts by picking up speed in -y direction
2017 Q2 (e) ● For an isolated system of particles, each of which is non-interacting, consider the expression where W i and p i are the energy and 3- momentum of a particle from the isolated system of particles, respectively.
2017 Q2 (e) ● Show that s 2 is invariant. s 2 is a contraction of a 4-vector with itself → scalar invariant
2017 Q2 (e) ● Assuming the isolated system of particles is a non-interacting photon gas, find s 2 .
Next week ● Will try to add further revision material on new material, such as group representations
Recommend
More recommend
