B2 Symmetry and Relativity Revision 1 TT 2020 Revision notes - - PowerPoint PPT Presentation

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 (already know it’s a 4-vector) 4-wave 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 xd,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 P1 and P2 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)

• |E/B| > c

– Note invariants – Choose frame S’ with B’ = 0

z y x

B E

2017 Q2 (d)

• Field transformations
• 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

z y x

B E

2017 Q2 (d)

• Select frame by

calculating v, then find E’

z y x

B E v

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
• scillating 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 Wi and pi are the energy and 3- momentum of a particle from the isolated system of particles, respectively.

2017 Q2 (e)

• Show that s2 is invariant.

s2 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 s2.

Next week

• Will try to add further revision material on new

material, such as group representations