(a) A hypothetical operator is proposed to reverse the direction of motion: and , and…
PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model
Symmetry and Flavour, Ch Parity, charge conjugation and time reversal as operators, and the CPT theorem
Problem
(a) A hypothetical operator T^lin is proposed to reverse the direction of motion: T^linx^T^lin−1=x^ and T^linp^T^lin−1=−p^, and is assumed linear and unitary. Apply T^lin to the canonical commutation relation [x^,p^]=iℏ (one Cartesian component) in two different ways and show that this assumption leads to a contradiction, so that the true reversal-of-motion operator cannot be linear and unitary. (b) The correct operator T^=U^K^, with K^ complex conjugation, satisfies T^2=(−1)2j on a state of total angular momentum j (you need not rederive this). Use it to show that for a system built from an odd number of spin-21 fermions, T^2=−1 on every energy eigenstate, whatever the orbital angular momenta involved. (c) Kramers' theorem: given [H^,T^]=0 and T^2=−1, show that every energy eigenstate ∣ψ⟩ is at least two-fold degenerate. (Consider what would have to be true if T^∣ψ⟩ were a multiple of ∣ψ⟩ itself.) (d) A Be+ ion (three electrons bound to a beryllium nucleus) sits in a static external electric field, with no magnetic field present. Treat the nucleus as a fixed spinless charge, ignoring its own spin. Using parts (b) and (c), explain why every electronic energy level of this ion remains at least doubly degenerate however the electric field is chosen, and state what kind of field would be needed to lift that degeneracy.