A superconducting circuit consists of an inductor of inductance and a capacitor of…
PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Quantum Mechanics · Wave mechanics in one dimension
The Structure of Quantum Mechanics, Ch The oscillator as an algebra
Problem
A superconducting circuit consists of an inductor of inductance L=1.00nH and a capacitor of capacitance C=1.00pF, joined to form a single closed loop with no resistance. Let Q^ be the charge on the capacitor and Φ^ the flux through the inductor; in the quantum treatment of the circuit these obey the canonical commutator [Q^,Φ^]=iℏ, exactly as x^ and p^ do for a particle, and the circuit's energy operator is H^=2LΦ^2+2CQ^2.
((a)) Say which of the mechanical oscillator's symbols (x^,p^,m) correspond to (Q^,Φ^,L), define ladder operators a^,a^† for this circuit by the same construction used for the mechanical oscillator, with ω=1/LC, verify [a^,a^†]=1 follows from [Q^,Φ^]=iℏ, and state what the eigenvalue n of a^†a^ counts physically for this circuit.
((b)) Compute ω in rad s−1, the energy quantum ℏω in electronvolts, and the zero-point energy, for the given L and C.