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Transcript
k g vz xz f Quantization of the electrical LC harmonic oscillator: parallel LC oscillator circuit: voltage across the oscillator: total energy (Hamiltonian): with the charge Q stored on the capacitor a flux f stored in the inductor properties of Hamiltonian written in variables Q and f Q and f are canonical variables see e.g.: Goldstein, Classical Mechanics, Chapter 8, Hamilton Equations of Motion Quantum version of Hamiltonian with commutation relation compare with particle in a harmonic potential: analogy with electrical oscillator: - charge Q corresponds to momentum p - flux f corresponds to position x Hamiltonian in terms of raising and lowering operators: with oscillator resonance frequency: Raising and lowering operators: number operator in terms of Q and f with Z c being the characteristic impedance of the oscillator charge Q and flux f operators can be expressed in terms of raising and lowering operators: Exercise: Making use of the commutation relations for the charge and flux operators, show that the harmonic oscillator Hamiltonian in terms of the raising and lowering operators is identical to the one in terms of charge and flux operators.
