Quantization Of LC Circuit

Date:

1 August 2019

What is the problem ?

Derivation

Step 1 Write down the Hamiltonian, and find correspondence:

\[\begin{split}\begin{align*} & H(I,Q) = {1\over 2}LI^2 + {1\over 2}{Q^2\over C}\\ & H(p,q) = {p^2\over 2m} + {1\over 2}m\omega^2 q^2\\ \end{align*}\end{split}\]

\[\begin{split}\begin{align*} & L\leftrightarrow m & C \leftrightarrow {1\over m\omega^2}\\ & I\leftrightarrow {p\over m} & Q \leftrightarrow q \end{align*}\end{split}\]

Step 2 Results from Quantum-Harmonic-Oscillator then follows