In this Chapter we apply the canonical quantization, meaning that the operators in our Hamiltonian themselves become quantum. The classical-quantum Analogy is the foundation upon which circuit quantization rests. In Devoret's words, the central mission is really:
"Quantum mechanically there's a huge difference between the variables, which are what the Hamiltonian depends on, and the parameters. This is your mission: to really distinguish between the two." [1]
In the comparison with a classical system we had the problem that $V$ and $I$ do not behave like generalized positions and velocities.
To solve it we defined a new dynamic variable $\phi$, as before this is the macroscopic phase.
$\phi(t) = \int_{-\infty}^{t} V(\tau) d\tau, \quad V = \dot{\phi} = L\dot{I} \Rightarrow I = \phi/L$
We need to identify the conjugate variables, this is done by differentiating the Lagrangian with respect to $\dot\Phi$.
Q and $\Phi$ are the dynamic variables.
What is $\dot\Phi?$
and what is Q , why is it the dynamical variable?
$Q = \frac{\partial \mathcal{L}}{\partial \dot{\Phi}}$
Lagrangian Mechanics By the principle of least action the Lagrangian $\mathcal{L} = \underbrace{T}{kinetic} - \underbrace{V}{potential}$ describes the dynamics of the system.
Typically, the Hamiltonian is formulated from the Lagrangian mechanics description as$H(t, \underbrace{q_i}{\text{position}}, \underbrace{p_i}{\text{conjugate momentum }}) = \sum_{i} p_i \dot{q}_i - L(t, q_i, \dot{q}_i)$
The conjugate momentum is often not the kinetic momentum!
$p = \partial L / \partial \dot{q}$
Legendre Transform
The goal is to write a workable Hamiltonian for a conserved energy system, since that is the natural language for analyzing conservation laws and the geometry of phase space.
By Noether’s theorem, we know that if we have a conserved quantity, we also have a symmetry: in the time translation $t \to t + \tau$, we have the requirements of the following conservation: $\partial L / \partial t = 0$.