Classical to Quantum in Hamiltonian Formalism

Schrodinger’s Equation is the short-wavelength limit of HJ

\begin{array}{r} H (q, p, t) ψ = i ℏ \frac{\partial}{\partial t} ψ \end{array}

So, $λ = \frac{h}{p} \to 0$ .

So, $ψ (q, t) \propto \exp (\frac{i}{ℏ} S (q, t))$ . Then,

\begin{aligned} H (q, p, t) ψ & = lim_{ℏ \to 0} i ℏ \frac{\partial}{\partial t} \exp (\frac{i}{ℏ} S (q, t)) \\ = i ℏ \frac{i}{ℏ} ψ \frac{\partial S}{\partial t} \\ = ψ (- \frac{\partial S}{\partial t}) \\ [H (q, p, t) + \frac{\partial S}{\partial t}] & = 0 \\ H (q, p, t) + \frac{\partial S}{\partial t} & = 0. \end{aligned}

For a seperable, periodic, system, we get quantized action,

\begin{array}{r} J_{σ} = \oint p_{σ} d q_{σ} = n_{σ} h \end{array}

Consider a particle in a central force potential.

$P_{θ} = ℓ_{z} = C$ . Then,

\begin{aligned} J & = \oint P_{θ} d θ \\ = 2 π ℓ_{z} = n h \\ \Rightarrow ℓ_{z} & = \frac{n h}{2 π} = n ℏ . \end{aligned}

For a Hamiltonian not dependent on time,

\begin{aligned} E & = H (J_{1}, \dots, J_{n}) \\ = H (n_{1} h, \dots, n_{n} h) . \end{aligned}

Suppose $n_{σ} \to n_{σ} + 1$ . Then,

\begin{aligned} E^{'} & = H (J_{1}, \dots, J_{σ}^{'}, \dots, J_{n}) \\ = H (n_{1} h, \dots, (n_{σ} + 1) h, \dots, n_{n} h) \\ Δ E & = H (n_{1} h, \dots, (n_{σ} + 1) h, \dots, n_{n} h) - H (n_{1} h, \dots, n_{n} h) \\ = (1 h) \frac{\partial H}{\partial J_{s} i g m a} \\ = h ν_{σ} . \end{aligned}

\begin{aligned} [A, B] & = \sum \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial q} \\ \frac{d F}{d t} & = [F, H] + \frac{\partial F}{\partial t} . \end{aligned}

For Quantum Mechanics, everything becomes an operator,

\begin{aligned} [\hat{A}, \hat{B}] & = \sum \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial q} \\ \frac{d \hat{F}}{d t} & = \frac{1}{i ℏ} [\hat{F}, \hat{H}] + \frac{\partial \hat{F}}{\partial t} \\ = \frac{i}{ℏ} [\hat{H}, \hat{F}] + \frac{\partial \hat{F}}{\partial t} . \end{aligned}