Some background information of the Quantization Recipe:

The purpose of this article is to give a summary of the theoretical background to the quantization recipe as it is presented in the lecture notes. In particular I want to point out, how the correspondence principle in the form of equation \eqref{eq:corr1}, which is used in the lecture notes, was originally found and why we have to find the conjugate momenta before applying the correspondence principle.

The following text is based on [1], whereas I didn´t always follow the same way of argumentation.

The Correspondence Principle:

"Between classical physics and quantum mechanics exists a formal analogy. A classical dynamic variable corresponds to a quantum mechanical hermitian operator."

This is mathematically expressed by the following equations:

\[ \begin{equation} \label{eq:corr1} \large\hat{\vec{X}} \rightarrow x, \hat{\vec{P}} \rightarrow \frac{\hbar}{i}\nabla_r \end{equation} \] \[ \begin{equation} \label{eq:corr2} \large \hat{\vec{P}}\rightarrow p, \hat{\vec{X}} \rightarrow -\frac{\hbar}{i} \nabla_p \end{equation} \] \[ \begin{equation} \label{eq:allgcorr} \large \left\{A,B\right\} = C \Leftrightarrow \left\{\hat A,\hat B\right\}_{QM} = \hat C = \frac{1}{i \hbar} \left[\hat A,\hat B\right]_{-} \end{equation} \] Where $\hbar$ is experimentally found to be the Planck quantum.

Equations \eqref{eq:corr1} and \eqref{eq:corr2} are specifications of equation \eqref{eq:allgcorr} which constitutes a general representation of the correspondence principle as it does not depend on whether the wave function is expressed in position space or in momentum space.

There is a couple of ways that lead to a quantum-mechanical description of a physical system, including path integrals. The Schrödinger equation can so far not be deduced from physical facts, without supposing analogies and implementation experimental results. There are two popular ways to get to the quantum mechanical equations of motion, described by Schrödinger and Heisenberg. Equations \eqref{eq:corr1} and \eqref{eq:corr2} are used in the Schrödinger formalism of quantization. Equation \eqref{eq:allgcorr} describes the correspondence principle in the Heisenberg formalism. Quantization in the Schrödinger Formalism: The Schrödinger formalism is based on analogies in the description of classical mechanics and optics.

In mechanics one can describe a problem by Newton, Lagrange or Hamilton formalism, but also via the concept of impact waves, which is an equivalent way, even if based on a different concept than the particle description.

De Broglie had the idea that classical mechanics should also be described by a more general wave mechanics considering that in electromagnetism the picture of rays of light particles as it is applied in optics also represents only a limiting case of the description of light as electromagnetic waves.

Comparing the mathematics for the mechanical impact waves with the Eikonal equation of optics he related the energy of a classical particle with a frequency \[ \begin{equation} \label{debroglie1} \large E=h\nu \end{equation} \] and the mechanical momentum with a wavelength, the well known de Broglie wave length,

\[ \begin{equation} \label{debroglie2} \large p=\frac{h}{\lambda} \end{equation} \] Where $\lambda$ and $h$ could be determined experimentally. Furthermore he found \[ \begin{equation} \label{debroglie3} \large E= i \hbar\frac{d}{dt}. \end{equation} \] Schrödinger expressed the idea of de Broglie mathematically and deduced the Schrödinger equation solving the wave equation using the above mentioned analogies.

Calculation of the expectation value of the quantum mechanical momentum operator in position space, using the de Broglie wave length and

\[ \begin{equation} \label{A62} \large pe^{\frac{i}{\hbar}px}=\frac{\hbar}{i}\frac{d}{dx} e^{\frac{i}{\hbar}px}, \end{equation} \] he found the correspondence principle eq.\eqref{eq:corr1}. In a similar way he found that the position operator actuates in momentum space as given by eq.\eqref{eq:corr2}. Even though the Schrödinger equation is the quantum mechanical equation of motion for the particular case of mechanical particles the found correspondence is generally valid since it can be deduced within the Heisenberg formalism of quantization without using the de Broglie analogies, which he found for mass carrying particles.

Quantization in the Heisenberg Formalism: Heisenberg parts from the fundamental uncertainty relation of quantum mechanics and introduces the Plancks quantum into the description of motion in quantum mechanics via substitution of the Poisson bracket in the Hamilton formalism (eq.\eqref{eq:klassBWGL}), as suggested in equation \eqref{eq:allgcorr}$^{*}$.

This access is more general as it says that any classical dynamic system can be quantized parting from the Hamilton formalism of classical motion. In specific the quantum mechanical dynamical equations of mass free systems as electro-magnetic waves can be found via the Heisenberg formalism.

The classical Hamiltonian equation of motion reads \[ \begin{equation} \label{eq:klassBWGL} \large \frac{dA}{dt} = \left\{A,H\right\} + \frac{\delta A}{\delta t} \end{equation} \] In the Hamilton formalism canonical coordinates are used, which can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation. Typical pairs of canonical coordinates are the usual Cartesian coordinates $q_i$ and their conjugate momenta $p_i$ [2]. The coordinates have to be substituted by operators that fulfill the correspondence principle, eq.\eqref{eq:allgcorr}.

That´s the same as substituting the Poisson bracket with a quantum mechanical bracket

$\left\{A,B\right\} \rightarrow \left\{\hat A,\hat B\right\}_{QM}$

that on the one hand should have equivalent properties as the Poisson bracket (satisfy the fundamental relations ${q_i,p_j}=\delta_{ij}$ etc.) and on the other hand has to be a measurable quantity.

These requirements are mathematically summed up in

\[ \begin{equation} \label{poissqm} \large \left\{\hat A,\hat B\right\}_{QM} = \hat C = i\alpha \left[\hat A,\hat B\right]_{-} \end{equation} \] where $\hat A,\hat B$ and $\hat C$ are hermitian operators. The constant $\alpha$ is experimentally found to be $1/\hbar$ (so far without contradictions) and this gives eq.\eqref{eq:allgcorr}.

Since $ \left\{\hat A,\hat B\right\}_{QM}$ is defined for canonical variables, we have to express the Hamiltonian in canonical coordinates when following the quantization recipe!

Heisenberg solves the so found quantum mechanical equation of motion developing the solution in plane waves and through this he gets to the de Broglie relations without the need of making analogies with mass carrying particles.

$^{*}$ The commutator is related to the product of the uncertainties of the expectation values as it is given by the generalized uncertainty relation, which is deduced via the Schwarz inequality:

$\Delta A \Delta B \geq 1/2 |\left\langle [A,B]\right\rangle|$

with the special case $\Delta_x \Delta_p \geq \hbar/2$.

Which is the consequence of measuring the commutator $[q_i, p_j]_{-} = i \hbar \delta_{ij}$ in a pure state. This commutator represents the Schrödinger correspondence principle, as it leads to equations \eqref{eq:corr1} and \eqref{eq:corr2} when applied to a pure state, that is represented in position space or momentum space respectively.

$\left[1\right]$ Grundkurs theoretische Physik 5/1, Wolfgang Nolting, ISBN3-540-40071-0

$\left[2\right]$, 11.06.2014