LSN/QFT-04001:: Second Quantization of Schrodinger Field

The essential components of classical and quantum theory of a point

particle are compared in the following table.  

The quantization of a classical system consists in identifying the commutators

of \(\mathbf q\,s\) and \(\mathbf p\,s\) with \(i\hbar \times \) Poisson bracket

\[\begin{equation}\big[q_j, p_k \big]_-= \big\{q_j, p_k\big\}_\text{PB} = i\hbar \delta_{jk}\end{equation}\]

In classical mechanics the states of a point particle are described by
generalized coordinates and canonical momenta. 
When we come to quantum description of a particle, we no longer describe the
states of a point particle by its coordinates and momentum. The dynamical
variables position and momentum become operators. The quantum
states are described by its wave function \(\psi(x,t)\). The wave function
obeys the Schrodinger equation
\[\begin{equation}\label{EQ01}i\hbar \pp[\psi(x,t)]{t} = -\frac{\hbar^2}{2m} \nabla^2 \psi(x,t) + V(x)\psi(x,t).\end{equation}\]

 This is the equation of motion of the quantum particle.

Second Quantization

The reason for introducing the language of second quantization is that it turns out to be
extremely convenient in the formulation of a quantum theory for many interacting particles.
The starting point of this chapter is the more familiar first quantized N-body Schrodinger
equation in the place representation, where the Hamiltonian of interest is motivated from
the study of ultracold atomic quantum gases. However, the resulting Hamiltonian is actually
seen to be much more general, such that it also applies to a large class of condensed-matter problems. 

Reference

Henk T.C., StoofKoos B., Gubbels Dennis B.M. and Dickerscheid, {\it Ultracold Quantum Fields}, Springer Netherlands (2009)  

 What is Second Quantization?

"Second quantization is a functor, first quantization is a mystery" ---E. Nelson 

as quoted by  Valter Moretti in

https://physics.stackexchange.com/questions/330428/first-quantization-vs-second-quantization 

 On Classical Quantum Connection --- A  Quote from Landau Lifshitz

Landau and Lifshitz in their book on Non-relativistic Quantum Mechanics write
Thus quantum mechanics occupies a very unusual place among physical theories:
it contains classical mechanics as a limiting case, yet at the same time it requires
this limiting case for its own formulation. 

An answer can be found in textbook by Schiff. We quote  

 " The quantization of a wave field imparts
to it some particle properties; in the case of the electromagnetic field, a
theory of light quanta (photons) results. The field quantization technique
can also be applied to a field, such as that described by the
nonrelativistic Schrodinger equation (6.16) or by one of the relativistic
equations (42.4)or (43.3). As we shall see (Sec. 46), it then converts a
one-particle theory into a many-particle theory; in the non relativistic case,
this is equivalent to the transition from Eq. (6.16) to (16.1) or (32.1).
Because of this equivalence, it might seem that the quantization of fields
merely provides another formal approach to the many-particle problem. However,
the new formalism can also deal as well with processes that involve the
creation or destruction of material particles (radioactive beta decay,meson-nucleon interaction" 

L.I. Schiff, ``Quantum Mechanics'', McGraw Hill Book Co. New York (1949).   

This lesson prepares the ground material for canonical quantization.
The act of quantization is complete with writing of
canonical commutation rules (CCR) in \eqref{qft-lec-04006;EQ06}
-\eqref{qft-lec-04006;EQ07}.
The CCR are very powerful and anything that we want to compute,
can now be computed.

Before we start computing quantities of physical interest,
we need to build the Hilbert space of all states of the system.
We also need to get physical interpretation of the mathematical objects and equations we have.