[LSN/QFT-04004] Bosons and Fermions

Connection between second quantized Schrodinger theory and
nonrelativistic quantum mechanics. Hilbert space of second quantized
theory;

Quantum mechanics of identical particles;
Symmetrization postulate
Spin statistics connection; Bosons and fermions.

1. To recall and discuss quantum mechanics of identical particles; Symmetrization postulate.
2. To show and establish connection of states in second quantized theory and wave function in quantum mechanics. To demonstrate that CCR quantization leads to correctly symmetrized wave function for a two particle state.
3. To briefly describe the connection between description of assembly of many identical bosons in quantum mechanics and Hilbert space of the second quantized Schrodinger theory}. To describe briefly the changes needed in {quantization of Schrodinger field to describe system of identical fermions}.
4. To give a brief about spin statistics connection

... 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).

The equivalence of the operator (second-quantized) description and the
ordinary Schrodinger theory description of a system of n particles was
established by Jordan and Klein [Jordan (1927)] for particles obeying
Bose statistics and by Jordan and Wigner [Jordan (1928b)] for particles
obeying Fermi statistics. The operator formalism was subsequently re-
formulated in " Fock space" by Fock (1932) who gave the generalization
of the ordinary Schrodinger wave mechanics to systems for which the
number of particles is not a constant of the motion. Fock's work was
based on some previous investigations along similar lines by Landau and
Peierls [Landau (1930)] who gave a configuration space treatment of the
quantized electromagnetic field interacting with matter. Landau and
Peierls' treatment was in turn suggested earlier by Oppenheimer,
Heisenberg, and Pauli [Heisenberg (1930)] who sketched a configuration space
treatment of quantized field 'theories.
\paragraph*{Reference:}

Schweber S. S., An Introduction to Relativistic Quantum Field Theory,  Row, Peterson And Company (1961).