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Syllabus
Why second quantization? Canonical quantization of Schrodinger field. Construction of Hilbert space (Fock Space) for second quantized Schrodinger
field. Quantization using anticommutators and identical fermions. Simple applications to computation of scattering cross section. Scattering in a
nonlinear second quantized Schrodinger equation.
Prerequisites
Postulates of quantum mechanics; Simulatneous measurement; Properties of commuting operators; Algebra of creation and annihilation operators. Harmonic oscillator energy eigenvalues. Transition amplitude and Fermi golden rule.
Instruction Goals
This unit basically teaches all the important tricks of the trade so that you can understand
- canonical quantization of fields
- utility of expansion of field operator in terms of a complete set of functions;
- introduction of creation annihilation and number operators;
- use of number operators as complete commuting set. Construction of Hilbert space (Fock Space);
- what corresponds to (Schrodinger) wave function for a system of particles;
- Using anticommutators and system of identical fermions
- Computation transition amplitudes and cross section for potential scattering.
The main aim is to build upon the knowledge of topics from quantum mechanics and to introduce all important ideas and applications in a familiar setting.
Lessons
- Canonical quantization
- Physical interpretation of second quantized theory. Expansion of fields in an orthonormal set;Fock space
- Examples of computing matrix elements; Second quantized fomalism as a many particle theory of identical bosons.
- Second quantization with anti-commutators and wave mechanics of a system of identical fermions.
Suggested Questions and Problems
- Computation of matrix elements
- Transition amplitudes as already done in wave mechanics
- Rutherford scattering; Potential scattering
- Scattering of identical particles, fermions and bosons, in a non-linear toy model.
Where do we need all, or some part, of this unit?
In this unit all the key ideas and techniques have been explained in simpler setting. The methods covered here will be repeated for second quantization of Klein Gordon, electromagnetic and Dirac fields.
