Second Quantized Representation of Quantum-Mechanical Operators

Zenonas Rudzikas

Bo‐Zang Li, WU JIAN-HUA, 中国科学院物理研究所,北京100080

In this work, a simple algebraic approach to the second quantization representation (SQR) for systems of many /identical particles is developed. In comparison with the traditional method, our method is simpler and elementary in mathematical operation, as well as is more transparent in exploring the mathematical structure and physical meaning of wave functions and mechanical quantities in SQR.

Philip Franklin

Ross Cressman

A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.

Donald H. Kobe

A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.