The relationship between the wavefunction-based approach in quantum mechanics (QM) and the field-based approach in quantum field theory (QFT) is intricate but connected. Here's a brief explanation of their relationship:
Quantum Mechanics (QM): In QM, particles are described by wavefunctions, which are solutions to the Schrödinger equation. The wavefunction represents the probability amplitude distribution of a particle in space and time. It provides information about the particle's position, momentum, and other observable properties. The wavefunction is a mathematical description of the quantum state of a single particle or a system of particles.
Quantum Field Theory (QFT): QFT extends the framework of QM to incorporate special relativity and describe systems with an arbitrary number of particles. In QFT, particles are treated as excitations or disturbances in underlying quantum fields. These fields permeate all of space and time and have associated operators. The quantum fields describe the behavior of elementary particles and their interactions.
Field Operators and Creation/Annihilation Operators: In QFT, the field operators represent the quantum fields, and they are expanded in terms of creation and annihilation operators. These operators create or annihilate particles in the field. The creation operator adds a particle to the field, while the annihilation operator removes a particle. The states of a QFT are described in terms of the occupation number of these particles.
Relation between QM and QFT: QFT can be seen as an extension of QM, where particles are treated as excitations of the underlying quantum fields. In this framework, particles described by QM are regarded as localized excitations or quanta of the corresponding quantum fields in QFT. The wavefunction in QM can be understood as the probability amplitude for finding a particle at a specific location, while the field in QFT describes the quantum state of the field at all points in space.
In summary, QFT provides a framework that unifies quantum mechanics and special relativity while describing particles as disturbances in quantum fields. It extends the concept of particles as described by wavefunctions in QM and provides a more comprehensive understanding of quantum behavior in systems with an arbitrary number of particles and interactions.