The choice between using a Quantum Field Theory (QFT)-style approach or a (non-relativistic) Quantum Mechanics (QM)-style approach depends on the specific physical situation and the scales involved. Here are some general considerations:
- Scale of the System: Quantum Mechanics is typically applicable to systems at non-relativistic scales, where the velocities involved are much smaller compared to the speed of light. It is commonly used to describe the behavior of individual particles, atoms, and molecules. Quantum Mechanics is particularly useful when dealing with systems that have discrete energy levels and are not subject to relativistic effects.
On the other hand, Quantum Field Theory is employed when dealing with systems at relativistic scales, such as high-energy particle interactions or the study of quantum fields. QFT incorporates both Quantum Mechanics and special relativity, allowing for the description of particles and fields that can have both particle-like and wave-like properties.
- Particle Interactions: If the primary focus is on describing the interactions and behavior of individual particles, Quantum Mechanics is usually sufficient. It provides a framework for calculating probabilities, wavefunctions, and observables for single-particle systems. It is applicable to situations such as atomic and molecular physics, condensed matter physics, and low-energy particle physics.
However, when studying systems with a large number of particles interacting at high energies or involving relativistic effects, Quantum Field Theory is essential. QFT enables the description of particle interactions by treating particles as excitations of underlying quantum fields. It is well-suited for high-energy particle physics, cosmology, and the study of fundamental forces.
- Time Evolution: In Quantum Mechanics, time evolution is described by the Schrödinger equation, which provides a deterministic evolution of the wavefunction. It describes how a quantum state changes with time, allowing the calculation of probabilities and expectation values.
Quantum Field Theory also incorporates time evolution but treats particles as excitations of quantum fields that span all of spacetime. The dynamics of these fields are governed by quantum field equations, such as the Klein-Gordon equation or the Dirac equation. Time evolution in QFT involves creating and annihilating particles, and the calculations often require advanced mathematical techniques.
- Relativistic Effects: Quantum Mechanics does not inherently incorporate special relativity. It treats time and space separately and does not account for effects such as time dilation or length contraction. If the system under study involves relativistic effects, such as high velocities or strong gravitational fields, then a QFT-style approach is necessary.
In summary, Quantum Mechanics is suitable for non-relativistic systems with a small number of particles, discrete energy levels, and low energies. Quantum Field Theory is employed for relativistic systems, high-energy particle interactions, and the study of quantum fields. The choice between QM-style and QFT-style approaches depends on the scale, particle interactions, time evolution, and presence of relativistic effects in the system being studied.