Mostly it means that we’re doing non-relativistic quantum mechanics, the only kind we’re likely to encounter in undergraduate classes. This is an approximation that is valid, accurate, and very useful as long as we’re dealing with energies small enough that relativistic effects are negligible.cosmanino2050 said:TL;DR Summary: What are the implications that the wavefunction is not Lorentz invariant?
What are the implications that the wavefunction is not Lorentz invariant?
A wavefunction in quantum mechanics is a mathematical description of the quantum state of a system. It contains all the information about a particle or system of particles and is typically represented by the symbol Ψ (Psi). The wavefunction's absolute square, |Ψ|², gives the probability density of finding a particle in a particular position or state.
Lorentz invariance is a principle from the theory of relativity stating that the laws of physics are the same for all observers, regardless of their relative motion. It implies that physical phenomena do not change under Lorentz transformations, which include rotations and boosts (changes in velocity) in spacetime.
The wavefunction is expected to transform in a specific way under Lorentz transformations to ensure that the probabilities it describes remain consistent for all observers. For spin-0 particles, the wavefunction transforms as a scalar field, while for particles with spin, it transforms according to representations of the Lorentz group appropriate to their spin (e.g., spinors for spin-1/2 particles).
No, the Schrödinger equation is not Lorentz invariant. It is formulated for non-relativistic quantum mechanics and does not incorporate the principles of special relativity. For relativistic scenarios, the Klein-Gordon equation or the Dirac equation is used, which are Lorentz invariant and describe spin-0 and spin-1/2 particles, respectively.
Lorentz invariance is crucial in quantum field theory because it ensures that the theory is consistent with special relativity. This invariance guarantees that the physical laws described by the theory are the same for all observers, which is essential for accurately describing high-energy processes and fundamental interactions in nature.