Quantum Mechanics: Infinitesimal Translation Time Evolution Operator

[ber70]

Two quantum mechanics operators are infinitesimal translation and time evulotion operators.Is there an infinitesimal translation time evolution operator similar to relativistic mechanics?

 

[Fredrik]

The operator that does translations in spacetime can be written as

[tex]e^{-ia_\mu P^\mu}[/itex]

where P is the four-momentum. (The 0th component is the Hamiltonian). Note that there's a sum over $\mu$ from 0 to 3 in the exponent. The notational convention is to not write any summation sigmas when the summation index occurs exactly twice.

 

[Entropia]

Yes, there is an infinitesimal translation time evolution operator in quantum mechanics, which is similar to the concept of infinitesimal translations in relativistic mechanics. In quantum mechanics, this operator is known as the momentum operator, which describes the infinitesimal change in position of a quantum particle over time. Similarly, in relativistic mechanics, the four-momentum operator describes the infinitesimal change in position and time of a relativistic particle. Both of these operators play a crucial role in understanding the dynamics and behavior of particles at the quantum and relativistic level. However, it is important to note that while the concepts of infinitesimal translations and time evolution operators are similar in both quantum and relativistic mechanics, their mathematical formulations and applications are different due to the fundamental differences between these two theories.