Rotation Operator (quantum Mechanics) - The Translation Operator

The Translation Operator

The rotation operator, with the first argument indicating the rotation axis and the second the rotation angle, can operate through the translation operator for infinitesimal rotations as explained below. This is why, it is first shown how the translation operator is acting on a particle at position x (the particle is then in the state according to Quantum Mechanics).

Translation of the particle at position x to position x+a:

Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing):

Taylor development gives:

with

From that follows:

This is a differential equation with the solution .

Additionally, suppose a Hamiltonian is independent of the position. Because the translation operator can be written in terms of, and, we know that . This result means that linear momentum for the system is conserved.