Question

I would like to know if there is any explanation at the level of a first introdutory course in quantum mechanics of the fact that if we have a conserved quantity under a certain transformation, then that quantity is the generator of that transformation. For example we can prove using Noether's theorem that the momentum is conserved under a spatial translation. But why does it imply that the momentum operator is the generator of the translation transformation? I would like to stress the fact that I'm looking for an undergraduate level explanation, if there's any, like of an introductory course to quantum mechanics.

A) The conservation law associated with a transformation implies a specific mathematical relationship between the conserved quantity and the transformation itself, making the conserved quantity the generator of the transformation.

B) According to the principles of quantum mechanics, any quantity that remains constant under a transformation automatically assumes the role of the generator of that transformation.

C) Through Noether's theorem, the mathematical structure of symmetries and conserved quantities reveals that the conserved quantity corresponds to the generator of the transformation, offering a fundamental link between the two.

D) The transformation operator and the conserved quantity possess identical mathematical properties, establishing the conserved quantity as the generator of the transformation as stipulated by Noether's theorem.

Answer 1

In quantum mechanics, momentum is the generator of spatial translations because of Noether's theorem, which links conserved quantities to corresponding symmetries. The mathematical structure defined by the theorem and the commutation relations of quantum operators show why momentum is conserved under spatial translations.

Step-by-step explanation:

The reason momentum is considered the generator of spatial translations in quantum mechanics can be attributed to Noether's theorem, a fundamental principle in theoretical physics that connects symmetries and conservation laws. Specifically, Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. In the context of quantum mechanics, the momentum operator generates infinitesimal translations because it is the conserved quantity associated with the continuous symmetry of space under translations. This link between symmetries and conservation laws is mathematical and is reflected in the commutation relations of quantum operators, which are part of the foundation of the Heisenberg picture of quantum mechanics.

Conservation laws, like that of momentum, energy, and angular momentum, play a critical role in understanding physical phenomena. These conserved quantities are universally conserved and offer insights into the fundamental organization of nature. For example, conservation of momentum is essential for understanding the behavior of particles in collisions and interactions, where momentum must be conserved in all inertial frames, in both classical and relativistic contexts.