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Wiemster
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Why do operators representing some symmetry commute with the Hamiltonian?
Symmetry in physics refers to the idea that certain physical laws and properties remain unchanged under certain transformations, such as rotations, translations, and reflections.
The Hamiltonian is a mathematical operator used in classical mechanics and quantum mechanics to describe the total energy of a system. It is defined as the sum of the kinetic and potential energies of all particles in the system.
In physics, symmetries are often associated with conserved quantities, which are physical properties that remain constant over time. The Hamiltonian, being a representation of the total energy of a system, is often related to symmetries through the Noether's theorem.
Some examples of symmetries in physics include translational symmetry (where the laws of physics are the same regardless of the location in space), rotational symmetry (where the laws of physics are the same regardless of the orientation in space), and time translation symmetry (where the laws of physics are the same regardless of the time).
The study of symmetry and Hamiltonian is important in physics because it helps us understand the fundamental principles that govern the behavior of physical systems. It also allows us to make predictions and calculations about the behavior of complex systems, such as atoms and molecules.