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How do I see that when my hamiltonian is translation invariant i.e. H = H(r-r') it means that it is diagonal in the momentum basis? I can see it intuitively but not mathematically.
Translation invariant refers to a property of an object or system that remains unchanged under translation, which is the process of moving or shifting the object or system without changing its shape or size.
In math, translation invariance is closely related to symmetry and is often used to simplify mathematical problems. It allows us to focus on the underlying structure or pattern of a system without being affected by its position or location.
One common example of translation invariance in the real world is a chessboard. The rules and strategies of the game remain the same regardless of where the pieces are positioned on the board. This is because the game is translation invariant, meaning the outcome is not affected by the position of the pieces but rather the moves they make.
While translation invariance refers to the preservation of an object or system under translation, rotational invariance refers to the preservation under rotation. In other words, a system is rotationally invariant if its properties remain unchanged when rotated around a fixed point.
Translation invariance plays a crucial role in science and engineering as it allows us to simplify and generalize problems, making them easier to solve. It also helps us to better understand the underlying patterns and structures of complex systems, leading to advancements in various fields such as physics, computer science, and image processing.