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Physical laws like conservation of linear and angular momentum are important. For example, angular momentum was key to the solution of the hydrogen atom in chapter 4.3. More generally, conservation laws are often the central element in the explanation for how simple systems work. And conservation laws are normally the most trusted and valuable source of information about complex, poorly understood, systems like atomic nuclei.

It turns out that conservation laws are related to fundamental symmetries of physics. A symmetry means that you can do something that does not make a difference. For example, if you place a system of particles in empty space, far from anything that might affect it, it does not make a difference where exactly you put it. There are no preferred locations in empty space; all locations are equivalent. That symmetry leads to the law of conservation of linear momentum. A system of particles in otherwise empty space conserves its total amount of linear momentum. Similarly, if you place a system of particles in empty space, it does not make a difference under what angle you put it. There are no preferred directions in empty space. That leads to conservation of angular momentum. See addendum {A.19} for the details.

Why is the relationship between conservation laws and symmetries important? One reason is that it allows for other conservation laws to be formulated. For example, for conduction electrons in solids all locations in the solid are not equivalent. For one, some locations are closer to nuclei than others. Therefore linear momentum of the electrons is not conserved. (The total linear momentum of the complete solid is conserved in the absence of external forces. In other words, if the solid is in otherwise empty space, it conserves its total linear momentum. But that does not really help for describing the motion of the conduction electrons.) However, if the solid is crystalline, its atomic structure is periodic. Periodicity is a symmetry too. If you shift a system of conduction electrons in the interior of the crystal over a whole number of periods, it makes no difference. That leads to a conserved quantity called crystal momentum, {A.19}. It is important for optical applications of semiconductors.

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