7.3 Conservation Laws and Symmetries

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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