How Do You Solve These Commutator Relations?

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SUMMARY

The discussion focuses on solving commutator relations, specifically the expression of the form [AB, C]. Participants emphasize the importance of using mathematical induction and the identity proposed by user @Orodruin to simplify the problem. The conversation highlights the need to express complex commutators in terms of simpler two-operator commutators. This approach is essential for effectively tackling commutator relations in quantum mechanics or related fields.

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  • Understanding of commutator relations in quantum mechanics
  • Familiarity with mathematical induction techniques
  • Knowledge of operator algebra
  • Experience with the expansion of commutators
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  • Study the properties of commutators in quantum mechanics
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  • Explore the identity suggested by @Orodruin for simplifying commutators
  • Practice solving commutator relations with various operator combinations
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Students and professionals in quantum mechanics, physicists dealing with operator algebra, and anyone interested in advanced mathematical techniques for solving commutator relations.

Juli
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Homework Statement
Solve ##[\hat{r}^{17}, \hat{p}]## and ##[\hat{r}, \hat{p}^{250}]##
Relevant Equations
##[\hat{p}, \hat{x}^{n}] = - i \hbar n x^{n-1}##
Hello, I need to solve the commutator relations above. I found the equation above for the last one, but I am not sure, if something similar applys to the first one. I am a little bit confused, because I know there has to be a trick and you don't solve it like other commutator.
Thanks for your help!
 
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Are you familiar with the expansion of a commutator on the form ##[AB,C]##? If not, take the time to write it out and see if you can express it in terms of commutators containing only two operators.
 
You understand, of course, that the relevant equation is what you need to prove. You can do this using mathematical induction and the identity suggested by @Orodruin.
 

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