Finding the Result of (a+a#)^n

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SUMMARY

The discussion focuses on calculating the expression \((a + a^\dagger)^n\), where \(a\) is the annihilation operator and \(a^\dagger\) is the creation operator. The user seeks a more efficient method than direct multiplication, especially for large \(n\), and suggests using Pascal's triangle. However, they acknowledge that the non-commuting nature of the operators complicates this approach, indicating that a general form for the result is not straightforward.

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Suppose a problem requires me to find the result of [(a+a#)^n ]ln> , where a is the annhilation operator , and a# is the creation operator , now n can be any number , suppose if it is 10 then will i write (a+a#) 10 times and then multiply each of these terms to see which are non zero , isn't there some general form of getting result of (a+a#)^n ?

The normal method of multiplying will become too cumbersome if n is a large number .
 
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Why not simply use Pascal's triangle?

edit: just realized that this might not actually works since the operators do not commute...
 

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