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# commutator - you can reverse the order of multiplication by...

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There are several rules for working with commutators and operators. First, you may move constants like 2, i (square root of minus 1) , h-bar, etc. in or out of the brackets at will. Note that constants include expectation values: in the formulas for standard deviations you will see things like <x> ... this is a constant. Next is the “associativity rule”: you can divide a commutator into two or more separate parts if the parts are separated by + or - signs. The order of sums does not matter: ] ˆ , ˆ [ ] ˆ , ˆ [ ] ˆ , ˆ [ ] ˆ , ˆ [ ] ˆ ˆ , ˆ [ ] ˆ ˆ , ˆ [ B A C A C A B A B C A C B A + = + = + = + Next are facts that apply to the operators in general:
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Unformatted text preview: you can reverse the order of multiplication by a variable, say x, and differentiation by a different one, say y dy y x df x dy y x xf d ) , ( )) , ( ( = What you can’t do is reverse the order of multiplication by x and differentiation by x (or y and differentiation by y, etc.) dx y x df x dx y x xf d ) , ( )) , ( ( ≠ Finally, while 2 2 2 2 2 ) ( and ) ˆ ( dx d dx d x x x x = = ⋅ = it is NOT true that 2 2 2 2 x is ) ( dx d dx d x , rather, you have to work it out...
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