I'm having difficulty trying to figure out which of the following is the correct method to properly evaluate the effect of the operators on f(x).(adsbygoogle = window.adsbygoogle || []).push({});

Given that,

[tex]\hat{A}f(x)=<x|\hat{A}|f>[/tex]

If the polarity operator, [tex]\hat{U_p}[/tex], and the translation operator, [tex]\hat{U_t}(a)[/tex], act as

[tex]\hat{U_p}f(x)=f(-x)[/tex]

[tex]\hat{U_t}(a)f(x)=f(x-a)[/tex]

Which of the following is the correct method of evaluating the commutator [tex][\hat{U_p},\hat{U_t}(a)]f(x)[/tex].

[tex]\begin{array}{rl}

\hat{U_p}\hat{U_t}(a)f(x)&=\hat{U_p}f(x-a)\\

&=f(-x+a)[/tex]

or

[tex]\begin{array}{rl}

\hat{U_p}\hat{U_t}(a)f(x)&=<x|\hat{U_p}\hat{U_t}(a)|f>\\

&=<-x|\hat{U_t}(a)|f>\\

&=<-x-a|f>\\

&=f(-x-a)[/tex]

Why do I get a different result? The order of the operators acting has obviously changed, but which is the correct order? I am tempted to believe the second case, but I can't really see the difference.

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# Commutator Troubles

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