- #1
joycey
- 8
- 0
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).
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.
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.