Question

For the operator A = a x + i b p where a and b are constants, calculate [A, x] and
[A, A].

Answer 1

Answer on Question #85009 Physics / Quantum Mechanics

For the operator $\hat{A} = a\hat{x} + ib\hat{p}$ where $a$ and $b$ are constants, calculate $[\hat{A}, \hat{x}]$ and $[\hat{A}, \hat{A}]$ .

Solution:

The commutation relation between position and momentum operators is as follows

$$[ \hat {p}, \hat {x} ] = - i \hbar$$

So

$$\left[ \hat {A}, \hat {x} \right] = \left[ a \hat {x} + i b \hat {p}, \hat {x} \right] = a \underbrace {\left[ \hat {x} , \hat {x} \right]} _ {0} + i b \underbrace {\left[ \hat {p} , \hat {x} \right]} _ {- i \hbar} = b \hbar$$

For any operator $\hat{A}$

$$\left[ \hat {A}, \hat {A} \right] = 0$$

Answer: $\left[\hat{A},\hat{x}\right] = b\hbar$ , $\left[\hat{A},\hat{A}\right] = 0$

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