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 #86636, Physics / Atomic and Nuclear Physics:

Given:

$$\mathrm {A} = a \hat {x} + i b \hat {p}$$

Where, a and b are constants and $\hat{x}$ and $\hat{p}$ position and momentum operator.

To find:

$$[ \mathrm {A}, \hat {x} ] \text { and } [ \mathrm {A}, \mathrm {A} ]$$

Solution:

(1) [A, $\hat{x}$]

$$\begin{array}{l} = [ a \hat {x} + i b \hat {p}, ] \\ = a [ \hat {x}, \hat {x} ] + i b [ \hat {p}, \hat {x} ] \end{array}$$

But $[\hat{x}, \hat{x}] = 0$ and $[\hat{p}, \hat{x}] = -i\hbar$

$$\begin{array}{l} = \mathrm {i b} ^ {*} (- i \hbar) \\ = \mathrm {b} ^ {*} \hbar \end{array}$$

(2) [A, A]

$$\begin{array}{l} = [ a \hat {x} + i b \hat {p}, a \hat {x} + i b \hat {p} ] \\ = a ^ {\wedge} 2 [ \hat {x}, \hat {x} ] + i a b [ \hat {x}, \hat {p} ] + i a b [ \hat {p}, \hat {x} ] - b ^ {\wedge} 2 [ \hat {p}, \hat {p} ] \\ \end{array}$$

But $[\hat{x}, \hat{x}] = 0, [\hat{x}, \hat{p}] = i\hbar, [\hat{p}, \hat{x}] = -i\hbar$ and $[\hat{p}, \hat{p}] = 0$

$$\begin{array}{l} = - a b + a b \\ = 0 \end{array}$$

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