Question

For the operator A=ax+ibp where a and b are constants, calculate [A,x] and [A,A]

Answer 1

Answer on Question #85019 – Math – Complex Analysis

Question

For the operator $A = ax + ibp$ where $a$ and $b$ are constants, calculate $[A,x]$ and $[A,A]$.

Solution

Fundamental commutation relation: $[x,p] = i\hbar$

$$[A, x] = [ax + ibp, x] = a[x, x] + ib[p, x] = ib(-i\hbar) = b\hbar$$ $$\begin{array}{l} [A, A] = [ax + ibp, ax + ibp] = a[x, ax + ibp] + ib[p, ax + ibp] \\ = a^{2}[x, x] + iab[x, p] + iab[p, x] - b^{2}[p, p] = 0 + iab(i\hbar - i\hbar) + 0 = 0 \end{array}$$

Answer: $[A,x] = b\hbar$ and $[A,A] = 0$.

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