Q1(k). Commutator of position and d/dx

Answer in brief

[x̂, d/dx] = −1; the operators do not commute, hence [x̂, p̂x] = iħ.

Key points

• An operator identity is found by acting on an arbitrary well-behaved function f(x).
• [x̂, d/dx]f = x df/dx − d(xf)/dx = x df/dx − (f + x df/dx) = −f (product rule).
• f is arbitrary ⇒ [x̂, d/dx] = −1; equally [d/dx, x̂] = +1.
• With p̂x = −iħ d/dx: [x̂, p̂x] = iħ ≠ 0 ⇒ non-commuting operators; uncertainty principle Δx Δpx ≥ ħ/2.