Extended uncertainty principle. The generalized uncertainty principle (Equation 3.62) states that
σA2σB2≥41⟨C⟩2,
where C^≡−i[A^,B^].
(a) Show that it can be strengthened to read
σA2σB2≥41(⟨C⟩2+⟨D⟩2),(3.115)
where D^≡A^B^+B^A^−2⟨A⟩⟨B⟩. *Hint:* Keep the Re(z) term in Equation 3.60.
(b) Check Equation 3.115 for the case B=A (the standard uncertainty principle is trivial, in this case, since C^=0; unfortunately, the extended uncertainty principle doesn't help much either).