1. Uncertainty Principle Let A and B be Hermitian operators. The uncertainty AA of  is defined as: (AA)² = ( = ((^- (^)

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1 Uncertainty Principle Let A And B Be Hermitian Operators The Uncertainty Aa Of A Is Defined As Aa 1 (72.66 KiB) Viewed 37 times

1. Uncertainty Principle Let A and B be Hermitian operators. The uncertainty AA of  is defined as: (AA)² = ( = ((^- (^))²), where <> denotes the expectation value with respect to a given state. The uncer- tainty principle states that (ΔΛ) (ΔΒ) Σ K[A, B]), valid for all states. = (a) Derive the canonical commutation relation for the position and momentum op- erators, [₁, P₁] ihd. You may assume that the position representation of the position and momentum operators are given by (rp) = -ih√(r) and (r) = r(r), respectively. [4 marks] (b) Assuming (1a) and the general uncertainty principle given above, state and prove Heisenberg's uncertainty principle. [4 marks] (c) Given the unit-normalised wavefunction -(x − x₁)²- ipor (x, t) = A exp exp cp (iPh.²) exp(-iwot) 4a² where To. Po, wo, a, and A are real constants, find the values of A, (î) = (v2), and (p) (p). Hint: you might find the following results useful: = Le e~1²³/²dy = √2π = ²y =y²e-²/²dy, -²/²dy = 0. ye