12b-prob5-10

12b-prob5-10 - 1 Ph 12b Homework Assignment No 5 Due 5pm...

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1 Ph 12b Homework Assignment No. 5 Due: 5pm, Thursday, 18 February 2010 1. Minimal uncertainty I: particle in one dimension (10 points). If we measure the Hermitian operator ˆ A in the state vector | ψ a , the variance of the measurement outcomes is A ) 2 = v v v v A ψ | p ˆ A -A ˆ A a P 2 | ψ a v v v v 2 , where A ˆ A a denotes A ψ | ˆ A | ψ a ; the standard deviation Δ A is also called the “uncertainty” of the observable ˆ A in the state | ψ a . The uncertainty principle , derived in class, is an inequality relating the product of the uncertainties for two Hermitian operators ˆ A and ˆ B to the expectation value of their commutator: Δ A Δ B 1 2 v v v A ψ | b ˆ A, ˆ B B | ψ a v v v . (1) The online lecture notes for the Feb. 4 lecture include a discussion of when eq.(1) is satis±ed as an equality — we have Δ A Δ B = 1 2 v v v A ψ | b ˆ A, ˆ B B | ψ a v v v . (2) if and only if there exists a real number γ and a complex number λ such that p ˆ A - ˆ B - λ P | ψ a = 0 . For a particle moving in one dimension, with position operator ˆ x and wave-number operator ˆ k = - i d dx , eq.(2) becomes Δ x Δ k = 1 / 2; we say that a wavefunction satisfying this condition has minimal un- certainty .

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12b-prob5-10 - 1 Ph 12b Homework Assignment No 5 Due 5pm...

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