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Unformatted text preview: Advanced Quantum Theory Amath 473, 673 / Phys 454 Problem Set 4 This assignment is due at the beginning of class on Monday, Nov. 29th, 2010. 1. Robertson Inequality aka Uncertainty Principle Define (Δ A ) 2 = h A 2 i - h A i 2 and (Δ B ) 2 = h B 2 i - h B i 2 , where the terms in the right hand side are quan- tum expectation values in the state | ψ i . Prove the Robertson inequality Δ A Δ B ≥ |h [ A,B ] i| / 2 and then explain the experimental procedures required to measure the product Δ A Δ B . (Hint: This proof can be found in most textbooks.) 2. Gaussian States and the Coordinate Representation A quantum state expressed in the position basis (also called the coordinate representation) is called a wave-function. Consider the following “Gaussian” wave-function, ψ ( x ) = h x | ψ i = A exp- ( x- x ) 2 4 σ 2 x exp( ik o x ) , where x is the position variable and all other parameters are constants. a) Show that h ˆ x m i = h ψ | ˆ x m | ψ i = R ∞-∞ dxx m | ψ ( x ) | 2 , were m is a positive integer, by using Dirac’s heuristic formula for the spectral resolution of the position operator, i.e., ˆ x = R dxx | x ih x | . b) Calculate (Δ x ) 2 ....
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- Fall '10
- Uncertainty Principle, unitary evolution