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Homework Assignment 04 Continuous Spectra, Translation and Momentum, Canonical Commutators (Due: 02/18/2016) Problem 1 (40%) Consider a particle of mass m in the n-th stationary states of a one-dimensional infinite potential well with width of l , whose wavefunction in position representation is ⎛ nπ ⎞ 2 ψn (x) ≡ x n = sin ⎜⎜⎜ x ⎟⎟⎟ . l ⎝ l ⎟⎠ (a) [15%] Find the wavefunction in momentum representation ψn (p) ≡ p n . (b) [25%] Evaluate Δx ⋅ Δp for this state, where Δx = (ΔX )2 and Δp = (ΔP)2 . Then 1 verify that the uncertainty principle, Δx ⋅ Δp ≥ ! , is held for this state. 2 [Useful integrals: ⎤ 1 ⎡⎢ 2 1 2 ⎥, x sin (x) dx = x − x sin(2x)− cos(2x) ∫ ⎥ 4 ⎢⎣ 2 ⎦ 3 2 ⎛ ⎞ x x 1⎟ x cos(2x) 2 2 ∫ x sin (x)dx = 6 − ⎜⎜⎜⎜⎝ 4 − 8 ⎟⎟⎟⎠ sin(2x)− 4 . ] Problem 2 (36%) For a Hamiltonian given by P2 H= +V(X ) 2m (a) [10%] Evaluate the commutators [H, X ] and [[H, X ], X ] . [Note: please use the fundamental commutator and the properties of commutators to evaluate these. You need to master this important skill.] (b) [26%] Then prove that 2 2 (E − E ) k X n = , ∑ k n 2m k where En and Ek are eigenvalues of H , while n and k are the corresponding eigenstates, respectively. Problem 3 (24%) (a) [6%] Let x and p be the coordinate and momentum in one dimension. Evaluate the classical Poisson bracket {x,F(p)} , with F(p) a function of p. PB (b) [6%] Let X and P be the corresponding quantum operators, evaluate the commutator ⎡ ⎛ ⎞⎤ ⎢X, exp ⎜⎜ iPa ⎟⎟⎥ , ⎢ ⎜⎝ ⎟⎟⎠⎥ ⎢⎣ ⎥⎦ where a is a constant. (c) [12%] Prove that ⎛ iPa ⎞⎟ ⎟⎟ x α ≡ exp ⎜⎜⎜ ⎝ ! ⎟⎠ is an eigenstate of operator X. What is the corresponding eigenvalue? [Notes: you might find that the result in part (b) is useful. In this problem, we use the fundamental commutator and momentum operator P to show that exp(iPa / ) is the space translation operator; while in class we used the property of space translation operator to show that P is the momentum operator.]