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hw06 - computed as a a m i p a a m x ˆ ˆ 2 ˆ ˆ ˆ 2 ˆ...

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1 Introduction to Quantum and Statistical Mechanics Homework 6 Prob. 6.1 For an infinite-wall with a small step potential defined as: ( ) = L x L x d V d x x x V 0 0 0 0 We would like to look at the formalism to find the ground-state energy E . Assume that E > V 0 . (a) Set up the eigenfunctions for 0 x d and d x L , respectively. (10 pts) (b) Derive the transcendental equation setup which can give the solution for the quantized ground energy E . No analytical solution for the transcendental equation needed. (10 pts) Prob. 6.2 For the simple harmonic oscillator described by the Hamiltonian: ) ( ) ( 2 1 ) ( 2 2 2 0 2 2 2 x E x x m x dx d m φ φ ω φ = + h (a) If the particle is at the ground state, use the direct integration to find Δ x and Δ p . Check your answer with Eq. (6.12). The integration table for Gaussian functions at http://mathworld.wolfram.com/GaussianIntegral.html can be helpful. (10 pts) (b) Use the promoter and the demoter to find Δ x and Δ p when the particle is at the ground state. Compare with the result in (a). The position and the momentum operators can be
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Unformatted text preview: computed as: ( ) ( ) a a m i p a a m x ˆ ˆ 2 ˆ ; ˆ ˆ 2 ˆ − = + = ⊥ ⊥ h h . (10 pts) (c) If the particle is at the first excited state, use the integration table to find x and p . (10 pts) (d) Use the promoter and the demoter to find x and p when the particle is at the first excited state. Compare the result in (c). (10 pts) (e) If the particle is only at the first three states with equal probability at t = 0 , write down the normalized wave function in the analytical form. (10 pts) (f) Following (e), use the promoter and demoter to compute x ˆ , p ˆ , x and p at t = 0 . Notice that the analytical form becomes much more manageable with ⊥ a ˆ and a ˆ . (20 pts) (g) Following (e), find x ˆ (t) and p ˆ (t) . Check whether m p dt x d ˆ ˆ = is always valid. (10 pts)...
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