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hw06sol

# hw06sol - Introduction to Quantum and Statistical Mechanics...

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Introduction to Quantum and Statistical Mechanics Homework 6 Solution Prob. 6.1 For an infinite-wall with a small step potential defined as: ( 29 = 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) ( 29 = L x L x d x k B d x x k A x x 0 sin 0 sin 0 0 2 1 ψ where 2 1 2 F mE k = and ( 29 2 0 2 2 F V E m k - = (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) Matching the boundary condition at d and L for both ψ and d ψ /dx and cancelling out A and B by division, we obtain: ( 29 ( 29 ( 29 d k V E L d k E 1 0 2 tan tan - = - 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 ϖ φ = + - (a) If the particle is at the ground state, use the direct integration to find x and p . Check your answer with Eq. (6.12). The integration table for Gaussian functions at

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hw06sol - Introduction to Quantum and Statistical Mechanics...

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