quiz01

# quiz01 - Prob 3 If a Hermitian operator Q ˆ has...

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ECE 4060: Introduction to Quantum and Statistical Mechanics Quiz 1 Prob. 1. Given a wave function of a free particle at t = 0 as: ( 29 ( 29 - = otherwise x x ik A x 0 1 1 exp 0 , 0 ψ (a) Find A . (5 pts) (b) Graphically illustrate ( 29 ( 29 2 0 , 0 , x x P = and Re ( (x,0)), assuming 0 < k 0 < 1. (10 pts) (c) Assuming a dispersion relation of m k 2 2 G = ϖ , graphically illustrate P(x, t) and Re ( (x,t)) (10 pts) 1 P(x,0) Re ( (x,0)) x x P(x,t) Re ( (x,t)) x x

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Prob. 2. Given a wave function of a free particle at t = 0 as: ( 29 ( 29 ( 29 2 1 2 1 exp 2 1 exp 2 3 0 , k k x ik x ik x + = ψ (a) What is the expectation value of momentum p ˆ at t = 0 ? (10 pts) (b) If we measure the momentum at t = 0 , what are the possible outcome? What are the probabilities of obtaining those outcome? Briefly explain. (10 pts) (c) Find the expression for (x,t) . (5 pts)
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Unformatted text preview: Prob. 3 If a Hermitian operator Q ˆ has eigenfunctions 1 and 2 with corresponding eigenvalues of a 1 and a 2 , but a 1 ≠ a 2 . Is = 1 + 2 also an eigenfunction of the operator Q ˆ ? Briefly explain. (10 pts) Prob. 4 Given a particle in the parabolic potential below, use the Ehrenfest’s theorem to evaluate 2 2 p dt d (15 pts) and p dt d ∆ (10 pts). 2 2 2 2 2 2 1 2 ˆ x m dx d m H ϖ +-= Prob. 5 Check if the following operators are Hermitian. A simple explanation will be sufficient and no proof necessary. ( 15 pts ) (a) ( 29 2 ˆ x ; (b) x i 2 2 ∂ ∂-; (c) 2 2 k i ∂ ∂-; 3...
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quiz01 - Prob 3 If a Hermitian operator Q ˆ has...

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