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hw1246 - H = 1 2 m p 2 1 2 mω 2 x 2 a Show that the...

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Physics 246, Spring 2007 Homework #1 Due in class, Wednesday, January 31, 2007 Feel free to discuss the problems with me and/or each other. Each student must write up his/her own solutions separately. Each problem is worth 10 points unless otherwise indicated. 1. Liboff, problem 3.3, p. 73. You may treat this as a short answer problem, i.e. for this problem only you can just give the answer without demonstrating it. 2. Liboff, problem 3.4, p. 73. 3. Liboff, problem 3.6, p. 76, any 4 parts. 4. Liboff, problem 3.16, p. 86. 5. (15 points) Consider the system with Hamiltonian
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Unformatted text preview: H = 1 2 m p 2 + 1 2 mω 2 x 2 . a) Show that the function φ = Ae-αx 2 is an eigenfunction of H for a particular α . What must α be? What are its units? b) What is the eigenvalue of H that corresponds to φ ? c) Show that xφ is also an eigenfunction of H . What is its eigenvalue? d) Compute Δ x and Δ p for the state in c). Compare the Δ x Δ p product with that of φ . (Hint: Note that some of the same integrals appear multiple times here. Also, be careful about normalization.)...
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