PS9 - Physics 115A Problem Set 9 Due Tuesday November 24...

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Physics 115A, Problem Set 9 Due Tuesday, November 24 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths Section 3.4-3.6. 1 Proof that 0 = 1 ? Consider a Hermitian operator ˆ A with some eigenvector | a i with real eigenvalue a , ˆ A | a i = a | a i . Suppose | a i is normalizable and properly normalized, h a | a i = 1. Consider a second observable B with corresponding Hermitian operator ˆ B , and suppose that the two operators are canonically conjugate, i.e. [ ˆ A, ˆ B ] = i ~ . 1. Compute 1 i ~ h a | [ ˆ A, ˆ B ] | a i using the value of the commutator. 2. Compute 1 i ~ h a | [ ˆ A, ˆ B ] | a i using the fact that ˆ A is Hermitian and ˆ A | a i = a | a i for real eigenvalue a . 3. Use the results of the above two computations to show that 0 = 1. Obviously that’s nonsense – so where did we go wrong? Hint: Consider the case ˆ A = ˆ x and ˆ B = ˆ p . You know a lot about the eigenvectors of these operators. 2 Spectra of Operators 1. Show that two noncommuting operators cannot have a complete set of common eigenfunctions. Hint: Show that if two operators ˆ A and ˆ B have a complete set of common eigenfunctions, then [ ˆ A, ˆ B ] f = 0 for any function in Hilbert space.
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  • Winter '03
  • Nelson
  • Physics, mechanics, Hilbert space, Aˆ, generalized uncertainty principle

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