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Problem%20Set%202%202008.1 - (scalar product is an integral...

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Problem Set #2 (Due Fri. Apr. 4, 12:00pm) 1. (10) Chap. 1, 12 2. (10) Chap. 1, 22 3. (10) Chap. 1, 23 4. (20) The normalized wave function of a one-dimensional particle is <x| ψ >= ψ (x)=N[exp( κ x) Θ (-x)+ exp(- κ x) Θ (x)] for some κ > 0. Here Θ (x) is the Heaviside step function, and N is real and positive. (a) What is N? (b) What is the expectation value of x? (c) What is the momentum space wavefunction <p| ψ >? (d) What is the expectation value of p 2 ? (e) Verify the expectation values of x 2 and p 2 satisfy the uncertainty principle inequality. 5. (20) Consider a vector space, where vectors are functions and the inner product
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Unformatted text preview: (scalar product) is an integral. For example, the collection of all complex functions f(x) of a real variable x with period 2 π forms a complex vector space. Define the inner (scalar) product by &amp;lt;f(x)|g(x)&amp;gt;= dx x g x f ) ( ) ( * ∫ − π a. Show that D=-id/dx is a Hermitian operator on this space. What are its eigenvalues and normalized eigenvectors? b. What are the eigenvalues and eigenvectors of D 2 =-d 2 /dx 2 ? c. Define the reflection operator R on this space by Rf(x)=f(-x) . Show that the commutator [ R,D ] ≠ 0. What is [ R,D 2 ]?...
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