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Unformatted text preview: PHYS 507 Homework III (Fall ’05) Assigned: October 17, 2005, Monday. Due: October 26, 2005, Wednesday, at 5:00 pm. 1. (a) Show that if X and Y are hermitian operators, then [ X,Y ] is anti-hermitian. (b) Show that for any operator A , the operator A † A is hermitian and has no negative eigenvalues. (Hint: First show that any expectation value is non-negative.) 2. Let | ψ i be the state of a certain particle in 1D which has the following position-space wavefunction ψ ( x ) = h x | ψ i = N exp- x 2 4 σ 2 + ikx ¶ . (a) Find the normalization factor N (based on h ψ | ψ i = 1). (b) Find the momentum-space wavefunction ˜ ψ ( p ) = h p | ψ i . (c) Using the momentum-space wavefunction compute h p i and h p 2 i . 3. Suppose that a particle in 1D has a position-space wavefunction which is real valued, i.e., ψ ( x ) * = ψ ( x ). This case is a frequently met in applications. (a) Show that the momentum-space wavefunction ˜ ψ ( p ) = h p | ψ i satisfies the property ˜ ψ ( p ) *...
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.
- Spring '11