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# hw2333 - PHYS 507 Homework II(Fall'06 Assigned Friday Due...

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PHYS 507 Homework II (Fall ’06) Assigned: November 17, 2006, Friday. Due: November 26, 2006, Sunday. 1. Consider a state of a particle in 1D having the following position-space wavefunction ψ ( x 0 ) = N exp - | x 0 | a , where a is a distance and N is an appropriate normalization factor. (a) Find N and then compute the momentum-space wavefunction ˜ ψ ( p 0 ). (b) Compute the probability density for momentum distribution ˜ P ( p 0 ) and sketch a plot of it. Are opposite momenta values ( p 0 and - p 0 ) equally probable? (c) Compute Δ x and Δ p and show that the uncertainty relation is satisfied. Hints: (1) You can compute expectation values by using either ψ or ˜ ψ . Use whichever is convenient. (2) Note that h p 2 i = h ψ | p 2 | ψ i = h | i = || || 2 . As a result, h p 2 i can be computed by using position-space wavefunction by taking only a single derivative. This is an extremely useful trick. (3) The Γ-function (factorial) integral might be useful in here Z 0 u n e - u du = n ! .

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hw2333 - PHYS 507 Homework II(Fall'06 Assigned Friday Due...

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