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hw2aa - PHYS 507 Homework II Assigned Wednesday Due Friday...

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PHYS 507 Homework II Assigned: October 15, 2003, Wednesday. Due: October 24, 2003, Friday, at 5:00 pm. 1. Let T ( a ) be the translation operator corresponding to displacement a . (a) Calculate T ( a ) | p 0 i where | p 0 i is a momentum eigenket. (b) Let | φ i be the state obtained by translating the state | ψ i by a distance a , i.e., | φ i = T ( a ) | ψ i . What is the relationship between the momentum-space wavefunctions of these kets. (Relate ˜ φ ( p 0 ) to ˜ ψ ( p 0 )). 2. Let A = ( xp + px ) / 2. Define the operator S ( μ ) by S ( μ ) = exp - i ¯ h μA . (a) Re-express S ( μ 1 ) S ( μ 2 ) and S ( μ ) in a suitable form. Is S ( μ ) unitary? (b) Calculate i ¯ h [ A, x ] and i ¯ h [ A, p ]. (c) Calculate S ( μ ) xS ( μ ) and S ( μ ) pS ( μ ) . (d) Let | φ i = S ( μ ) | ψ i . Express the expectation values of the position and momentum in the state | φ i in terms of expectation values in | ψ i . (i.e., what is h x i φ = h φ | x | φ i in terms of h x i ψ and what is h p i φ in terms of h p i ψ ?) 3. Let the position-space wavefunction for a ket | ψ i be ψ ( x 0 ) = h x 0 | ψ i = Ne ikx 0 for 0 < x < a , 0 otherwise .
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  • Spring '11
  • starg
  • Complex number, Schwarz, schwarz inequality, uncertainty relation, position-space wavefunction, uncertainty relation be1

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