Time evolution of a one-dimensional free particle wavepacket

3 Time evolution of a one-dimensional free particle wavepacket 

Consider now the time evolution of a free particle moving in one dimension that starts at Z = 0 in the normalized state 

We refer to the wavefunction (2.98) in the present context as a wavepacket It is a local function in position space that can obviously be expanded in the complete set of free particle waves (2.82). Below we see that this function is also localized in momentum space. 

Problem 2.15. Show that for the wavefunction (2.98) the expectation values of the position and momentum are 

Also show that the position variance associated with this wavefunction is 

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