A path contributing to ⟨ ⟨ x ^ ^ ⟩ ⟩ − − at some time must have negative momentum − − ℏ ℏ k 0 by definition; since every part of the wavepacket originated at the same positive momentum ℏ ℏ k 0 behavior, this path from the origin to x and from initial ℏ ℏ k 0 momentum to final − − ℏ ℏ k 0 momentum can be time-reversed and translated to create another path from x back to the origin with the same initial and final momenta. This second, time-reversed path is equally weighted in the calculation of n − − ( x , t ) and ultimately results in ⟨ ⟨ x ^ ^ ⟩ ⟩ − − = 0 . The same logic does not apply to ⟨ ⟨ x ^ ^ ⟩ ⟩ + because there is no initial population in the momentum state − − ℏ ℏ k 0 . Thus, the wavepacket variance only has the first term: