Show that . That is, verify that unless the wave function is an eigen function of the mome...

Show that . That is, verify that unless the wave function is an eigen function of the momentum operator, there will be a nonzero uncertainty in momentum. Start by showing that the quantity

is (Δp)2. Then, using the differential operator form of  and integration by parts, show that it is also

(Note: Because momentum is real,  is real.) Together these show that if Ap is 0, then the preceding quantity must be 0. However, the integral of the complex square of a function (the quantity in brackets) can only be 0 if the function is identically 0, so the assertion is proved.