Please see attached file for full problem statement.
Suppose a particle is in some state |psi1>. Then, the probability P that it is found to be in some (other) state |psi2> during a measurement is given by the square of the absolute value of the inner product:
P = |<psi2|psi1>|^2
The inner product can be expanded in the position basis as:
<psi2|psi1> = Integral over x of <psi2|x><x|psi1> dx
= Integral over x of <x|psi2>*<x|psi1> dx
= Integral over x of psi2(x)*psi1(x) dx, (1)
where the star denotes complex conjugation and we used that, in the position basis, the components of a state constitute, by definition, the wavefunction. Note that, for a continuum of states, like the possible positions a particle can be in, the square of the absolute value of the inner product yields a probability density, not a probability. E.g.:
|<x|psi>|^2 = |psi(x)|^2 is the probability ...
We give a detailed proof of the fact that if the wavefunction is real valued, then the expectation value of the momentum is zero.