Robin, you are correct about the sign mixup in the definition of the freely-falling frame. The exercise should actually say:

(Possibly Penrose forgot which set of coordinates were which: If you switch them around, the sign is correct as written - i.e. changing the sign actually gives you the

inverse coordinate transformation. However, the given expression for the wavefunction does indeed require the lower-case coordinates to be the original (non-falling) ones in order to make sense.)

You are also correct that there is a minor error in the expression for the wavefunction, however it isn't at all what your solution indicates. The error is just a missing

in the exponent of the

e. The equation in the exercise statement should actually look like this:

-----------------------------------------------------------------------------------------------

The method you used in your posted solution is a good one, however your result is

incorrect because you made an error expanding out the term

. It should be:

But you omitted the

term. I re-did your calculations with this term included, and you do indeed get:

Which differs from the book only by the missing factor of

.