commutators and translation operators

Hi,

I'm currently working through Julian Schwinger's "Quantum Mechanics"
book and there's an exercise I cannot solve. Below q and p are
operators with [q,p] = i and q' is a number.

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Ex. 1-55: Arrange

[1]: exp(-i f(q)) exp(i q' p) exp(i f(q))

in different ways to show that:

[2]: exp(i q' (p + df/dq (q)))
= exp(i q' p) exp(i (f(q) - f(q - q')))
= exp(i (f(q + q') - f(q))) exp(i q' p)

where

[3]: df/dq (q) = f'(q) = i[p, f(q)] (to show this is the exercise
before this one)
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I could easily show that [1] is equal to the second and third term of
[2] by inserting

1 = exp(-i q' p) exp(i q' p)

on either the left or the right side of of equation [1].

But I'm not at all able to show that this is equal to the left hand
side of [2]. The proof is probably very basic. At this point in the
book he hasn't even shown that:

<q'|p'> = 1/sqrt(2 pi) exp(i q' p')

On a more general note: Is the following also true (for arbitrary g)?

g(p + df/dq)(q)) = exp(-i f(q)) g(p) exp(i f(q))

If it is, then this solves the exercise above of course, but I couldn't
show it either.

I would appreciate any help on this.

Regards,
Boris

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