I have to find the inverse of the 1-D momentum operator.
The Attempt at a Solution
Here's my solution:
Pψ(x) = -iħ dψ/dx
P-1[Pψ(x)] = P-1[-iħ dψ/dx]
[P-1P]ψ(x) = -iħ [P-1 dψ/dx]
Iψ(x) = -iħ [P-1 dψ/dx]
ψ(x) = -iħ [P-1 dψ/dx]
By induction, P-1f(x) = (i/ħ) ∫f(x)dx.
On a secondary note, I am not sure what property guarantees the validity of the second sentence: why should two equal expressions remain equal if they are being operated on by the same operator, that is, is an operator allowed to map an element from a vector space to only a single element (and not multiple elements) of the same vector space?
Also, what property of an operator guarantees the validity of the third sentence: does the property of associaitivity hold for operators?
Finally, I am wondering if I had started by calculating the expectation value of P-1 in momentum space and then use the Fourier transform of the wave functions, would I have got the right answer?