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## Homework Statement

I have to find the inverse of the 1-D momentum operator.

## Homework Equations

## The Attempt at a Solution

Here's my solution:

Pψ(x) = -iħ dψ/dx

P

^{-1}[Pψ(x)] = P

^{-1}[-iħ dψ/dx]

[P

^{-1}P]ψ(x) = -iħ [P

^{-1}dψ/dx]

Iψ(x) = -iħ [P

^{-1}dψ/dx]

ψ(x) = -iħ [P

^{-1}dψ/dx]

By induction, P

^{-1}f(x) = (i/ħ) ∫f(x)dx.

Any mistakes?

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?