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It's my first semester with quantum mechanics and I'm uncertain if my solution of this problem is correct, would be nice if someone could check and let me know

1. Homework Statement

1. Homework Statement

I have to calculate the representation of the state:

[itex]|\alpha \rangle \equiv exp[-i \textbf b \cdot \textbf x]|\phi\rangle [/itex], in momentum space.

Where [itex]\textbf b[/itex] is a constant vector.

## Homework Equations

- Dirac notation

- Position operator in momentum space: [itex]\textbf x=i\hbar\nabla [/itex](1)

- [itex]\langle \textbf p | \textbf p' \rangle = \delta(\textbf p-\textbf p') [/itex] (2)

3. The Attempt at a Solution

3. The Attempt at a Solution

First I multiply by [itex]\langle \textbf p |[/itex]

[itex]\langle \textbf p |\alpha \rangle = \langle \textbf p | exp[-i \textbf b \cdot \textbf x]|\phi\rangle [/itex]

Then I add a one to the equation:

[itex] =\int \langle \textbf p |exp[-i \textbf b \cdot \textbf x]|\textbf p'\rangle \langle \textbf p'|\phi\rangle d^3p'[/itex]

With (1):

[itex] =\int \langle \textbf p |exp[-i \textbf b \cdot \textbf i\hbar\nabla]|\textbf p'\rangle \langle \textbf p'|\phi\rangle d^3p'[/itex]

I'm unsure about this step. Can I move the bra [itex]\langle \textbf p |[/itex] past the operator [itex]exp[-i \textbf b \cdot \textbf i\hbar\nabla] [/itex] like this:

[itex] \int exp[-i \textbf b \cdot \textbf i\hbar\nabla] \delta(\textbf p-\textbf p') \langle \textbf p'|\phi\rangle d^3p'[/itex] ?

If I can then I would get:

[itex]= exp[-i \textbf b \cdot \textbf i\hbar\nabla] \langle \textbf p|\phi\rangle [/itex]

[itex]\Rightarrow \alpha(\textbf p) = exp[-i \textbf b \cdot \textbf i\hbar\nabla]\phi(\textbf p) [/itex]

I think I'm doing something wrong somewhere since I didn't do a fouriertransform anywhere.

Is this even the right approach?

Any hints/tips or corrections are very appreciated.

Kind regards

Alex