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I hope that this addresses your question(s). In the following we set $\hbar=1$.

Let us first clarify that things like $\hat{x}\,\psi(x)$ in fact mean

$$ (\hat{x}\psi)(x) \equiv \langle x| \hat{x} |\psi\rangle \quad .$$ We further define the wave function as $$ \psi(x) \equiv \langle x|\psi\rangle \quad. $$

Similar equations hold for the case of the momentum operator / basis.

The position representation of the position and momentum operator is (very nicely explained here):

\begin{align} \langle x|\hat{x} &= \langle x| \,x \\ \langle x| \hat{p} &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \langle x| \quad . \end{align} This should look familiar to you: Evaluated in the position basis, the position operator acts as a multiplicative operator while the momentum operator acts as a derivative. Using the conventions from the beginning we then find \begin{align} (\hat{x}\psi)(x) &= x\, \psi(x) \\ (\hat{p}\psi)(x) &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \psi(x) \quad , \end{align}

which means for example that the projection of the vector $\hat{x}|\psi\rangle$ onto a state of the position basis is the number $ x\, \psi(x)$.

However, this does not mean that the momentum operator is a derivative; in the same sense, the position operator is not an operator multiplying 'everything' it acts on with $x$. Indeed, evaluated in the momentum basis, we find that the momentum operator is multiplicative and the position operator acts as a derivative: \begin{align} \langle p | \hat{x} &= i\, \frac{\mathrm{d}}{\mathrm{d}p} \langle p|\\ \langle p | \hat{p } &= \langle p |\,p \quad . \end{align} See also this question and the links therein. This tells us that e.g. the projection of the vector $\hat{x}|\psi\rangle$ onto an element of the momentum basis is the number $i\, \frac{\mathrm{d}}{\mathrm{d}p} \psi(p)$.

In conclusion, we can express the action of the position operator $\hat{x}$ on a state $|\psi\rangle$ either as $$ \hat{x}|\psi\rangle = \int \mathrm{d}x\, x\, |x\rangle \langle x|\psi\rangle$$ or $$\hat{x}|\psi\rangle = i\,\int \mathrm{d}p \, |p\rangle \frac{\mathrm{d}}{\mathrm{d}p}\langle p|\psi\rangle \quad .$$

Regarding your example: We know that $\langle x |p\rangle = e^{ipx}$ (up to some constant factor which I'll omit) and let's say we want to calculate $ \langle x |\hat{x}|p\rangle$, which is I think you meant by $\hat{x}\, e^{ipx}$.

Using the position representation of the position operator, we easily find $$\langle x |\hat{x}|p\rangle = x \, \langle x|p\rangle = x \,e^{ipx} \quad . $$

Performing the calculation within the momentum representation of the position operator yields:

$$\langle x |\hat{x}|p\rangle = \left({\langle p |\hat{x}^\dagger|x\rangle }\right)^* = \left({\langle p |\hat{x}|x\rangle }\right)^* = \left(i \frac{\mathrm{d}}{\mathrm{d}p} e^{-ipx}\right)^* = x \, e^{ipx} \quad ,$$ as expected.

Let us first clarify that things like $\hat{x}\,\psi(x)$ in fact mean

$$ (\hat{x}\psi)(x) \equiv \langle x| \hat{x} |\psi\rangle \quad .$$ We further define the wave function as $$ \psi(x) \equiv \langle x|\psi\rangle \quad. $$

Similar equations hold for the case of the momentum operator / basis. In the following we set $\hbar=1$.

The position representation of the position and momentum operator is (very nicely explained here):

\begin{align} \langle x|\hat{x} &= \langle x| \,x \\ \langle x| \hat{p} &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \langle x| \quad . \end{align} This should look familiar to you: Evaluated in the position basis, the position operator acts as a multiplicative operator while the momentum operator acts as a derivative. Using the conventions from the beginning we then find \begin{align} (\hat{x}\psi)(x) &= x\, \psi(x) \\ (\hat{p}\psi)(x) &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \psi(x) \quad , \end{align}

which means for example that the projection of the vector $\hat{x}|\psi\rangle$ onto a state of the position basis is the number $ x\, \psi(x)$.

However, this does not mean that the momentum operator is a derivative; in the same sense, the position operator is not an operator multiplying 'everything' it acts on with $x$. Indeed, evaluated in the momentum basis, we find that the momentum operator is multiplicative and the position operator acts as a derivative: \begin{align} \langle p | \hat{x} &= i\, \frac{\mathrm{d}}{\mathrm{d}p} \langle p|\\ \langle p | \hat{p } &= \langle p |\,p \quad . \end{align} See also this question and the links therein. This tells us that e.g. the projection of the vector $\hat{x}|\psi\rangle$ onto an element of the momentum basis is the number $i\, \frac{\mathrm{d}}{\mathrm{d}p} \psi(p)$.

Finally, by making use of the completeness relations $\displaystyle \mathbb I = \int \mathrm dx \, |x\rangle \langle x| = \int \mathrm dp \, |p\rangle\langle p|$, we can express the action of the position operator $\hat{x}$ on a state $|\psi\rangle$ either as $$ \hat{x}|\psi\rangle = \int \mathrm{d}x\, x\, |x\rangle \langle x|\psi\rangle$$ or $$\hat{x}|\psi\rangle = i\,\int \mathrm{d}p \, |p\rangle \frac{\mathrm{d}}{\mathrm{d}p}\langle p|\psi\rangle \quad .$$

Regarding your example: We know that $\langle x |p\rangle = e^{ipx}$ (up to some constant factor which I'll omit) and let's say we want to calculate $ \langle x |\hat{x}|p\rangle$, which is I think you meant by $\hat{x}\, e^{ipx}$.

Using the position representation of the position operator, we easily find $$\langle x |\hat{x}|p\rangle = x \, \langle x|p\rangle = x \,e^{ipx} \quad . $$

Performing the calculation within the momentum representation of the position operator yields:

$$\langle x |\hat{x}|p\rangle = \left({\langle p |\hat{x}^\dagger|x\rangle }\right)^* = \left({\langle p |\hat{x}|x\rangle }\right)^* = \left(i \frac{\mathrm{d}}{\mathrm{d}p} e^{-ipx}\right)^* = x \, e^{ipx} \quad ,$$ as expected.

I hope that this addresses your question(s). If not, let me know. In the following we set $\hbar=1$.

Let us clarify that things like $\hat{x}\,\psi(x)$ in fact mean

$$ (\hat{x}\psi)(x) \equiv \langle x| \hat{x} |\psi\rangle \quad .$$ We further define the wave function as $$ \psi(x) \equiv \langle x|\psi\rangle \quad. $$

The position representation of the position and momentum operator is (very nicely explained here):

\begin{align} \langle x|\hat{x} &= \langle x| \,x \\ \langle x| \hat{p} &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \langle x| \quad . \end{align} This should look familiar to you: Evaluated in the position basis, the position operator acts as a multiplicative operator while the momentum operator acts as a derivative. Using the conventions from the beginning we then find \begin{align} (\hat{x}\psi)(x) &= x\, \psi(x) \\ (\hat{p}\psi)(x) &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \psi(x) \quad , \end{align}

which means for example that the projection of the vector $\hat{x}|\psi\rangle$ onto a state of the position basis is the number $ x\, \psi(x)$.

However, this does not mean that the momentum operator is a derivative; in the same sense, the position operator is not an operator multiplying 'everything' it acts on with $x$. Indeed, evaluated in the momentum basis, we find that the momentum operator is multiplicative and the position operator acts as a derivative: \begin{align} \langle p | \hat{x} &= i\, \frac{\mathrm{d}}{\mathrm{d}p} \langle p|\\ \langle p | \hat{p } &= \langle p |\,p \quad . \end{align} See also this question and the links therein. This tells us that e.g. the projection of the vector $\hat{x}|\psi\rangle$ onto an element of the momentum basis is the number $i\, \frac{\mathrm{d}}{\mathrm{d}p} \psi(p)$.

In conclusion, we can express the action of the position operator $\hat{x}$ on a state $|\psi\rangle$ either as $$ \hat{x}|\psi\rangle = \int \mathrm{d}x\, x\, |x\rangle \langle x|\psi\rangle$$ or $$\hat{x}|\psi\rangle = i\,\int \mathrm{d}p \, |p\rangle \frac{\mathrm{d}}{\mathrm{d}p}\langle p|\psi\rangle \quad .$$

Regarding your example: We know that $\langle x |p\rangle = e^{ipx}$ (up to some constant factor which I'll omit) and let's say we want to calculate $$ \langle x |\hat{x}|p\rangle \quad, $$

which is I think you meant by $\hat{x}\, e^{ipx}$.

Using the position representation of the position operator, we easily find $$\langle x |\hat{x}|p\rangle = x \, \langle x|p\rangle = x \,e^{ipx} \quad . $$

Now if we want to use the momentum representation, we calculate:

$$\langle x |\hat{x}|p\rangle = \left({\langle p |\hat{x}^\dagger|x\rangle }\right)^* = \left({\langle p |\hat{x}|x\rangle }\right)^* = \left(i \frac{\mathrm{d}}{\mathrm{d}p} e^{-ipx}\right)^* = x \, e^{ipx} \quad ,$$ as expected.

I hope that this addresses your question(s). In the following we set $\hbar=1$.

Let us first clarify that things like $\hat{x}\,\psi(x)$ in fact mean

$$ (\hat{x}\psi)(x) \equiv \langle x| \hat{x} |\psi\rangle \quad .$$ We further define the wave function as $$ \psi(x) \equiv \langle x|\psi\rangle \quad. $$

Similar equations hold for the case of the momentum operator / basis.

The position representation of the position and momentum operator is (very nicely explained here):

\begin{align} \langle x|\hat{x} &= \langle x| \,x \\ \langle x| \hat{p} &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \langle x| \quad . \end{align} This should look familiar to you: Evaluated in the position basis, the position operator acts as a multiplicative operator while the momentum operator acts as a derivative. Using the conventions from the beginning we then find \begin{align} (\hat{x}\psi)(x) &= x\, \psi(x) \\ (\hat{p}\psi)(x) &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \psi(x) \quad , \end{align}

which means for example that the projection of the vector $\hat{x}|\psi\rangle$ onto a state of the position basis is the number $ x\, \psi(x)$.

However, this does not mean that the momentum operator is a derivative; in the same sense, the position operator is not an operator multiplying 'everything' it acts on with $x$. Indeed, evaluated in the momentum basis, we find that the momentum operator is multiplicative and the position operator acts as a derivative: \begin{align} \langle p | \hat{x} &= i\, \frac{\mathrm{d}}{\mathrm{d}p} \langle p|\\ \langle p | \hat{p } &= \langle p |\,p \quad . \end{align} See also this question and the links therein. This tells us that e.g. the projection of the vector $\hat{x}|\psi\rangle$ onto an element of the momentum basis is the number $i\, \frac{\mathrm{d}}{\mathrm{d}p} \psi(p)$.

In conclusion, we can express the action of the position operator $\hat{x}$ on a state $|\psi\rangle$ either as $$ \hat{x}|\psi\rangle = \int \mathrm{d}x\, x\, |x\rangle \langle x|\psi\rangle$$ or $$\hat{x}|\psi\rangle = i\,\int \mathrm{d}p \, |p\rangle \frac{\mathrm{d}}{\mathrm{d}p}\langle p|\psi\rangle \quad .$$

Regarding your example: We know that $\langle x |p\rangle = e^{ipx}$ (up to some constant factor which I'll omit) and let's say we want to calculate $ \langle x |\hat{x}|p\rangle$, which is I think you meant by $\hat{x}\, e^{ipx}$.

Using the position representation of the position operator, we easily find $$\langle x |\hat{x}|p\rangle = x \, \langle x|p\rangle = x \,e^{ipx} \quad . $$

Performing the calculation within the momentum representation of the position operator yields:

$$\langle x |\hat{x}|p\rangle = \left({\langle p |\hat{x}^\dagger|x\rangle }\right)^* = \left({\langle p |\hat{x}|x\rangle }\right)^* = \left(i \frac{\mathrm{d}}{\mathrm{d}p} e^{-ipx}\right)^* = x \, e^{ipx} \quad ,$$ as expected.

I hope that this addresses your question(s). If not, let me know.

Let us clarify that things like $\hat{x}\,\psi(x)$ in fact mean

$$ (\hat{x}\psi)(x) \equiv \langle x| \hat{x} |\psi\rangle \quad .$$ We further define the wave function as $$ \psi(x) \equiv \langle x|\psi\rangle \quad. $$

In the following we set $\hbar=1$.

The position representation of the position and momentum operator is (very nicely explained here):

\begin{align} \langle x|\hat{x} &= \langle x| \,x \\ \langle x| \hat{p} &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \langle x| \quad . \end{align} This should look familiar to you: Evaluated in the position basis, the position operator acts as a multiplicative operator while the momentum operator acts as a derivative. Using the conventions from the beginning we then find \begin{align} (\hat{x}\psi)(x) &= x\, \psi(x) \\ (\hat{p}\psi)(x) &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \psi(x) \quad , \end{align}

which means for example that the projection of the vector $\hat{x}|\psi\rangle$ onto a state of the position basis is the number $ x\, \psi(x)$.

However, this does not mean that the momentum operator is a derivative; in the same sense, the position operator is not an operator multiplying 'everything' it acts on with $x$. Indeed, evaluated in the momentum basis, we find that the momentum operator is multiplicative and the position operator acts as a derivative: \begin{align} \langle p | \hat{x} &= i\, \frac{\mathrm{d}}{\mathrm{d}p} \langle p|\\ \langle p | \hat{p } &= \langle p |\,p \quad . \end{align} See also this question and the links therein. This tells us that e.g. the projection of the vector $\hat{x}|\psi\rangle$ onto an element of the momentum basis is the number $i\, \frac{\mathrm{d}}{\mathrm{d}p} \psi(p)$.

In conclusion, we can express the action of the position operator $\hat{x}$ on a state $|\psi\rangle$ either as $$ \hat{x}|\psi\rangle = \int \mathrm{d}x\, x\, |x\rangle \langle x|\psi\rangle$$ or $$\hat{x}|\psi\rangle = i\,\int \mathrm{d}p \, |p\rangle \frac{\mathrm{d}}{\mathrm{d}p}\langle p|\psi\rangle \quad .$$

Regarding your example: We know that $\langle x |p\rangle = e^{ipx}$ (up to some constant factor which I'll omit) and let's say we want to calculate $$ \langle x |\hat{x}|p\rangle \quad, $$

which is I think you meant by $\hat{x}\, e^{ipx}$.

Using the position representation of the position operator, we easily find $$\langle x |\hat{x}|p\rangle = x \, \langle x|p\rangle = x \,e^{ipx} \quad . $$

Now if we want to use the momentum representation, we calculate:

$$\langle x |\hat{x}|p\rangle = \left({\langle p |\hat{x}^\dagger|x\rangle }\right)^* = \left({\langle p |\hat{x}|x\rangle }\right)^* = \left(i \frac{\mathrm{d}}{\mathrm{d}p} e^{-ipx}\right)^* = x \, e^{ipx} \quad ,$$ as expected.

I hope that this addresses your question(s). If not, let me know. In the following we set $\hbar=1$.

Let us clarify that things like $\hat{x}\,\psi(x)$ in fact mean

$$ (\hat{x}\psi)(x) \equiv \langle x| \hat{x} |\psi\rangle \quad .$$ We further define the wave function as $$ \psi(x) \equiv \langle x|\psi\rangle \quad. $$

The position representation of the position and momentum operator is (very nicely explained here):

\begin{align} \langle x|\hat{x} &= \langle x| \,x \\ \langle x| \hat{p} &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \langle x| \quad . \end{align} This should look familiar to you: Evaluated in the position basis, the position operator acts as a multiplicative operator while the momentum operator acts as a derivative. Using the conventions from the beginning we then find \begin{align} (\hat{x}\psi)(x) &= x\, \psi(x) \\ (\hat{p}\psi)(x) &= - i \,\frac{\mathrm{d}}{\mathrm{d}x} \psi(x) \quad , \end{align}

which means for example that the projection of the vector $\hat{x}|\psi\rangle$ onto a state of the position basis is the number $ x\, \psi(x)$.

However, this does not mean that the momentum operator is a derivative; in the same sense, the position operator is not an operator multiplying 'everything' it acts on with $x$. Indeed, evaluated in the momentum basis, we find that the momentum operator is multiplicative and the position operator acts as a derivative: \begin{align} \langle p | \hat{x} &= i\, \frac{\mathrm{d}}{\mathrm{d}p} \langle p|\\ \langle p | \hat{p } &= \langle p |\,p \quad . \end{align} See also this question and the links therein. This tells us that e.g. the projection of the vector $\hat{x}|\psi\rangle$ onto an element of the momentum basis is the number $i\, \frac{\mathrm{d}}{\mathrm{d}p} \psi(p)$.

In conclusion, we can express the action of the position operator $\hat{x}$ on a state $|\psi\rangle$ either as $$ \hat{x}|\psi\rangle = \int \mathrm{d}x\, x\, |x\rangle \langle x|\psi\rangle$$ or $$\hat{x}|\psi\rangle = i\,\int \mathrm{d}p \, |p\rangle \frac{\mathrm{d}}{\mathrm{d}p}\langle p|\psi\rangle \quad .$$

Regarding your example: We know that $\langle x |p\rangle = e^{ipx}$ (up to some constant factor which I'll omit) and let's say we want to calculate $$ \langle x |\hat{x}|p\rangle \quad, $$

which is I think you meant by $\hat{x}\, e^{ipx}$.

Using the position representation of the position operator, we easily find $$\langle x |\hat{x}|p\rangle = x \, \langle x|p\rangle = x \,e^{ipx} \quad . $$

Now if we want to use the momentum representation, we calculate:

$$\langle x |\hat{x}|p\rangle = \left({\langle p |\hat{x}^\dagger|x\rangle }\right)^* = \left({\langle p |\hat{x}|x\rangle }\right)^* = \left(i \frac{\mathrm{d}}{\mathrm{d}p} e^{-ipx}\right)^* = x \, e^{ipx} \quad ,$$ as expected.