Definition of the potential energy operator

[Lebnm]

In quantum mechanics, I can write the hamiltonian as $\hat{H} = \hat{p}^{2}/2m + \hat{V}$. I am confusing with the definition of the operator $\hat{V}$, who represents the potential energy. If the potential energy depend only on the position, is it correct write $\hat{V} = V(\hat{x})$? And, assuming that $V$ can be expanded in a power series of $\hat{x}$, and using $\hat{x} | x \rangle = x | x \rangle$, can I write $\hat{V} | x \rangle = V(x) | x \rangle$? I would like know if this things I wrote are correct. Does anyone know a book or article that talk about this?

 

[vanhees71]

Yes, you are right. A good introductory QT textbook is

J.J. Sakurai, Modern Quantum Mechanics, Revised Edition, Addison-Wesley 1994

 

[A. Neumaier]

assuming that $V$ can be expanded in a power series of $\hat{x}$, and using $\hat{x} | x \rangle = x | x \rangle$, can I write $\hat{V} | x \rangle = V(x) | x \rangle$?

By definition of the position representation, this is correct for all $V$, even if no series expansion exists.

 

[Lebnm]

By definition of the position representation, this is correct for all $V$, even if no series expansion exists.

Why? Would not this imply that all potentials are local?

Do you have references about this? it has been difficult find books that talk about this.

 

[A. Neumaier]

Why? Would not this imply that all potentials are local?

Do you have references about this? it has been difficult find books that talk about this.

For all $V(x)$, as in your formula. Usually only such interactions get the name potential energy.