# I 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?

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#### vanhees71

Gold Member
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?

#### A. Neumaier

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

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

"Definition of the potential energy operator"

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