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I am now reading my lecture notes on dipole moment and there's a point which confused me. It says that:

Let us consider the $|1s\rangle$ and $|2p_x\rangle$ states of a hydrogen atom. The atom has inversion symmetry, so all the eigenstates have either even (such as $|1s\rangle$ ) or odd (such as $|2p_x\rangle$). Therefore, when the atom is in an eigenstate, its dipole moment vanishes, i.e.

$$\langle 1s | d |1s \rangle = \langle 2_x | d |2p_x\rangle = 0$$

Can someone please further explain this line?

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1 Answer 1

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To start with note that the dipole moment operator is an odd function.

Suppose $\psi$ is even, like $|1s\rangle$, then $\hat d\psi$ is odd and therefore $\psi^* \hat d \psi$ is odd. When we integrate an odd function from $-\infty$ to $\infty$ the result is zero.

Now suppose $\psi$ is odd, like $|2p_x\rangle$, then $\hat d\psi$ is even and therefore $\psi^* \hat d \psi$ is odd. And as before when we integrate from $-\infty$ to $\infty$ we get zero.

So the result of $\langle\psi|\hat d|\psi\rangle$ is zero if $\psi$ is either an even function or an odd function.

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