Issue with the Scalar Potential for an electric field

Question

I've been given a question saying to give the time-dependent Hamiltonian for the atomic electron in the E-field in the form of a scalar potential.
My question is: as scalar potential is defined $E = -nabla V$ and the given E-field is $E = E(t) hat{z}$, I've written my scalar potential $V(t)$ as $V(t) = E(t)z$.
However, the mark scheme denotes it as $V(t) = -qE(t)z$, and I don't know why the -q is necessary. If you could explain it, that'd be great thank you.

oliver · Accepted Answer

You have to differentiate between electrostatic potential $Phi$ and potential energy $V$ used in the Hamiltonian. Since the electric field $vec E$ is a force per charge, the electrostatic potential must also be potential energy per charge:
$$vec F=qvec E$$
$$vec F=-vec nabla V$$
Therefore,
$$vec E=-frac{1}{q}vec nabla V=-vec nabla Phi$$
and finally
$$V=qPhi$$
So if you have $vec E=(0,0,E(t))$, you need $Phi=-E(t)z$ and hence $V=-qE(t)z$.

Issue with the Scalar Potential for an electric field

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