Hidden momentum in a hydrogen atom and the Lorentz-force law

By using perturbation theory, we show that a hydrogen atom with magnetic moment due to the orbital angular momentum of the electron has so-called hidden momentum in the presence of an external electric field. This means that the atomic electronic cloud has a nonzero linear momentum in its center-of-mass rest frame due to a relativistic effect. This is completely analogous to the hidden momentum that a classical current loop has in the presence of an external electric field. We discuss how this effect is essential for the validity of the Lorentz-force law in quantum systems. We also connect our results to the long-standing Abraham-Minkowski debate about the momentum of light in material media.

Figure 1

A rectangular closed circuit carrying a stationary electric current $I$ in the presence of a constant and uniform external electric field $\mathbf{E}=E\widehat{\mathbf{x}}$ has hidden momentum. ${\mathbf{V}}_{\mathbf{t}}$ $\left({\mathbf{V}}_{\mathbf{b}}\right)$ is the velocity of the particles in the top (bottom) segment.

Figure 2

The electronic cloud of a hydrogen atom initially in the unperturbed state $|{\psi }_{2,1,1}^{\left(0\right)}\rangle$ has hidden momentum in the presence of a perturbation electric field $\mathbf{E}=E\widehat{\mathbf{x}}$. The plot represents the surface with constant ${\left|\psi \right|}^2$, where $\psi$ is the wave function for this unperturbed state.

Figure 3

The blue (gray) straight line is $y=-m_l$ and the red (gray) dots represent the values of $\left(P_{\mathrm{hid}}^{\prime \left(1\right)}-P_{\mathrm{hid}}^{\prime \left(2b\right)}\right)c^2/{\mu }_{B}E$ for the following quantum states $|{\psi }_{n,l,m}\rangle$: $\left\{n,l,m\right\}=$ $\left\{13,12,-5\right\},$ $\left\{11,7,-4\right\},$ $\left\{6,5,-3\right\},$ $\left\{12,8,-2\right\},$ $\left\{3,1,-1\right\},$ $\left\{1,0,0\right\},$ $\left\{2,1,1\right\},$ $\left\{9,2,2\right\},$ $\left\{8,4,3\right\},$ $\left\{5,4,4\right\},$ $\left\{7,6,5\right\}$.

Figure 4

The blue (gray) line is $y=cos\theta$ and the red (gray) dots represent the values of $\left(P_{\mathrm{hid}}^{\prime \left(1\right)}-P_{\mathrm{hid}}^{\prime \left(2b\right)}\right)c^2/{\mu }_{B}E$ for the quantum state $|{\psi }_{3,1,-1}\rangle$ perturbed by an electric field $\mathbf{E}=E\left(\widehat{\mathbf{x}}cos\theta +\widehat{\mathbf{z}}sin\theta \right)$ for different values of $\theta$.