More Electron Traps — Question 6

Question 6

A circulating ring state has uniform density and a nonzero electric current

An electron on a ring of radius R has $\psi_\ell(\phi)=e^{i\ell\phi}/\sqrt{2\pi}$, integer $\ell$, angular momentum $L_z=\ell\hbar$ and energy $E_\ell=\ell^2\hbar^2/(2mR^2)$. The electron charge is $-e$.

1. Find the angular speed and electric current of a traveling mode.

2. Compare modes $+\ell$ and $-\ell$, and find the mean current of an equal incoherent mixture.

Question 6 - Solution

A circulating ring state has uniform density and a nonzero electric current

(a) Method and calculation

$$\Omega_\ell=L_z/(mR^2)=\ell\hbar/(mR^2).$$$$\boxed{I_\ell=(-e)\Omega_\ell/(2\pi)=-\frac{e\ell\hbar}{2\pi mR^2}}.$$

(b) Interpretation and further calculation

$$\boxed{E_{-\ell}=E_\ell,\qquad I_{-\ell}=-I_\ell},\quad|\psi_{\pm\ell}|^2=1/(2\pi).$$$$\boxed{\langle I\rangle_{\mathrm{mixture}}=\tfrac12I_\ell+\tfrac12I_{-\ell}=0}.$$

Physical check

Uniform probability density does not imply zero current. Conventional electric current points opposite to the direction of electron motion.