The Landau levels do not look like classical orbits, just like harmonic oscillator eigenstates do not look like classical oscillations. The expectation values of ##\vec{r}## and ##\vec{v}## do not change with time in the Landau levels, since they are stationary states. Thus, a Landau level is not an orbit. It's just an energy eigenstate.
Recall the symmetric-gauge Hamiltonian from Post #6, $$H = \frac{\mathbf{p}^2}{2m} - \frac{1}{2} \omega_c \mathbf{L}_z + \frac{1}{8} m \omega_c^2 \rho^2$$ where ##\omega_c = qB_z/m## is the cyclotron frequency. This Hamiltonian conserves angular momentum. That's a good starting point. A 1D gaussian wavepacket like you might find in Griffiths will not give you cyclotron orbits, because it doesn't have a well defined "radius"; such a wavepacket is only confined in the longitudinal axis. You need wavepacket that is confined in both the longitudinal and transverse directions to see cyclotron motion, and even then you need to assume that the Larmor radius is much larger than the transverse confinement or you will see very weird interference phenomena. And you probably have ignore dispersion. It's a mess. Just have a little faith in ol' Ehrenfest.
There is a mathematically clean way to derive states that look like the cyclotron orbits. They're called
coherent states. I'm going to use the notation from the
Landau levels wikipedia's section on the symmetric gauge. A coherent state ##|\alpha,\beta \rangle## is an eigenstate of both ladder operators: $$ \begin{align} \hat{a} |\alpha \beta \rangle &= \alpha |\alpha \beta \rangle \\ \hat{b} |\alpha \beta \rangle &= \beta |\alpha \beta \rangle \end{align}$$ My gut feeling (this stuff is tricky) is that a coherent state with large ##\alpha## and ##\beta## will look like a gaussian wavepacket with transverse confinement in a cyclotron orbit with large Larmor radius.
I couldn't derive the coherent states to prove it to you. That stuff's hard! Just getting the coherent states for the harmonic oscillator is usually an exercise for a graduate course. I looked for open-access papers on the topic, but the only articles I found had math that was way above my pay grade. If you're feeling ambitious, knock yourself out:
here's the link. Maybe someone here knows a better article or can summarize the results?