I How does QM explain that we see electrons circulating in a magnetic field?

[George Jones]

I do not see how to visualize latex formulas in this forum.

See "Delimiting your LaTeX code" in Physics Forums help for LaTeX,

https://www.physicsforums.com/help/latexhelp/

 

[Twigg]

If you're not sure how to evaluate the energy from the Hamiltonian $H = \hbar \omega (a^\dagger a + \frac{1}{2})$, then you may want to review the harmonic oscillator in the ladder operator picture. It's discussed in pretty much any serious quantum textbook, undergraduate or graduate level. Griffiths (2nd Ed.) section 2.3.1 is a good place to start. (You can later derive the coherent states step-by-step with hints in Problem 3.35.)

there are also the formulas in the Landau book. n and m are definite and i read
E = hbar omega (n + m + 1/2)

are they talking about the same energy?

Yes, just in terms of quantum numbers that describe different basis states (sounds like eigenmodes in cartesian coordinates?). The energies are always the same, in any basis and any gauge. If you're quoting a result from Landau, please share which volume and section you are reading from so we can see what you're looking at.

 

[Heidi]

It is in vol 3 quantum mechanics (non relativistic)
chapter xv motion in a magetic field . problem 1 page 460
we have E depending on n and m and it is written that it is equivalent to 112.7 where there is no m.
we have here |m| + m which is null if m négative but is it so here?

 

[Heidi]

j bought the first part of the paper written by Kuo-Ho Yang . Thank you VanHees71. I ignored that an hamiltonian could be cut in 2 parts. A "basis" part with a basis of orthonormal eigenvectors and a residual part. Here the résidual part is omega times Lz i suppose.