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When calculating the orbital energy of an electron in a hydrogen-like atom, the orbital velocity is required. However, to derive this value, the equation given is:

\begin{equation*} ZK_{e} e^{2}/r^{2} \end{equation*}

And \begin{equation*}K_{e}\end{equation*}is the atomic number. However, there are plenty of systems that behave with an electron orbital which are non-atomic, like those have an quantum potential well. In these cases, there is no single defined atomic number, so how would one go off calculating this?

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First of all, an observation on terminology. Bohr's approach to electron dynamics was to use classical mechanics plus some additional rules. In classical mechanics, speaking about orbital energy or velocity is acceptable, where orbital means of the orbit. Orbit is a classical concept. However, in the second part of the question, the name orbital appears. This is out of the proper context. Orbitals are not orbits, and the name orbital is nothing but the name of a quantum one-particle wavefunction.

Bohr's original approach to the mechanics of the atom did not end with the quantization of the hydrogen atom. Actually, in the decade following his breakthrough paper on the hydrogen atom, he and other physicists (Sommerfeld, Born) tried to extend the quantization ideas to completely general systems. That evolved into the theory named Old Quantum Mechanics (OQM), well summarized in Born's book The mechanics of the atom.

OQM was a pretty general theory based on the principle of assigning integer values to the mechanical adiabatic invariants of the system. In this form, it was completely general and applicable to every mechanical system. In principle, it could be applied to quantum wells too.

Unfortunately, we know that the theory did not work. Frustration for this failure was the main driving force for finding alternative approaches. Heisenberg and Jordan's matrix mechanics and Schrödinger's wave mechanics were the starting point of modern Quantum Mechanics (QM). Nowadays, Bohr's approach (and its generalizations) are historical milestones of Physics, sometimes useful as heuristic tools but definitely replaced by QM.

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    $\begingroup$ It is perhaps worth noting that the Bohr-Sommerfeld condition $\oint p \mathrm dq = nh$ applied to a classical particle with momentum $\pm p$ bouncing back and forth between the walls of a box of length $L$ yields $2pL = nh \implies E= \frac{p^2}{2m} = \frac{n^2 h^2}{8mL^2}$, and so does correctly predict the energy levels of a particle in a box. $\endgroup$
    – J. Murray
    Commented Oct 19, 2021 at 23:30
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If you have a quantum system, you use the mechanism of quantum mechanics. You solve the Schrodinger equatons with the specific potential. For hydrogen atom you use the Coulomb potential, for a potential well you use whatever function describes your potential well. No orbital velocity is required in either case. The solutions of the Schrodinger equation are the wave functions for all possible states. With the wave functions you can calcuate the energies for any specific state, if this is what you want. If you mean to use the calssical approach used in the Bohr model of the atom, this is not valid for a quantum system. The main reason the QM was developed.

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