In summary, the 1D time-independent Schrodinger equation allows for the mixing of potentials and energies in the same equation, as long as the potential is converted to an energy first. This is due to the relationship between potential and potential energy for charged particles. However, some confusion may arise if the relationship between the quantum mechanical Hamiltonian and classical Hamiltonian dynamics is not well understood.
Well,Tom,that's from the DEFINITION.If they don't do it in CM,
I know, and that's my point. If you show QM students the Hamiltonian without explaining what it is, then I would think they would either not understand an important facet of the science, or be entirely lost. As Galileo said in this thread, one can be taught to solve problems in QM without learning Hamiltonian dynamics. Of course that's true, but why stop there? It also would be possible to teach a math student with no physics background how to mechanically solve problems in QM. But there's an important layer of understanding that would be lost, which is why I agree with you that CM should be compulsory prior to QM.