![]() Spin is probably best considered a magnetic moment that would arise if a charged particle were to rotate around itself. Spin and charge are, however, two different properties. The energy of a single electron consists of its kinetic energy (described with the $\hat T$ operator) and its potential energy towards the nucleus ($\hat V_\ce$). Unfortunately, it is not possible to solve the Hamiltonian for two electrons. The quantisation of the electron’s energy provides a nice and succinct explanation of the hydrogen spectral lines. The nice thing is that the electron’s energy is quantitised and dependent only on the principal quantum number $n$ while other quantum numbers are also required to fully describe the wave function ($l, m_l, s, m_s$). ![]() This long and tangled sentence boils down to: we can calculate the wave function of an electron in the vicinity of a nucleus. Using this equation alongside the derivation of the correct Hamiltonian, the wave function for a singly negatively charged particle rotating and translating freely in a spherical electric field of a given positive charge can be determined. Naturally, the wave function may be a different one depending on the operator in question. Operators and observables observe the Schrödinger equation $(2)$ in which the Hamiltonian operator $\hat H$ can be replaced by any other operator to observe a different observable than energy $E$. A key part of the theory is the existence of operators which can be applied to the wave function to observe observables - physical parameters such as a particle’s energy or magnetic moment. The key principle is that each particle is described by a wave function, typically labelled $\Psi$, dependent on space and/or time coordinates. Quantum mechanics is the underlying theory that attempts to shed light into how electrons and other microscopic particles behave according to their wave nature. This discovery earned de Broglie a Nobel Prize for the theory and Thompson and Davisson for the supporting diffraction experiments. The probably most frequently cited connection between the particle nature and the wave nature of electrons and others is the de Broglie wavelength given by equation $(1)$. However, particles as small as electrons can also be considered waves as is evident by electron diffraction. This is not entirely incorrect for example, the cathode ray tube is best explained by electrons being considered negatively charged particles that can travel through vacuum and be deflected by (electro)magnetic fields. Throughout your question it is clear that you are considering electrons as particles.
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