Why is the energy of an electron quantised?

The quantum mechanics that are interpreted in a probabilistic manner along with the boundary conditions contribute to quantifying energy. If you apply an equation like the Schrodinger waves equation on an electron that is free restricted to a well or an enclosure, etc. The solution to the equation due to the boundary conditions applied automatically result in the quantization of energy. Quantum numbers, like n, l as well as ml as well as n, k, and so on. are derived from the solution of a second-order differential equation that acts on a probability wave function, subject to the limitations of boundary conditions. The quantization of electron energy can also be demonstrated through spectroscopic research using various studies currently available.

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