system
Given the following post, respond as if you're commenting on Reddit:
human
Allowed electron orbits seem odd and purely a mathematical construction. Are they just theory but reality gives more laxitude to where an electron might be?

In A Brief History of Time, Hawkins explains the allowed orbits by saying only integer multiples of an electron's wavelength are allowed, otherwise after a certain number of orbits the electron's wave will destructivley interfere and the e- will disappear and you restrict your solution-space to discrete allowed orbits.

Does this mean we limit ourselves further to electron energies that have wavelengths longer than or equal to their orbital circumference so they can intefere? If the wavelength is shorter than the circumference would we not have a band of orbits which will never destructivley interfere and hence not limit ourselves to the simpler situation of discrete energy levels for the e-? 

Also, as a mathematician, if space around the e- is compact these allowed orbits (to me) represent the limit points of allowed orbits and that in the real world you would have a band of 'effectively' allowed orbits which would be stable for long periods of time without ever collapsing into the nucleus or flying out into space.

The band might be small, but a band of any size at all would give an uncountabley infinite number of possible orbits in the physical world, hence when you come to describe the e- in the world of quantum mechanics you can't use the simplification of discrete energy levels which will messes up their representation when expressed as vectors.
gpt
If you're a mathematician, I'd recommend reading more rigorous sources than A Brief History of Time.

Electron orbitals are the stationary solutions of the time-independent Schrodinger equation.

The electron having negative total energy (where "zero" energy is defined by a particle at r = infinity, with zero kinetic energy) imposes boundary conditions which lead to a discrete spectrum of bound states.

The modulus-squared of the stationary solutions gives the probability density for where you'd find the electron in space if you measured its position.