Point one :
The discrete energy levels for an electron in an atom is due to boundary conditions on
Schrodinger equation.Consider a particle in a one dimensional box with potential at
boundaries set at infinity, and solve it! Because the potential is infinite outside the
box, the wave function must be zero outside the box.The wave function must be continuous,
so the wave function at the boundaries must be zero too.From this boundary condition there
is a condition on energies which the electron can have.I answered the origin of discrete
energy levels.Thus, energy levels do not exist by themselves.
When we solve the Schrodinger equation for hydrogen atom, we go to center of mass frame.In
this frame our nuclei and electron kinetic operator convert to center of mass kinetic
operator and the relative motion kinetic operator.Then, because the mass of nuclei is
around 1850 bigger than that of electron the kinetic operator of relative motion reduces
to that of electron.Finally, we have the center of mass kinetic operator and relative
motion kinetic operator.We assume that the total wave function is the product of wave
function of center of mass and relative motion.At the end, we get a discrete energy level
for relative motion, which almost is due to motion of electron, and a continuous energy
for the center of mass.If we did not use the approximation that the nuclei is much more
heavier than electron then those discrete levels would be that of nuclei too.There is
another twist to the problem.The nuclei has an internal structure.It is made of quarks.So,
those quarks similar to the electron can have their own discrete level of energy.However,
in the elementary text books the internal structure of nuclei is not considered.
1 - Discrete energy levels are due to boundary conditions on the Schrodinger' s equation.
2 - We separate the kinetic operators of nuclei and electron into center of mass kinetic
operator and relative motion operator.
3 - The mass of the nuclei is much more than that of
electron.Thus, the relative motion is almost due to the electron.
4 - Now the equation is divided in two parts, center of mass and relative motion and we
can assume that the total wave function is the multiplication of the wave function of the
5 - The solution to the center of mass gives rise to the continuous energy level.The
solution to relative motion gives rise to a discrete energy level.
6 - Nuclei has internal structure.They are made of quarks.So, the nuclei can has its own
discrete levels similar to the electron.
Perhaps the link below is useful: