Discrete and continuous confusion

[qbslug]

I don't understand how in quantum mechanics we have discrete and exact energy states for electron orbits but then at the same time we have a continuous probability density function for the position of an electron.
This seems like a paradox (although I know it can't be) since considering a continuous position distribution you would expect the different positions of the electron to create different energies of the electron in a continuous way. How is this explained? How can you have one energy for different positions?

 

[dx]

Definite energies are not associated to each position state, but to certain wavefunctions called energy eigenstates (eigenvectors of the Hamiltonian operator). In certain situations, the energy spectrum associated with the energy eigenvectors is discrete. In other situations it may be continuous.

 

[facenian]

May be is becouse you're thinking semi-classicaly like in Bhor's rule where only certain orbits are allowed while others are forbidden.

 

[f95toli]

I don't understand how in quantum mechanics we have discrete and exact energy states for electron orbits but then at the same time we have a continuous probability density function for the position of an electron.

I don't think it is quite correct to say that we have "exact" discrete energy states for real electrons. A soon as you allow for coupling to the vacuum the energies are "smeared put" and the eigeneneriges are then just the centres of Lorentzian energy distributions.

It is pretty much analogues to resonances in an electrical LC circuit, in an ideal LC circuit there is a single discrete resonance frequency but as soon as you add some damping the resonance is broadened.

Note that this is just a consequences of the "mathematical uncertainty principle", in order to have an exact energy the electrons would have to stay in a single energy state for an infinite amount of time; a system with a finite lifetime can not have a truly discrete energy spectrum.