Does energy quantisation entail spatial orbit quantisation?

First and foremost, what is meant by 'orbit quantisation'?

Is there a relation between the spatial orbits of, say, a bound electron in a Hydrogen atom or in a Landau level, and the fact that its energy levels are quantised?

Is momentum quantised?

'Orbit quantisation' is the central concept of a now defunct atomic model called Bohr's model of the hydrogen atom. In it, electrons orbit the nucleus in orbits that are an integer number of wavelengths of the electron (hence the quantisation: the electron was understood to form standing waves around the nucleus).

Despite the model's success in explaining the electromagnetic absorption/emission spectrum of hydrogen, Bohr et al understood it to be incorrect, leading as it would to constant and catastrophic radiation loss of the atom.

Today the leading model that explains the energy levels of the hydrogen atom is quantum mechanics. Central to that theory are the Schrödinger equations that describe the quantum state of a quantum system.

Solutions of the time independent Schrödinger equation are called wave functions ($\Psi$) and contain all the information about the quantum system (applying so-called quantum operators to $\Psi$ allows to extract that information, such as system momentum). For bound systems (like the hydrogen and other atoms or molecules) the wave functions are quantized by one or more quantum numbers.

The electrons in hydrogen atoms occupy quantized orbitals (a word chosen specifically to distinguish it from "orbits") which show the spatial probability density distribution of the electron with respect to the nucleus. This probability density distribution is computed directly from the wave function $\Psi$, using Born's interpretation.

Look at the ground state orbital of hydrogen e.g., its wave function and some wave function formulas at The Orbitron. Explore the same for the quantised excitations (higher energy levels) by clicking around.

• Thanks. But in the Hydrogen atom (and Landau levels) the expectation value of r and r^2 now have also an 'n', which means that the energy quantisation is also impacting the spatial configuration of the system. Is this a thing? Is there anything more to interpret? – SuperCiocia Sep 4 '16 at 17:54
• The expectation values $\langle r \rangle$ and $\langle r^2 \rangle$ are calculated directly from $\Psi_n$, so the quantum numbers are 'carried over': $\langle r_n \rangle$ and $\langle r_n^2 \rangle$. – Gert Sep 4 '16 at 18:03
• So in some sense energy quantisation does entail spatial quantisation...? – SuperCiocia Sep 4 '16 at 23:16
• The spatial probability density distribution is quantised by the same quantum numbers that quantises energy but there is degeneracy involved. See Principal Quantum Number and Azimuthal Quantum Number for example: same energy, different orbital shape. – Gert Sep 5 '16 at 0:22
• Disregarding for simplicity's sake the so-called fine structure of the H spectrum, it only depends on $n$ but the orbital shapes depend on $n$ and $\ell$. – Gert Sep 5 '16 at 0:38