The energy of a state with quantum number $n$ is
$$E_n=-\frac{\mu Z^2e^4}{8\epsilon_0^2h^2n^2}\qquad.$$
For
*transitions*
between two states, the energy difference between the two states must either be supplied (excitation) or emitted.
Since the states have discrete energy levels, only fitting
*energy quanta*
can be involved in the transition. The energy difference between two adjacent states is
$$\Delta E_{n_1,n_2}=-\frac{\mu Z^2e^4}{8\epsilon_0^2h^2}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\qquad;$$
thus spectral lines occur only at the corresponding energies (frequencies, wavelengths, wavenumbers,...).

This had been observed long before quantum mechanics - the first observations concern the Sun's spectrum.
The constants in front of the bracket are collectively known as the
*Rydberg constant*, $R_H$;
its value was determined experimentally before quantum physics provided the link to the fundamental constants.

The different series of spectral lines discovered by Lyman, Balmer, and Paschen, correspond to the excitations from the $n$=1,2,3 states.

The wave function itself, a complex function with positive and negative values, doesn't tell us much about the
structure of the atom or any connectivity it may have with other atoms. The complex square of the wave function
represents the
*probability density*
of finding the electron at a given point in space when one looks (*i.e.* does an experiment). It does not
say anything about where the electron actually *is* at any moment, the solutions of the Schrödinger
equation only refer to which states are observable. The act of measuring throws the system into one of the
*eigenstates*
that are solutions of the Schrödinger equation.

A recurrent idea of statistics is that temporal averages are equivalent to ensemble averages, *i.e.*
look for a while at one object moving randomly and you will see the same set of orientations that you
would see when taking a snapshot of many identical objects moving randomly. In the same way, we can
interpret the probability density as a representation of the
*electron density*
in an atom, molecule, or solid.

The radial solutions of the Schrödinger equation of the hydrogen atom, $R(r)$, are plotted
on the right. Each time the quantum number $n$ increases, an additional node is created. At
$n$=1, the radial function is all positive. Its maximum is at $r$=0, *i.e.* the point in
space with the highest probability density of finding the electron is actually *inside* the nucleus!
That is why the term
*probability density*
is used: As we move outward along the radius, the volume
of a shell of equal thickness is getting larger and larger, thereby spreading out the probability over
a larger volume.

Each time the quantum number $l$ is increased, one of the spherical nodes disappears again. It is replaced by a planar node that goes through the nucleus. Therefore, only $l$=0 electrons have a finite probability density at the nucleus.

The diagram on the right shows cross sections of the full wave function $\psi(r,\vartheta,\varphi)$ in the polar ($r$-$\vartheta$) plane. This representation highlights the transformation of spherical nodes into planar nodes as $l$ increases.

It is also apparent that the wave function is spreading out into space as $n$ increases, *i.e.*
that electrons with a small $n$ are, on balance, nearer the nucleus. Given that the energy eigenvalue
increases with $n$, that matches the semi-classical expectation that electrons have a lower energy if
they are deep down in the Coulomb potential.

One more observation: As $l$ increases, the additional planar nodes cause the wave function to become less and less symmetric. This is compensated by the increasing number of equivalent states having the same $n$ and $l$ but different $m$. (In the diagram, only $m$=0 states are shown.) All (2$l$+1) states with the same value of $n$ and $l$ together form a perfectly spherical distribution of probability density.

Note that since the probability density is the square of the wave function, it makes no difference if the wave function is positive or negative.

l= |
0 | 1 | 2 | 3 | 4... |

letter: | s | p | d | f | g (then continue alphabetically) |

Instead of describing a state by listing the values of its quantum numbers, a common practice is to refer to them by main quantum number, $n$, followed by a letter representing the value of $l$ as shown in the table.

Thus, a 2p state is one with $n$=2 and $l$=1 (the $m$=-1,0,+1 cases are sometimes
distinguished as 2p_{x}, 2p_{y}, and 2p_{z}); a 3d state is one with $n$=3 and
$l$=2.

We now have accurate wave functions and their energies for hydrogen-like atoms. Solving the Schrödinger equation analytically is impossible for more complex systems. Instead, we can use the known system as a base and add complexity gradually, adjusting the wave functions and energies step by step. This perturbation approach also allows us to calculate the effect of interactions with the electron system, such as the interaction of radiation with matter on which spectroscopy is based.