Common experience tells us that the behavior of waves is much different than the behavior of particles. Wave phenomema has many common examples, but all waves share some common features. Waves have a frequency, a wavelength, a wave velocity, and an amplitude, which may be examined in the following figure:

For a given type of
wave
in a given medium, the wavelength
l and the frequency n
can be related to the speed of propagation of the wave(wave velocity) as follows:

l n = c

For Light (electromagnetic waves) travelling in a vacuum, this speed of propagation is mighty quick: 2.99792 x 10

Light is just one portion (one range of frequencies) of the EM spectrum, which spans vastly diverse types of radiation:

A device that separates light by its frequency is said to 'disperse' the light. Prisms and raindrops disperse light by refraction, gratings and holograms by diffraction.

E = h n

Where h== Planck's constant = 6.626 x 10^{-34} Js. When light strikes matter, in particular a molecule, the entire energy of the
photon must be absorbed or emitted. Thus the color of the light that interacts with a
particular piece of matter tells you about the change in energy that is possible in
that matter,

When one confines a wave to a particular region of space, the edges of the containment place a constraint on the wavelength due to 'boundary conditions'. This is how you play different notes on the same guitar string by moving the position in which you make the wave displacement zero, i.e. where your finger touches the fret.

Note that the 'boundary conditions' can be satisfied by many different waves (called

(Sometimes we distinguish two types of waves, travelling waves and standing waves, by whether the nodes of the wave move or not. Our discussion of the atom will pretty much rely on the standing wave picture of the electron.)

If electrons are waves, then the wavelength of the electron must 'fit' into any orbit that it makes around the nucleus in an atom. This is the 'boundary condition' for a one electron atom. All orbits that do not have the electrons wavelength 'fit' are not possible, because wave interference will rapidly destroy the wave amplitude and the electron wouldn't exist anymore. This 'interference' effect leads to discrete (quantized) energy levels for the atom. Since light interacts with the atom by causing a transition between these levels, the color(spectrum) of the atom is observed to be a series of sharp lines. For the hydrogen atom:

This equation works for all one-electron atoms, not just hydrogen. (Here is a more complete derivation of the Bohr Atom's Properties. Danger: Advanced). This is precisely the pattern of energy levels that are observed to exist in the Hydrogen atom. Transitions between these levels give the pattern in the absorption or emission spectrum of the atom.

The frequency of the transitions between the energy levels should be given by

Where the constant

A Note on Signs:

Obviously the change in energy of an atom can be either positive or negative
depending on whether energy is absorbed by the atom from the light field
or emitted by the atom to its surroundings.
Yet frequencies (or wavelengths) of light MUST be positive numbers.
Thus, in the equation above, n_{2} must be **greater** than n_{1}
for the resulting frequency to be a positive number.
Yet, if the initial principle quantum number of the atom (**n _{initial}**)
is smaller than the final principle quantum number
(

To represent the observed spectra of one electron atoms using the above energy spacing,
it is useful to relate the energy of the photon to its wavelength through
E = hn
and
l n = c:

(1/l) = -Z

Where

The energy patterns of atoms give the elements their characteristic 'flame' colors because the light they emit when heated has specific photon energy:

We know, too, that Sodium is yellow (Streetlights), and Neon is red (Fluorescent Signs), etc...

The wave nature of the electron is what makes atoms have the properties that they do. It explains the colors of the atoms and also their size. It also means that we cannot think about the electron in an atom as a little ball whirling about the nucleus, but a cloud of probability that is smeared out over the orbit.

This cloud is only spherical for the lowest energy level of the atom. As the energy of the electron in the atom increases, its wavelength decreases, and the number of times the wave amplitude crosses zero per orbit increases. Again, these zero crossings are called

The greater number of nodes, the greater the energy of the system.

The lowest level of the H atom has no (zero) nodes, the next higher level has 1 node, but that node can either be an

If you have a radial node, then you have a 2s orbital (3s shown also)

The number of nodes determines the energy.
The Principle quantum number, **n**, is
equal to the number of nodes plus 1, i.e. **nodes = n-1**.
For a hydrogen atom,
the energy it takes to make a radial node is
equal to the the energy it takes to make an angular node.

For higher **n**, you can have a greater numbers of nodes. For **n**>=3,
you can have 2 angular nodes, and these are called **d orbitals**

Here are some more pictures of the atomic
orbital shapes

The following shows the angular nodes for a 2p orbital:

Similarly for a 3d orbital (with two angular nodes):

- s: no angular nodes
- p: one angular node
- d: two angular nodes
- f: three angular nodes
- etc...

PJ Brucat || University of Florida