The Quantum Atom

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 108 m/s.
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.

Newton and Einstein both thought that light, although a wave, can also have some properties of a particle. In fact, they were right. Light actually is a stream of particles called photons. The amplitude of the light can be related to the number of photons in a given volume, and the energy of each photon is related to the frequency of the light:

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,

If Light has properties of a particle, surely particles have properties of a wave. DeBroglie showed that this was in fact the case, in the first equation that has both wave and particle properties related:

p = h / l

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 harmonics) if each of those waves has a position of zero displacement at the right place. These positions where the value of the wave is zero are called nodes
(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 pattern is described by discrete energy levels of the atom that have energy inversely proportional to the square of the number of waves in the orbit. We will call this number n, which is an integer from one to infinity. This integer, n, is more precisely described as the number of nodes in the wavefunction of the electron plus one.
En = -Z2 RH /n2

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

hn = DE = En2 - En1 = -Z2 RH (1/n22 - 1/n12)

Where the constant RH has the value 2.180 x 10-18 J.

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, n2 must be greater than n1 for the resulting frequency to be a positive number. Yet, if the initial principle quantum number of the atom (ninitial) is smaller than the final principle quantum number (nfinal), photon absorption has taken place and if nfinal is smaller than ninitial,   photon emission has occurred.

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) = -Z2 ( RH / hc ) (1/n22 - 1/n12) = -Z2 Rl (1/n22 - 1/n12)

Where Rl is called the Rydberg Constant and has been precisely measured as 1.096776 x 107 m-1.

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 nodes. The number of nodes is related to the frequency of the wave and therefore its energy.
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 angular node or a radial node. If it is an angular node, then you have a 2p orbital

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):

In short, the energy of the atom is determined by the number of nodes which is related to the principal quantum number n by: nodes = n-1.
The number of angular nodes is labelled by a letter (s, p, d, f, g, h, i, ....)

s: no angular nodes
p: one angular node
d: two angular nodes
f: three angular nodes
Note: The number of radial nodes is the total number of nodes minus the number of angular nodes.
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PJ Brucat || University of Florida