Quantum Atomica Introduction

Many people will know what atoms are; some may understand what the quantum atom is - virtually no one will have seen it. Quantum Atomica can take you down to the atomic level to see what's really happening within the atom. It focuses on the hydrogen atom, the simplest atom possible, constructed from one proton and one electron.

What is the Quantum Atom?

To help explain what the quantum atom is, paradoxically, it is necessary to explain what it isn't. There is a general misconception surrounding the structure of atoms. Many people would describe them as having a massively heavy nucleus consisting of the positively charged protons and electrically neutral neutrons surrounded by orbiting electrons of negative charge.

This model is simple to understand because of the analogy with the familiar concept of the planets orbiting the sun. Instead of gravity holding things together, the forces of attraction are electrostatic owing to the differing charges of the nucleus and the electrons. This is a classical model of the atom, as it is based on the classical Newtonian laws of mechanics.

The description of the classical atoms given earlier is incorrect in one way - the use of the word "orbiting". Whilst to many this may seem a point of pedantry, to quantum physicists, it's closer to heresy. When considering the electron to be simply a particle of negative charge and very little mass, compared to a proton, it would seem logical to think of the electron as "orbiting" the nucleus. However, a number of problems arise. Unlike the planets, which are not charged, the electron is, and therefore, owing to the laws of electromagnetism, must constantly radiate electromagnetic radiation. This because it is accelerating - not in the sense of changing speed, but direction, since it moves along a circular path. Therefore whilst it moves it will dissipate its energy and thus would gradually spiral inward towards the nucleus until it finally crashes on top of it. This would cause all atoms to be unstable, which clearly is not the case, since we are still here.

Bohr This law of physics was not included by Bohr when he developed his model of the atom. He stated that at certain discrete radial distances, the electron would be stable. Each distance corresponded to a different energy of the electron. Therefore only discrete energy levels were allowed. When an electron decreases in energy it emits a photon whose energy is the difference between the two electron states. This can be seen on emission spectra, since for each energy photon, there is a corresponding wavelength seen as different colours.

Bohr's model was able to predict the emission spectra for hydrogen but not for the more complex atoms, with more than one electron. This was because it was built on the false principle that the electron exists as a solid particle within the atom. It is not however a forgotten theory, since it formed the foundations of the theories that followed. Additionally, the radial distance Bohr predicted the electron to be found is still called the Bohr radius and is the unit of measurement used by Quantum Atomica.

de Broglie, Heisenberg & Schrödinger

Louis de Broglie, Werner Heisenberg and Erwin Schrödinger all developed theories relevant to the quantum atom. De Broglie, in 1924, developed the idea that matter can behave as a wave. Extending this idea to the electron, which had been shown to be a particle by J. J. Thomson, led de Broglie to believe the electron would exhibit wave-like properties such as diffraction and interference. This was subsequently demonstrated by J. J. Thomson's son, G. P. Thomson. This led to the idea that the electron had a wave-particle duality, and therefore, scientists could try to model the electron in the atom as a wave, rather than a particle in the classical sense.

Heisenberg certainly deserves mention because his uncertainty principle dispels all ideas of classical Bohr-like atomic models, where the position of the electron is defined exactly. This principle states that both the position and velocity of a particle such as the electron cannot be accurately measured. The more accurately you know one, the less accurately you can know the other. This is clearly not the case with the Bohr model, because, at any time, it is assumed you will know the exact position and velocity of the electron.

It is however Erwin Schrödinger who is best known for the quantum atom. Developing de Broglie's ideas, he created a mathematical technique called wave mechanics. This can be used to model how the electron behaves as a standing wave within the atom. This means that its energy is not dissipated, and the atoms formed are stable. It is described by the Schrödinger equation. This describes the wave function, Y, of the electron.

When squared, this will give rise to what is known as the probability density function:

The infinite number of solutions to this equation account for the different discrete energies the electron may have and for the different sub-shells. It can be solved numerically for all atoms, but can only be solved analytically for the case of hydrogen. The solutions do not give exact points were the electron will be, as this would violate the uncertainty principle. Instead, they describe the probability throughout space of finding the electron at any particular location. Thus there are no locations where it is certain to be; merely where it is likely to be. This might be thought of as quite a "fuzzy" solution for indeed it is often best to imagine the solution as being a "cloud of probability", densest in the regions where the electron's presence is most likely. It is this cloud that is the quantum atom, and this is what we must try to visualise.

Nomenclature

As we have seen, there are an infinite number of states that the electron can be in, depending on the energy it possesses. A system is needed to label each of these states, and it requires only three pieces of information.

1. The electron can be in one of an infinite number of discrete energy levels. There is a minimum energy level called the ground state, in which the electron will normally be found unless is has been excited by collision with a photon or electron. In which case, depending on the size of the energy boost it receives, it will move to a higher level. The number of the level it is at is described by the quantum number n, the principle quantum number. n can take integer values from 1 (the lowest level) upwards. In the case of the hydrogen atom, for any value of n, the energy needed by the electron the break free of the attraction of the proton is given by:

where a-subscript-0 is the Bohr radius.

2. The electron may also have angular momentum. This is described by l, the azimuthal quantum number. For any value of n, l can take integer values from 0 to n-1. Thus, for n = 1, no angular momentum is possible, but for anything higher it is possible.

3. In the presence of a magnetic/electric field, there is a clear way of establishing a direction for the angular momentum. Therefore, the magnetic quantum number m can take integer values from -l to +l.

All these numbers arise naturally from the solutions to the Schrödinger equation. Each different combination of n, l, m is called an orbital and is expressed as (n,l,m). The ground state orbital is (1,0,0). Another orbital could be (3,2,0): this would be a much more excited electron. However, this very logical and extensible naming system is less commonly used than an older, more arcane and non-extensible system - the s, p, d, f notation. Under spdf notation, the (1,0,0) orbital is named 1s and the (3,2,0) is 3dz².

1. The first number is the same as n, the energy level.

2. The non sub-scripted letter represents the azimuthal number l. The words they stand for: sharp, principle, diffuse and fundamental, date back to scientists finding evidence of sub-shells from emission spectra. The lines made by s orbitals were sharp, those by d orbital clearly diffuse. This system is incredibly limited, as you can only describe up to n = 4 orbitals.

Letter	Numerical Equivalent
s	0
p	1
d	2
f	3

3. The last part, present only when l does not equal 0 (which it does for s orbitals), describes the magnetic quantum number m. It does this in a very esoteric way, different for each value of l. For all p orbitals:

Letter	Numerical Equivalent
y	-1
z	0
x	1

It soon though appears to become very strange, and is best shown again in the tables for the d orbitals:

Letters	Numerical Equivalent
xy	-2
yz	-1
z²	0
xz	1
x²-y²	2

And for the f orbitals:

Letters	Numerical Equivalent
y(3x²-y²)	-3
xyz	-2
yz²	-1
z³	0
xz²	1
z(x²-y²)	2
x(x²-3y²)	3

Equations for the 2P orbitals There is however a certain logic to this system. When looking at the wave function, Y, the part that varies for differing values of m are the combinations of letters listed above. The simplest examples are the p orbitals.

These are Y² or PDFs, but the differing factor is still very clear.

It is this older system that is in predominant use in Quantum Atomica, since it includes 30 orbitals including all up to n = 4. However, for each orbital, there are two things it can display. One is the PDF: the probability density function, which is essentially the wave function squared, Y². The other is called a DRPDF: this is more of a proprietary term and stands for direction radial probability density function. It is formed by multiplying the PDF by r²sinq. This represents the physical problem of where the electron is likely to be in space around the proton in the presence of a magnetic/electric field. If such a field is not present, Quantum Atomica includes averaged PDFs, created by averaging the PDFs of a certain n and l across all the possible corresponding m values. These averaged PDFs may form DRPDFs if multiplied by r².

The Four Dimensions of the Electron

A Cloud As it is now clear, the electron can no longer be thought of in terms of classical mechanics as one particle orbiting another, which requires only three dimensions to describe it at any point in time. The correct way of representing the electron as a probability cloud introduces the fourth dimension to the problem in the form of the probability density at every point in three-dimensional space. It is this four dimensional object of a "3D cloud" with varying density that must some how be represented on a two-dimensional computer screen (three-dimensional if colour is taken to be a dimension). In essence, this is the main problem in visualising the hydrogen atom.

Visualising the orbital functions is the task that Quantum Atomica was designed to do. It is able to do this in 3 main ways: by the use of transects, weighted dot distributions and raytraced volumetric images. More information on how these images are formed and what they represent are given on the features page.